Integrand size = 22, antiderivative size = 332 \[ \int \frac {c+d x}{(e x)^{5/2} \left (a+b x^2\right )^3} \, dx=-\frac {77 c}{48 a^3 e (e x)^{3/2}}-\frac {45 d}{16 a^3 e^2 \sqrt {e x}}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}+\frac {11 c+9 d x}{16 a^2 e (e x)^{3/2} \left (a+b x^2\right )}+\frac {\sqrt [4]{b} \left (77 \sqrt {b} c+45 \sqrt {a} d\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{15/4} e^{5/2}}-\frac {\sqrt [4]{b} \left (77 \sqrt {b} c+45 \sqrt {a} d\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{15/4} e^{5/2}}-\frac {\sqrt [4]{b} \left (77 \sqrt {b} c-45 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{32 \sqrt {2} a^{15/4} e^{5/2}} \] Output:
-77/48*c/a^3/e/(e*x)^(3/2)-45/16*d/a^3/e^2/(e*x)^(1/2)+1/4*(d*x+c)/a/e/(e* x)^(3/2)/(b*x^2+a)^2+1/16*(9*d*x+11*c)/a^2/e/(e*x)^(3/2)/(b*x^2+a)+1/64*b^ (1/4)*(77*b^(1/2)*c+45*a^(1/2)*d)*arctan(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^( 1/4)/e^(1/2))*2^(1/2)/a^(15/4)/e^(5/2)-1/64*b^(1/4)*(77*b^(1/2)*c+45*a^(1/ 2)*d)*arctan(1+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(15/ 4)/e^(5/2)-1/64*b^(1/4)*(77*b^(1/2)*c-45*a^(1/2)*d)*arctanh(2^(1/2)*a^(1/4 )*b^(1/4)*(e*x)^(1/2)/e^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(15/4)/e^(5/2 )
Time = 1.03 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.65 \[ \int \frac {c+d x}{(e x)^{5/2} \left (a+b x^2\right )^3} \, dx=\frac {x \left (-\frac {4 a^{3/4} \left (32 a^2 (c+3 d x)+b^2 x^4 (77 c+135 d x)+a b x^2 (121 c+243 d x)\right )}{\left (a+b x^2\right )^2}+3 \sqrt {2} \sqrt [4]{b} \left (77 \sqrt {b} c+45 \sqrt {a} d\right ) x^{3/2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+3 \sqrt {2} \left (-77 b^{3/4} c+45 \sqrt {a} \sqrt [4]{b} d\right ) x^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{192 a^{15/4} (e x)^{5/2}} \] Input:
Integrate[(c + d*x)/((e*x)^(5/2)*(a + b*x^2)^3),x]
Output:
(x*((-4*a^(3/4)*(32*a^2*(c + 3*d*x) + b^2*x^4*(77*c + 135*d*x) + a*b*x^2*( 121*c + 243*d*x)))/(a + b*x^2)^2 + 3*Sqrt[2]*b^(1/4)*(77*Sqrt[b]*c + 45*Sq rt[a]*d)*x^(3/2)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqr t[x])] + 3*Sqrt[2]*(-77*b^(3/4)*c + 45*Sqrt[a]*b^(1/4)*d)*x^(3/2)*ArcTanh[ (Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(192*a^(15/4)*( e*x)^(5/2))
Time = 1.22 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.29, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {551, 27, 551, 27, 553, 27, 553, 27, 554, 1482, 27, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {c+d x}{(e x)^{5/2} \left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}-\frac {\int -\frac {11 c+9 d x}{2 (e x)^{5/2} \left (b x^2+a\right )^2}dx}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {11 c+9 d x}{(e x)^{5/2} \left (b x^2+a\right )^2}dx}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}-\frac {\int -\frac {77 c+45 d x}{2 (e x)^{5/2} \left (b x^2+a\right )}dx}{2 a}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {77 c+45 d x}{(e x)^{5/2} \left (b x^2+a\right )}dx}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {\frac {-\frac {2 \int -\frac {3 (45 a d-77 b c x)}{2 (e x)^{3/2} \left (b x^2+a\right )}dx}{3 a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {45 a d-77 b c x}{(e x)^{3/2} \left (b x^2+a\right )}dx}{a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \int \frac {a b (77 c+45 d x)}{2 \sqrt {e x} \left (b x^2+a\right )}dx}{a e}-\frac {90 d}{e \sqrt {e x}}}{a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {-\frac {b \int \frac {77 c+45 d x}{\sqrt {e x} \left (b x^2+a\right )}dx}{e}-\frac {90 d}{e \sqrt {e x}}}{a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 b \int \frac {77 c e+45 d x e}{b x^2 e^2+a e^2}d\sqrt {e x}}{e}-\frac {90 d}{e \sqrt {e x}}}{a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 b \left (\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}-45 d\right ) \int \frac {\sqrt {b} \left (\sqrt {a} e-\sqrt {b} e x\right )}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 b}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+45 d\right ) \int \frac {\sqrt {b} \left (\sqrt {b} x e+\sqrt {a} e\right )}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 b}\right )}{e}-\frac {90 d}{e \sqrt {e x}}}{a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 b \left (\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}-45 d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+45 d\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}\right )}{e}-\frac {90 d}{e \sqrt {e x}}}{a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 b \left (\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}-45 d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+45 d\right ) \left (\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}\right )}{2 \sqrt {b}}\right )}{e}-\frac {90 d}{e \sqrt {e x}}}{a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 b \left (\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}-45 d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+45 d\right ) \left (\frac {\int \frac {1}{-e x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int \frac {1}{-e x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )}{e}-\frac {90 d}{e \sqrt {e x}}}{a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 b \left (\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}-45 d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+45 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )}{e}-\frac {90 d}{e \sqrt {e x}}}{a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 b \left (\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}-45 d\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+45 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )}{e}-\frac {90 d}{e \sqrt {e x}}}{a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 b \left (\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}-45 d\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+45 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )}{e}-\frac {90 d}{e \sqrt {e x}}}{a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 b \left (\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}-45 d\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {e}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+45 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )}{e}-\frac {90 d}{e \sqrt {e x}}}{a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 b \left (\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+45 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}-45 d\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )}{e}-\frac {90 d}{e \sqrt {e x}}}{a e}-\frac {154 c}{3 a e (e x)^{3/2}}}{4 a}+\frac {11 c+9 d x}{2 a e (e x)^{3/2} \left (a+b x^2\right )}}{8 a}+\frac {c+d x}{4 a e (e x)^{3/2} \left (a+b x^2\right )^2}\) |
Input:
Int[(c + d*x)/((e*x)^(5/2)*(a + b*x^2)^3),x]
Output:
(c + d*x)/(4*a*e*(e*x)^(3/2)*(a + b*x^2)^2) + ((11*c + 9*d*x)/(2*a*e*(e*x) ^(3/2)*(a + b*x^2)) + ((-154*c)/(3*a*e*(e*x)^(3/2)) + ((-90*d)/(e*Sqrt[e*x ]) - (2*b*((((77*Sqrt[b]*c)/Sqrt[a] + 45*d)*(-(ArcTan[1 - (Sqrt[2]*b^(1/4) *Sqrt[e*x])/(a^(1/4)*Sqrt[e])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e])) + ArcTan [1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])]/(Sqrt[2]*a^(1/4)*b^(1/ 4)*Sqrt[e])))/(2*Sqrt[b]) + (((77*Sqrt[b]*c)/Sqrt[a] - 45*d)*(-1/2*Log[Sqr t[a]*e + Sqrt[b]*e*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e]*Sqrt[e*x]]/(Sqrt[2] *a^(1/4)*b^(1/4)*Sqrt[e]) + Log[Sqrt[a]*e + Sqrt[b]*e*x + Sqrt[2]*a^(1/4)* b^(1/4)*Sqrt[e]*Sqrt[e*x]]/(2*Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e])))/(2*Sqrt[b ])))/e)/(a*e))/(4*a))/(8*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[(-(e*x)^(m + 1))*(c + d*x)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1) )), x] + Simp[1/(2*a*(p + 1)) Int[(e*x)^m*(c*(m + 2*p + 3) + d*(m + 2*p + 4)*x)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p , -1] && LtQ[m, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp [1/(a*e*(m + 1)) Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[m, -1]
Int[((c_) + (d_.)*(x_))/(Sqrt[(e_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*c + d*x^2)/(a*e^2 + b*x^4), x], x, Sqrt[e*x]], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Time = 0.48 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.14
| method | result | size |
| risch | \(-\frac {2 \left (3 d x +c \right )}{3 a^{3} x \sqrt {e x}\, e^{2}}-\frac {b \left (\frac {\frac {13 b d \left (e x \right )^{\frac {7}{2}}}{16}+\frac {15 b c e \left (e x \right )^{\frac {5}{2}}}{16}+\frac {17 a d \,e^{2} \left (e x \right )^{\frac {3}{2}}}{16}+\frac {19 a c \,e^{3} \sqrt {e x}}{16}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {77 c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 e a}+\frac {45 d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} e^{2}}\) | \(377\) |
| derivativedivides | \(2 e^{4} \left (-\frac {c}{3 e^{5} a^{3} \left (e x \right )^{\frac {3}{2}}}-\frac {d}{a^{3} e^{6} \sqrt {e x}}-\frac {b \left (\frac {\frac {13 b d \left (e x \right )^{\frac {7}{2}}}{32}+\frac {15 b c e \left (e x \right )^{\frac {5}{2}}}{32}+\frac {17 a d \,e^{2} \left (e x \right )^{\frac {3}{2}}}{32}+\frac {19 a c \,e^{3} \sqrt {e x}}{32}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {77 c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 e a}+\frac {45 d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} e^{6}}\right )\) | \(387\) |
| default | \(2 e^{4} \left (-\frac {c}{3 e^{5} a^{3} \left (e x \right )^{\frac {3}{2}}}-\frac {d}{a^{3} e^{6} \sqrt {e x}}-\frac {b \left (\frac {\frac {13 b d \left (e x \right )^{\frac {7}{2}}}{32}+\frac {15 b c e \left (e x \right )^{\frac {5}{2}}}{32}+\frac {17 a d \,e^{2} \left (e x \right )^{\frac {3}{2}}}{32}+\frac {19 a c \,e^{3} \sqrt {e x}}{32}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {77 c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 e a}+\frac {45 d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} e^{6}}\right )\) | \(387\) |
| pseudoelliptic | \(-\frac {45 \left (\frac {77 x \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) b \sqrt {2}\, \left (b \,x^{2}+a \right )^{2} c \sqrt {e x}\, \sqrt {\frac {a \,e^{2}}{b}}}{45}+\left (d \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) x \sqrt {2}\, \left (b \,x^{2}+a \right )^{2} \sqrt {e x}+\frac {256 \left (\left (3 d x +c \right ) a^{2}+\frac {121 x^{2} b \left (\frac {243 d x}{121}+c \right ) a}{32}+\frac {77 x^{4} b^{2} \left (\frac {135 d x}{77}+c \right )}{32}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{135}\right ) e a \right )}{128 \sqrt {e x}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} e^{3} a^{4} x \left (b \,x^{2}+a \right )^{2}}\) | \(427\) |
Input:
int((d*x+c)/(e*x)^(5/2)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
-2/3*(3*d*x+c)/a^3/x/(e*x)^(1/2)/e^2-1/a^3*b*(2*(13/32*b*d*(e*x)^(7/2)+15/ 32*b*c*e*(e*x)^(5/2)+17/32*a*d*e^2*(e*x)^(3/2)+19/32*a*c*e^3*(e*x)^(1/2))/ (b*e^2*x^2+a*e^2)^2+77/128*c/e*(a*e^2/b)^(1/4)/a*2^(1/2)*(ln((e*x+(a*e^2/b )^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2))/(e*x-(a*e^2/b)^(1/4)*(e*x)^(1 /2)*2^(1/2)+(a*e^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2) +1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1))+45/128*d/b/(a*e^2/b)^ (1/4)*2^(1/2)*(ln((e*x-(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2) )/(e*x+(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2)))+2*arctan(2^(1 /2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^ (1/2)-1)))/e^2
Leaf count of result is larger than twice the leaf count of optimal. 1170 vs. \(2 (240) = 480\).
Time = 0.15 (sec) , antiderivative size = 1170, normalized size of antiderivative = 3.52 \[ \int \frac {c+d x}{(e x)^{5/2} \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)/(e*x)^(5/2)/(b*x^2+a)^3,x, algorithm="fricas")
Output:
1/192*(3*(a^3*b^2*e^3*x^6 + 2*a^4*b*e^3*x^4 + a^5*e^3*x^2)*sqrt(-(a^7*e^5* sqrt(-(35153041*b^3*c^4 - 24012450*a*b^2*c^2*d^2 + 4100625*a^2*b*d^4)/(a^1 5*e^10)) + 6930*b*c*d)/(a^7*e^5))*log(-(35153041*b^3*c^4 - 4100625*a^2*b*d ^4)*sqrt(e*x) + (45*a^12*d*e^8*sqrt(-(35153041*b^3*c^4 - 24012450*a*b^2*c^ 2*d^2 + 4100625*a^2*b*d^4)/(a^15*e^10)) + 77*(5929*a^4*b^2*c^3 - 2025*a^5* b*c*d^2)*e^3)*sqrt(-(a^7*e^5*sqrt(-(35153041*b^3*c^4 - 24012450*a*b^2*c^2* d^2 + 4100625*a^2*b*d^4)/(a^15*e^10)) + 6930*b*c*d)/(a^7*e^5))) - 3*(a^3*b ^2*e^3*x^6 + 2*a^4*b*e^3*x^4 + a^5*e^3*x^2)*sqrt(-(a^7*e^5*sqrt(-(35153041 *b^3*c^4 - 24012450*a*b^2*c^2*d^2 + 4100625*a^2*b*d^4)/(a^15*e^10)) + 6930 *b*c*d)/(a^7*e^5))*log(-(35153041*b^3*c^4 - 4100625*a^2*b*d^4)*sqrt(e*x) - (45*a^12*d*e^8*sqrt(-(35153041*b^3*c^4 - 24012450*a*b^2*c^2*d^2 + 4100625 *a^2*b*d^4)/(a^15*e^10)) + 77*(5929*a^4*b^2*c^3 - 2025*a^5*b*c*d^2)*e^3)*s qrt(-(a^7*e^5*sqrt(-(35153041*b^3*c^4 - 24012450*a*b^2*c^2*d^2 + 4100625*a ^2*b*d^4)/(a^15*e^10)) + 6930*b*c*d)/(a^7*e^5))) - 3*(a^3*b^2*e^3*x^6 + 2* a^4*b*e^3*x^4 + a^5*e^3*x^2)*sqrt((a^7*e^5*sqrt(-(35153041*b^3*c^4 - 24012 450*a*b^2*c^2*d^2 + 4100625*a^2*b*d^4)/(a^15*e^10)) - 6930*b*c*d)/(a^7*e^5 ))*log(-(35153041*b^3*c^4 - 4100625*a^2*b*d^4)*sqrt(e*x) + (45*a^12*d*e^8* sqrt(-(35153041*b^3*c^4 - 24012450*a*b^2*c^2*d^2 + 4100625*a^2*b*d^4)/(a^1 5*e^10)) - 77*(5929*a^4*b^2*c^3 - 2025*a^5*b*c*d^2)*e^3)*sqrt((a^7*e^5*sqr t(-(35153041*b^3*c^4 - 24012450*a*b^2*c^2*d^2 + 4100625*a^2*b*d^4)/(a^1...
Result contains complex when optimal does not.
Time = 99.27 (sec) , antiderivative size = 6887, normalized size of antiderivative = 20.74 \[ \int \frac {c+d x}{(e x)^{5/2} \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)/(e*x)**(5/2)/(b*x**2+a)**3,x)
Output:
c*(128*a**(19/4)*exp(I*pi/4)*gamma(-3/4)/(256*a**(31/4)*e**(5/2)*x**(3/2)* exp(I*pi/4)*gamma(1/4) + 768*a**(27/4)*b*e**(5/2)*x**(7/2)*exp(I*pi/4)*gam ma(1/4) + 768*a**(23/4)*b**2*e**(5/2)*x**(11/2)*exp(I*pi/4)*gamma(1/4) + 2 56*a**(19/4)*b**3*e**(5/2)*x**(15/2)*exp(I*pi/4)*gamma(1/4)) + 612*a**(15/ 4)*b*x**2*exp(I*pi/4)*gamma(-3/4)/(256*a**(31/4)*e**(5/2)*x**(3/2)*exp(I*p i/4)*gamma(1/4) + 768*a**(27/4)*b*e**(5/2)*x**(7/2)*exp(I*pi/4)*gamma(1/4) + 768*a**(23/4)*b**2*e**(5/2)*x**(11/2)*exp(I*pi/4)*gamma(1/4) + 256*a**( 19/4)*b**3*e**(5/2)*x**(15/2)*exp(I*pi/4)*gamma(1/4)) + 792*a**(11/4)*b**2 *x**4*exp(I*pi/4)*gamma(-3/4)/(256*a**(31/4)*e**(5/2)*x**(3/2)*exp(I*pi/4) *gamma(1/4) + 768*a**(27/4)*b*e**(5/2)*x**(7/2)*exp(I*pi/4)*gamma(1/4) + 7 68*a**(23/4)*b**2*e**(5/2)*x**(11/2)*exp(I*pi/4)*gamma(1/4) + 256*a**(19/4 )*b**3*e**(5/2)*x**(15/2)*exp(I*pi/4)*gamma(1/4)) + 308*a**(7/4)*b**3*x**6 *exp(I*pi/4)*gamma(-3/4)/(256*a**(31/4)*e**(5/2)*x**(3/2)*exp(I*pi/4)*gamm a(1/4) + 768*a**(27/4)*b*e**(5/2)*x**(7/2)*exp(I*pi/4)*gamma(1/4) + 768*a* *(23/4)*b**2*e**(5/2)*x**(11/2)*exp(I*pi/4)*gamma(1/4) + 256*a**(19/4)*b** 3*e**(5/2)*x**(15/2)*exp(I*pi/4)*gamma(1/4)) - 231*a**4*b**(3/4)*x**(3/2)* log(1 - b**(1/4)*sqrt(x)*exp_polar(I*pi/4)/a**(1/4))*gamma(-3/4)/(256*a**( 31/4)*e**(5/2)*x**(3/2)*exp(I*pi/4)*gamma(1/4) + 768*a**(27/4)*b*e**(5/2)* x**(7/2)*exp(I*pi/4)*gamma(1/4) + 768*a**(23/4)*b**2*e**(5/2)*x**(11/2)*ex p(I*pi/4)*gamma(1/4) + 256*a**(19/4)*b**3*e**(5/2)*x**(15/2)*exp(I*pi/4...
Exception generated. \[ \int \frac {c+d x}{(e x)^{5/2} \left (a+b x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)/(e*x)^(5/2)/(b*x^2+a)^3,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.15 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.32 \[ \int \frac {c+d x}{(e x)^{5/2} \left (a+b x^2\right )^3} \, dx=-\frac {13 \, \sqrt {e x} b^{2} d e^{3} x^{3} + 15 \, \sqrt {e x} b^{2} c e^{3} x^{2} + 17 \, \sqrt {e x} a b d e^{3} x + 19 \, \sqrt {e x} a b c e^{3}}{16 \, {\left (b e^{2} x^{2} + a e^{2}\right )}^{2} a^{3} e^{2}} - \frac {2 \, {\left (3 \, d e x + c e\right )}}{3 \, \sqrt {e x} a^{3} e^{3} x} - \frac {\sqrt {2} {\left (77 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e + 45 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} b^{2} e^{4}} - \frac {\sqrt {2} {\left (77 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e + 45 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} b^{2} e^{4}} - \frac {\sqrt {2} {\left (77 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e - 45 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{128 \, a^{4} b^{2} e^{4}} + \frac {\sqrt {2} {\left (77 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e - 45 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{128 \, a^{4} b^{2} e^{4}} \] Input:
integrate((d*x+c)/(e*x)^(5/2)/(b*x^2+a)^3,x, algorithm="giac")
Output:
-1/16*(13*sqrt(e*x)*b^2*d*e^3*x^3 + 15*sqrt(e*x)*b^2*c*e^3*x^2 + 17*sqrt(e *x)*a*b*d*e^3*x + 19*sqrt(e*x)*a*b*c*e^3)/((b*e^2*x^2 + a*e^2)^2*a^3*e^2) - 2/3*(3*d*e*x + c*e)/(sqrt(e*x)*a^3*e^3*x) - 1/64*sqrt(2)*(77*(a*b^3*e^2) ^(1/4)*b^2*c*e + 45*(a*b^3*e^2)^(3/4)*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*e^ 2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(a^4*b^2*e^4) - 1/64*sqrt(2)*(7 7*(a*b^3*e^2)^(1/4)*b^2*c*e + 45*(a*b^3*e^2)^(3/4)*d)*arctan(-1/2*sqrt(2)* (sqrt(2)*(a*e^2/b)^(1/4) - 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(a^4*b^2*e^4) - 1 /128*sqrt(2)*(77*(a*b^3*e^2)^(1/4)*b^2*c*e - 45*(a*b^3*e^2)^(3/4)*d)*log(e *x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(a^4*b^2*e^4) + 1/ 128*sqrt(2)*(77*(a*b^3*e^2)^(1/4)*b^2*c*e - 45*(a*b^3*e^2)^(3/4)*d)*log(e* x - sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(a^4*b^2*e^4)
Time = 0.54 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.45 \[ \int \frac {c+d x}{(e x)^{5/2} \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((c + d*x)/((e*x)^(5/2)*(a + b*x^2)^3),x)
Output:
- ((2*c*e^3)/(3*a) + (2*d*e^3*x)/a + (121*b*c*e^3*x^2)/(48*a^2) + (81*b*d* e^3*x^3)/(16*a^2) + (77*b^2*c*e^3*x^4)/(48*a^3) + (45*b^2*d*e^3*x^5)/(16*a ^3))/(b^2*(e*x)^(11/2) + a^2*e^4*(e*x)^(3/2) + 2*a*b*e^2*(e*x)^(7/2)) - 2* atanh((194281472*a^9*b^5*c^2*e^8*(e*x)^(1/2)*((5929*b*c^2*(-a^15*b)^(1/2)) /(4096*a^15*e^5) - (3465*b*c*d)/(2048*a^7*e^5) - (2025*d^2*(-a^15*b)^(1/2) )/(4096*a^14*e^5))^(1/2))/(46656000*a^7*b^4*d^3*e^6 + (233744896*b^5*c^3*e ^6*(-a^15*b)^(1/2))/a^2 - 136604160*a^6*b^5*c^2*d*e^6 - (79833600*b^4*c*d^ 2*e^6*(-a^15*b)^(1/2))/a) - (66355200*a^10*b^4*d^2*e^8*(e*x)^(1/2)*((5929* b*c^2*(-a^15*b)^(1/2))/(4096*a^15*e^5) - (3465*b*c*d)/(2048*a^7*e^5) - (20 25*d^2*(-a^15*b)^(1/2))/(4096*a^14*e^5))^(1/2))/(46656000*a^7*b^4*d^3*e^6 + (233744896*b^5*c^3*e^6*(-a^15*b)^(1/2))/a^2 - 136604160*a^6*b^5*c^2*d*e^ 6 - (79833600*b^4*c*d^2*e^6*(-a^15*b)^(1/2))/a))*(-(2025*a*d^2*(-a^15*b)^( 1/2) - 5929*b*c^2*(-a^15*b)^(1/2) + 6930*a^8*b*c*d)/(4096*a^15*e^5))^(1/2) - 2*atanh((194281472*a^9*b^5*c^2*e^8*(e*x)^(1/2)*((2025*d^2*(-a^15*b)^(1/ 2))/(4096*a^14*e^5) - (3465*b*c*d)/(2048*a^7*e^5) - (5929*b*c^2*(-a^15*b)^ (1/2))/(4096*a^15*e^5))^(1/2))/(46656000*a^7*b^4*d^3*e^6 - (233744896*b^5* c^3*e^6*(-a^15*b)^(1/2))/a^2 - 136604160*a^6*b^5*c^2*d*e^6 + (79833600*b^4 *c*d^2*e^6*(-a^15*b)^(1/2))/a) - (66355200*a^10*b^4*d^2*e^8*(e*x)^(1/2)*(( 2025*d^2*(-a^15*b)^(1/2))/(4096*a^14*e^5) - (3465*b*c*d)/(2048*a^7*e^5) - (5929*b*c^2*(-a^15*b)^(1/2))/(4096*a^15*e^5))^(1/2))/(46656000*a^7*b^4*...
Time = 0.19 (sec) , antiderivative size = 1029, normalized size of antiderivative = 3.10 \[ \int \frac {c+d x}{(e x)^{5/2} \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((d*x+c)/(e*x)^(5/2)/(b*x^2+a)^3,x)
Output:
(sqrt(e)*(270*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq rt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*d*x + 540*sqr t(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x) *sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*d*x**3 + 270*sqrt(x)*b**(1/4)*a **(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**( 1/4)*a**(1/4)*sqrt(2)))*b**2*d*x**5 + 462*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2 )*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)* sqrt(2)))*a**2*c*x + 924*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)* a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*c*x **3 + 462*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2 ) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*c*x**5 - 270*sqrt (x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)* sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*d*x - 540*sqrt(x)*b**(1/4)*a**( 3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4 )*a**(1/4)*sqrt(2)))*a*b*d*x**3 - 270*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*at an((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt (2)))*b**2*d*x**5 - 462*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a **(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*c*x - 924*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*c*x**3 - 462*sqrt(...