Integrand size = 37, antiderivative size = 567 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {\left (a \left (B+\frac {a D}{b}\right )+(A b+a C) x\right ) (c+d x)^{5/2}}{a b \sqrt {a-b x^2}}+\frac {\left (105 A b^2 c d+a \left (225 a d^2 D+b \left (217 c C d+175 B d^2+30 c^2 D\right )\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{105 a b^3}+\frac {(35 A b d+49 a C d+10 a c D) (c+d x)^{3/2} \sqrt {a-b x^2}}{35 a b^2}+\frac {2 D (c+d x)^{5/2} \sqrt {a-b x^2}}{7 b^2}+\frac {\left (105 A b d \left (b c^2+3 a d^2\right )+a \left (9 a d^2 (49 C d+110 c D)+b c \left (427 c C d+700 B d^2+30 c^2 D\right )\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 \sqrt {a} b^{5/2} d \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (105 A b^2 c d+a \left (225 a d^2 D+b \left (217 c C d+175 B d^2+30 c^2 D\right )\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 \sqrt {a} b^{7/2} d \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
(a*(B+a*D/b)+(A*b+C*a)*x)*(d*x+c)^(5/2)/a/b/(-b*x^2+a)^(1/2)+1/105*(105*A* b^2*c*d+a*(225*a*d^2*D+b*(175*B*d^2+217*C*c*d+30*D*c^2)))*(d*x+c)^(1/2)*(- b*x^2+a)^(1/2)/a/b^3+1/35*(35*A*b*d+49*C*a*d+10*D*a*c)*(d*x+c)^(3/2)*(-b*x ^2+a)^(1/2)/a/b^2+2/7*D*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/b^2+1/105*(105*A*b* d*(3*a*d^2+b*c^2)+a*(9*a*d^2*(49*C*d+110*D*c)+b*c*(700*B*d^2+427*C*c*d+30* D*c^2)))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^( 1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(1/ 2)/b^(5/2)/d/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-1/105* (-a*d^2+b*c^2)*(105*A*b^2*c*d+a*(225*a*d^2*D+b*(175*B*d^2+217*C*c*d+30*D*c ^2)))*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticF (1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^( 1/2)*d))^(1/2))/a^(1/2)/b^(7/2)/d/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 30.22 (sec) , antiderivative size = 796, normalized size of antiderivative = 1.40 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (225 a^3 d^2 D+105 A b^3 c^2 x+a b^2 \left (105 A d (2 c+d x)+15 c^2 x (7 C-6 D x)-6 d^2 x^3 (7 C+5 D x)-2 c d x^2 (77 C+45 D x)+35 B \left (3 c^2+6 c d x-2 d^2 x^2\right )\right )+a^2 b \left (195 c^2 D+4 c d (91 C+75 D x)+d^2 (175 B+3 x (49 C-30 D x))\right )\right )}{a b^3 \left (a-b x^2\right )}-\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (105 A b d \left (b c^2+3 a d^2\right )+a \left (9 a d^2 (49 C d+110 c D)+b c \left (427 c C d+700 B d^2+30 c^2 D\right )\right )\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (105 A b d \left (b c^2+3 a d^2\right )+a \left (9 a d^2 (49 C d+110 c D)+b c \left (427 c C d+700 B d^2+30 c^2 D\right )\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (105 A b^2 c d-105 \sqrt {a} b^{3/2} \left (2 c^2 C+5 B c d+3 A d^2\right )+225 a^2 d^2 D-9 a^{3/2} \sqrt {b} d (49 C d+85 c D)+a b \left (217 c C d+175 B d^2+30 c^2 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{a b^3 d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{105 \sqrt {c+d x}} \] Input:
Integrate[((c + d*x)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(a - b*x^2)^(3/2),x]
Output:
(Sqrt[a - b*x^2]*(((c + d*x)*(225*a^3*d^2*D + 105*A*b^3*c^2*x + a*b^2*(105 *A*d*(2*c + d*x) + 15*c^2*x*(7*C - 6*D*x) - 6*d^2*x^3*(7*C + 5*D*x) - 2*c* d*x^2*(77*C + 45*D*x) + 35*B*(3*c^2 + 6*c*d*x - 2*d^2*x^2)) + a^2*b*(195*c ^2*D + 4*c*d*(91*C + 75*D*x) + d^2*(175*B + 3*x*(49*C - 30*D*x)))))/(a*b^3 *(a - b*x^2)) - (d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(105*A*b*d*(b*c^2 + 3* a*d^2) + a*(9*a*d^2*(49*C*d + 110*c*D) + b*c*(427*c*C*d + 700*B*d^2 + 30*c ^2*D)))*(a - b*x^2) + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(105*A*b*d*(b*c^2 + 3*a*d^2) + a*(9*a*d^2*(49*C*d + 110*c*D) + b*c*(427*c*C*d + 700*B*d^2 + 30*c^2*D)))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/ Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + ( Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - S qrt[a]*d)] + I*Sqrt[a]*d*(Sqrt[b]*c - Sqrt[a]*d)*(105*A*b^2*c*d - 105*Sqrt [a]*b^(3/2)*(2*c^2*C + 5*B*c*d + 3*A*d^2) + 225*a^2*d^2*D - 9*a^(3/2)*Sqrt [b]*d*(49*C*d + 85*c*D) + a*b*(217*c*C*d + 175*B*d^2 + 30*c^2*D))*Sqrt[(d* (Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/ Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(a*b^3*d ^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(105*Sqrt[c + d*x])
Time = 1.27 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.03, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {2176, 27, 2185, 27, 687, 27, 687, 27, 600, 509, 508, 327, 512, 511, 321}
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.
| \(\displaystyle \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2176 |
| \(\displaystyle \frac {\int -\frac {(c+d x)^{3/2} \left (2 a d D x^2+(5 A b d+7 a C d+2 a c D) x+\frac {a (2 b c C+5 b B d+5 a d D)}{b}\right )}{2 \sqrt {a-b x^2}}dx}{a b}+\frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\int \frac {(c+d x)^{3/2} \left (2 a d D x^2+(5 A b d+7 a C d+2 a c D) x+\frac {a (2 b c C+5 b B d+5 a d D)}{b}\right )}{\sqrt {a-b x^2}}dx}{2 a b}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {-\frac {2 \int -\frac {d^2 (c+d x)^{3/2} (a (14 b c C+35 b B d+45 a d D)+b (35 A b d+49 a C d+10 a c D) x)}{2 \sqrt {a-b x^2}}dx}{7 b d^2}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b}}{2 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\int \frac {(c+d x)^{3/2} (a (14 b c C+35 b B d+45 a d D)+b (35 A b d+49 a C d+10 a c D) x)}{\sqrt {a-b x^2}}dx}{7 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b}}{2 a b}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {-\frac {2 \int -\frac {b \sqrt {c+d x} \left (a \left (70 b C c^2+175 b B d c+255 a d D c+105 A b d^2+147 a C d^2\right )+\left (105 A c d b^2+a \left (225 a D d^2+b \left (30 D c^2+217 C d c+175 B d^2\right )\right )\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{5 b}-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} (10 a c D+49 a C d+35 A b d)}{7 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b}}{2 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{5} \int \frac {\sqrt {c+d x} \left (a \left (70 b C c^2+175 b B d c+255 a d D c+105 A b d^2+147 a C d^2\right )+\left (105 A c d b^2+a \left (225 a D d^2+b \left (30 D c^2+217 C d c+175 B d^2\right )\right )\right ) x\right )}{\sqrt {a-b x^2}}dx-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} (10 a c D+49 a C d+35 A b d)}{7 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b}}{2 a b}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{5} \left (-\frac {2 \int -\frac {a \left (225 a^2 D d^3+a b \left (795 D c^2+658 C d c+175 B d^2\right ) d+105 b^2 c \left (2 C c^2+5 B d c+4 A d^2\right )\right )+b \left (105 A b d \left (b c^2+3 a d^2\right )+a \left (9 a (49 C d+110 c D) d^2+b c \left (30 D c^2+427 C d c+700 B d^2\right )\right )\right ) x}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} (10 a c D+49 a C d+35 A b d)}{7 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b}}{2 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{5} \left (\frac {\int \frac {a \left (225 a^2 D d^3+a b \left (795 D c^2+658 C d c+175 B d^2\right ) d+105 b^2 c \left (2 C c^2+5 B d c+4 A d^2\right )\right )+b \left (105 A b d \left (b c^2+3 a d^2\right )+a \left (9 a (49 C d+110 c D) d^2+b c \left (30 D c^2+427 C d c+700 B d^2\right )\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} (10 a c D+49 a C d+35 A b d)}{7 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b}}{2 a b}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{5} \left (\frac {\frac {b \left (105 A b d \left (3 a d^2+b c^2\right )+a \left (9 a d^2 (110 c D+49 C d)+b c \left (700 B d^2+30 c^2 D+427 c C d\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} (10 a c D+49 a C d+35 A b d)}{7 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b}}{2 a b}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{5} \left (\frac {\frac {b \sqrt {1-\frac {b x^2}{a}} \left (105 A b d \left (3 a d^2+b c^2\right )+a \left (9 a d^2 (110 c D+49 C d)+b c \left (700 B d^2+30 c^2 D+427 c C d\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} (10 a c D+49 a C d+35 A b d)}{7 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b}}{2 a b}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{5} \left (\frac {-\frac {\left (b c^2-a d^2\right ) \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (105 A b d \left (3 a d^2+b c^2\right )+a \left (9 a d^2 (110 c D+49 C d)+b c \left (700 B d^2+30 c^2 D+427 c C d\right )\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} (10 a c D+49 a C d+35 A b d)}{7 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b}}{2 a b}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{5} \left (\frac {-\frac {\left (b c^2-a d^2\right ) \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (105 A b d \left (3 a d^2+b c^2\right )+a \left (9 a d^2 (110 c D+49 C d)+b c \left (700 B d^2+30 c^2 D+427 c C d\right )\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} (10 a c D+49 a C d+35 A b d)}{7 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b}}{2 a b}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{5} \left (\frac {-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (105 A b d \left (3 a d^2+b c^2\right )+a \left (9 a d^2 (110 c D+49 C d)+b c \left (700 B d^2+30 c^2 D+427 c C d\right )\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} (10 a c D+49 a C d+35 A b d)}{7 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b}}{2 a b}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{5} \left (\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (105 A b d \left (3 a d^2+b c^2\right )+a \left (9 a d^2 (110 c D+49 C d)+b c \left (700 B d^2+30 c^2 D+427 c C d\right )\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} (10 a c D+49 a C d+35 A b d)}{7 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b}}{2 a b}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {(c+d x)^{5/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{5} \left (\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (105 A b d \left (3 a d^2+b c^2\right )+a \left (9 a d^2 (110 c D+49 C d)+b c \left (700 B d^2+30 c^2 D+427 c C d\right )\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{3 b}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (a \left (225 a d^2 D+b \left (175 B d^2+30 c^2 D+217 c C d\right )\right )+105 A b^2 c d\right )}{3 b}\right )-\frac {2}{5} \sqrt {a-b x^2} (c+d x)^{3/2} (10 a c D+49 a C d+35 A b d)}{7 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b}}{2 a b}\) |
Input:
Int[((c + d*x)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(a - b*x^2)^(3/2),x]
Output:
((a*(B + (a*D)/b) + (A*b + a*C)*x)*(c + d*x)^(5/2))/(a*b*Sqrt[a - b*x^2]) - ((-4*a*D*(c + d*x)^(5/2)*Sqrt[a - b*x^2])/(7*b) + ((-2*(35*A*b*d + 49*a* C*d + 10*a*c*D)*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/5 + ((-2*(105*A*b^2*c*d + a*(225*a*d^2*D + b*(217*c*C*d + 175*B*d^2 + 30*c^2*D)))*Sqrt[c + d*x]*Sqr t[a - b*x^2])/(3*b) + ((-2*Sqrt[a]*Sqrt[b]*(105*A*b*d*(b*c^2 + 3*a*d^2) + a*(9*a*d^2*(49*C*d + 110*c*D) + b*c*(427*c*C*d + 700*B*d^2 + 30*c^2*D)))*S qrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqr t[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(d*Sqrt[(Sqrt[b]*(c + d* x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2) *(105*A*b^2*c*d + a*(225*a*d^2*D + b*(217*c*C*d + 175*B*d^2 + 30*c^2*D)))* Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*Elli pticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sq rt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2]))/(3*b))/5)/(7*b))/( 2*a*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 , x], x, 1]}, Simp[(d + e*x)^m*(a + b*x^2)^(p + 1)*((a*S - b*R*x)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*(d + e*x)*Qx - a*e*S*m + b*d*R*(2*p + 3) + b*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x ] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && !(IGtQ[m, 0] && R ationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(1492\) vs. \(2(491)=982\).
Time = 6.29 (sec) , antiderivative size = 1493, normalized size of antiderivative = 2.63
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1493\) |
| default | \(\text {Expression too large to display}\) | \(4666\) |
Input:
int((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(3/2),x,method=_RETURNVER BOSE)
Output:
1/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*((d*x+c)*(-b*x^2+a))^(1/2)*(-2*(-b*d*x-b* c)*(1/2*(A*a*b*d^2+A*b^2*c^2+2*B*a*b*c*d+C*a^2*d^2+C*a*b*c^2+2*D*a^2*c*d)/ b^3/a*x+1/2*(2*A*b^2*c*d+B*a*b*d^2+B*b^2*c^2+2*C*a*b*c*d+D*a^2*d^2+D*a*b*c ^2)/b^4)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2/7*D*d^2/b^2*x^2*(-b*d*x^3-b*c*x^ 2+a*d*x+a*c)^(1/2)-2/5*(-d^2*(C*d+3*D*c)/b+6/7*D*d^2/b*c)/b/d*x*(-b*d*x^3- b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(-1/b^2*d*(B*b*d^2+3*C*b*c*d+D*a*d^2+3*D*b*c^ 2)-5/7*D*d^3/b^2*a-4/5*(-d^2*(C*d+3*D*c)/b+6/7*D*d^2/b*c)/d*c)/b/d*(-b*d*x ^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(-(3*A*b^2*c*d^2+B*a*b*d^3+3*B*b^2*c^2*d+3*C *a*b*c*d^2+C*b^2*c^3+D*a^2*d^3+3*D*a*b*c^2*d)/b^3+1/b^3*(3*A*a*b^2*c*d^2+A *b^3*c^3+B*a^2*b*d^3+3*B*a*b^2*c^2*d+3*C*a^2*b*c*d^2+C*a*b^2*c^3+D*a^3*d^3 +3*D*a^2*b*c^2*d)/a-1/2/b^3*d*(2*A*b^2*c*d+B*a*b*d^2+B*b^2*c^2+2*C*a*b*c*d +D*a^2*d^2+D*a*b*c^2)-1/b^2*c*(A*a*b*d^2+A*b^2*c^2+2*B*a*b*c*d+C*a^2*d^2+C *a*b*c^2+2*D*a^2*c*d)/a+2/5*(-d^2*(C*d+3*D*c)/b+6/7*D*d^2/b*c)/b/d*a*c+1/3 *(-1/b^2*d*(B*b*d^2+3*C*b*c*d+D*a*d^2+3*D*b*c^2)-5/7*D*d^3/b^2*a-4/5*(-d^2 *(C*d+3*D*c)/b+6/7*D*d^2/b*c)/d*c)/b*a)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/ d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/ 2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a* d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b* (a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+2*(-1/b^2*(A*b*d^3+3*B*b*c*d^2 +C*a*d^3+3*C*b*c^2*d+3*D*a*c*d^2+D*b*c^3)-1/2*(A*a*b*d^2+A*b^2*c^2+2*B*...
Time = 0.10 (sec) , antiderivative size = 819, normalized size of antiderivative = 1.44 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(3/2),x, algorithm= "fricas")
Output:
1/315*((30*D*a^2*b^2*c^4 - 7*(29*C*a^2*b^2 - 15*A*a*b^3)*c^3*d - 5*(279*D* a^3*b + 175*B*a^2*b^2)*c^2*d^2 - 21*(73*C*a^3*b + 45*A*a^2*b^2)*c*d^3 - 75 *(9*D*a^4 + 7*B*a^3*b)*d^4 - (30*D*a*b^3*c^4 - 7*(29*C*a*b^3 - 15*A*b^4)*c ^3*d - 5*(279*D*a^2*b^2 + 175*B*a*b^3)*c^2*d^2 - 21*(73*C*a^2*b^2 + 45*A*a *b^3)*c*d^3 - 75*(9*D*a^3*b + 7*B*a^2*b^2)*d^4)*x^2)*sqrt(-b*d)*weierstras sPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3) , 1/3*(3*d*x + c)/d) + 3*(30*D*a^2*b^2*c^3*d + 7*(61*C*a^2*b^2 + 15*A*a*b^ 3)*c^2*d^2 + 10*(99*D*a^3*b + 70*B*a^2*b^2)*c*d^3 + 63*(7*C*a^3*b + 5*A*a^ 2*b^2)*d^4 - (30*D*a*b^3*c^3*d + 7*(61*C*a*b^3 + 15*A*b^4)*c^2*d^2 + 10*(9 9*D*a^2*b^2 + 70*B*a*b^3)*c*d^3 + 63*(7*C*a^2*b^2 + 5*A*a*b^3)*d^4)*x^2)*s qrt(-b*d)*weierstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9* a*c*d^2)/(b*d^3), weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27 *(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) + 3*(30*D*a*b^3*d^4*x^4 - 15*(13*D*a^2*b^2 + 7*B*a*b^3)*c^2*d^2 - 14*(26*C*a^2*b^2 + 15*A*a*b^3)*c *d^3 - 25*(9*D*a^3*b + 7*B*a^2*b^2)*d^4 + 6*(15*D*a*b^3*c*d^3 + 7*C*a*b^3* d^4)*x^3 + 2*(45*D*a*b^3*c^2*d^2 + 77*C*a*b^3*c*d^3 + 5*(9*D*a^2*b^2 + 7*B *a*b^3)*d^4)*x^2 - 3*(35*(C*a*b^3 + A*b^4)*c^2*d^2 + 10*(10*D*a^2*b^2 + 7* B*a*b^3)*c*d^3 + 7*(7*C*a^2*b^2 + 5*A*a*b^3)*d^4)*x)*sqrt(-b*x^2 + a)*sqrt (d*x + c))/(a*b^5*d^2*x^2 - a^2*b^4*d^2)
Timed out. \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(5/2)*(D*x**3+C*x**2+B*x+A)/(-b*x**2+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{\frac {5}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(3/2),x, algorithm= "maxima")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*(d*x + c)^(5/2)/(-b*x^2 + a)^(3/2), x)
\[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{\frac {5}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(3/2),x, algorithm= "giac")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*(d*x + c)^(5/2)/(-b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:
int(((c + d*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(3/2),x)
Output:
int(((c + d*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(3/2), x)
\[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\left (d x +c \right )^{\frac {5}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (-b \,x^{2}+a \right )^{\frac {3}{2}}}d x \] Input:
int((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(3/2),x)
Output:
int((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(3/2),x)