Integrand size = 22, antiderivative size = 612 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {d x}} \, dx=-\frac {2 \sqrt {d x} \left (4 b^4-27 a b^2 c-180 a^2 c^2+12 b c \left (b^2-8 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{693 c^2 d}+\frac {10 \sqrt {d x} \left (3 \left (b^2+6 a c\right )+7 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{693 c d}+\frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}+\frac {b \left (8 b^4-93 a b^2 c+372 a^2 c^2\right ) \sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{693 \sqrt {2} c^{7/2} \sqrt {d} \sqrt {a+x (b+c x)}}-\frac {\sqrt {-b+\sqrt {b^2-4 a c}} \left (8 b^6-101 a b^4 c+456 a^2 b^2 c^2-720 a^3 c^3+b \sqrt {b^2-4 a c} \left (8 b^4-93 a b^2 c+372 a^2 c^2\right )\right ) \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{693 \sqrt {2} c^{7/2} \sqrt {d} \sqrt {a+x (b+c x)}} \] Output:
-2/693*(d*x)^(1/2)*(4*b^4-27*a*b^2*c-180*a^2*c^2+12*b*c*(-8*a*c+b^2)*x)*(c *x^2+b*x+a)^(1/2)/c^2/d+10/693*(d*x)^(1/2)*(7*b*c*x+18*a*c+3*b^2)*(c*x^2+b *x+a)^(3/2)/c/d+2/11*(d*x)^(1/2)*(c*x^2+b*x+a)^(5/2)/d+1/1386*b*(372*a^2*c ^2-93*a*b^2*c+8*b^4)*(-b+(-4*a*c+b^2)^(1/2))^(1/2)*(b+(-4*a*c+b^2)^(1/2))* (1+2*c*x/(b-(-4*a*c+b^2)^(1/2)))^(1/2)*(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1 /2)*EllipticE(2^(1/2)*c^(1/2)*(d*x)^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/2)/d^ (1/2),((b-(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2)/c^(7/ 2)/d^(1/2)/(a+x*(c*x+b))^(1/2)-1/1386*(-b+(-4*a*c+b^2)^(1/2))^(1/2)*(8*b^6 -101*a*b^4*c+456*a^2*b^2*c^2-720*a^3*c^3+b*(-4*a*c+b^2)^(1/2)*(372*a^2*c^2 -93*a*b^2*c+8*b^4))*(1+2*c*x/(b-(-4*a*c+b^2)^(1/2)))^(1/2)*(1+2*c*x/(b+(-4 *a*c+b^2)^(1/2)))^(1/2)*EllipticF(2^(1/2)*c^(1/2)*(d*x)^(1/2)/(-b+(-4*a*c+ b^2)^(1/2))^(1/2)/d^(1/2),((b-(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^ (1/2))*2^(1/2)/c^(7/2)/d^(1/2)/(a+x*(c*x+b))^(1/2)
Result contains complex when optimal does not.
Time = 25.15 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {\sqrt {x} \left (\frac {4 \sqrt {x} (a+x (b+c x)) \left (-4 b^4+3 b^3 c x+b^2 c \left (42 a+113 c x^2\right )+b c^2 x \left (347 a+161 c x^2\right )+9 c^2 \left (37 a^2+24 a c x^2+7 c^2 x^4\right )\right )}{c^2}+\frac {x \left (\frac {4 b \left (8 b^4-93 a b^2 c+372 a^2 c^2\right ) (a+x (b+c x))}{x^{3/2}}-\frac {i b \left (8 b^4-93 a b^2 c+372 a^2 c^2\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}+\frac {i \left (-8 b^6+101 a b^4 c-456 a^2 b^2 c^2+720 a^3 c^3+8 b^5 \sqrt {b^2-4 a c}-93 a b^3 c \sqrt {b^2-4 a c}+372 a^2 b c^2 \sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}\right )}{c^3}\right )}{1386 \sqrt {d x} \sqrt {a+x (b+c x)}} \] Input:
Integrate[(a + b*x + c*x^2)^(5/2)/Sqrt[d*x],x]
Output:
(Sqrt[x]*((4*Sqrt[x]*(a + x*(b + c*x))*(-4*b^4 + 3*b^3*c*x + b^2*c*(42*a + 113*c*x^2) + b*c^2*x*(347*a + 161*c*x^2) + 9*c^2*(37*a^2 + 24*a*c*x^2 + 7 *c^2*x^4)))/c^2 + (x*((4*b*(8*b^4 - 93*a*b^2*c + 372*a^2*c^2)*(a + x*(b + c*x)))/x^(3/2) - (I*b*(8*b^4 - 93*a*b^2*c + 372*a^2*c^2)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*Sqrt[(2*a + b*x - Sqrt [b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticE[I*ArcSinh[(Sqrt[2] *Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - S qrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])] + (I*(-8*b^6 + 101*a*b ^4*c - 456*a^2*b^2*c^2 + 720*a^3*c^3 + 8*b^5*Sqrt[b^2 - 4*a*c] - 93*a*b^3* c*Sqrt[b^2 - 4*a*c] + 372*a^2*b*c^2*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqr t[b^2 - 4*a*c]*x)]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a *c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/ (b + Sqrt[b^2 - 4*a*c])]))/c^3))/(1386*Sqrt[d*x]*Sqrt[a + x*(b + c*x)])
Time = 1.40 (sec) , antiderivative size = 514, normalized size of antiderivative = 0.84, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {1162, 25, 27, 1231, 27, 1231, 27, 1241, 1240, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {d x}} \, dx\) |
| \(\Big \downarrow \) 1162 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}-\frac {5 \int -\frac {d (2 a+b x) \left (c x^2+b x+a\right )^{3/2}}{\sqrt {d x}}dx}{11 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \int \frac {d (2 a+b x) \left (c x^2+b x+a\right )^{3/2}}{\sqrt {d x}}dx}{11 d}+\frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{11} \int \frac {(2 a+b x) \left (c x^2+b x+a\right )^{3/2}}{\sqrt {d x}}dx+\frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5}{11} \left (\frac {2 \sqrt {d x} \left (3 \left (6 a c+b^2\right )+7 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 c d}-\frac {2 \int \frac {d^2 \left (a \left (b^2-36 a c\right )+4 b \left (b^2-8 a c\right ) x\right ) \sqrt {c x^2+b x+a}}{2 \sqrt {d x}}dx}{21 c d^2}\right )+\frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{11} \left (\frac {2 \sqrt {d x} \left (3 \left (6 a c+b^2\right )+7 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 c d}-\frac {\int \frac {\left (a \left (b^2-36 a c\right )+4 b \left (b^2-8 a c\right ) x\right ) \sqrt {c x^2+b x+a}}{\sqrt {d x}}dx}{21 c}\right )+\frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5}{11} \left (\frac {2 \sqrt {d x} \left (3 \left (6 a c+b^2\right )+7 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 c d}-\frac {\frac {2 \sqrt {d x} \left (-180 a^2 c^2+12 b c x \left (b^2-8 a c\right )-27 a b^2 c+4 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {2 \int \frac {d^2 \left (2 a \left (2 b^4-21 a c b^2+180 a^2 c^2\right )+b \left (8 b^4-93 a c b^2+372 a^2 c^2\right ) x\right )}{2 \sqrt {d x} \sqrt {c x^2+b x+a}}dx}{15 c d^2}}{21 c}\right )+\frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{11} \left (\frac {2 \sqrt {d x} \left (3 \left (6 a c+b^2\right )+7 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 c d}-\frac {\frac {2 \sqrt {d x} \left (-180 a^2 c^2+12 b c x \left (b^2-8 a c\right )-27 a b^2 c+4 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {\int \frac {2 a \left (2 b^4-21 a c b^2+180 a^2 c^2\right )+b \left (8 b^4-93 a c b^2+372 a^2 c^2\right ) x}{\sqrt {d x} \sqrt {c x^2+b x+a}}dx}{15 c}}{21 c}\right )+\frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}\) |
| \(\Big \downarrow \) 1241 |
| \(\displaystyle \frac {5}{11} \left (\frac {2 \sqrt {d x} \left (3 \left (6 a c+b^2\right )+7 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 c d}-\frac {\frac {2 \sqrt {d x} \left (-180 a^2 c^2+12 b c x \left (b^2-8 a c\right )-27 a b^2 c+4 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {\sqrt {x} \int \frac {2 a \left (2 b^4-21 a c b^2+180 a^2 c^2\right )+b \left (8 b^4-93 a c b^2+372 a^2 c^2\right ) x}{\sqrt {x} \sqrt {c x^2+b x+a}}dx}{15 c \sqrt {d x}}}{21 c}\right )+\frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {5}{11} \left (\frac {2 \sqrt {d x} \left (3 \left (6 a c+b^2\right )+7 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 c d}-\frac {\frac {2 \sqrt {d x} \left (-180 a^2 c^2+12 b c x \left (b^2-8 a c\right )-27 a b^2 c+4 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {2 \sqrt {x} \int \frac {2 a \left (2 b^4-21 a c b^2+180 a^2 c^2\right )+b \left (8 b^4-93 a c b^2+372 a^2 c^2\right ) x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{15 c \sqrt {d x}}}{21 c}\right )+\frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {5}{11} \left (\frac {2 \sqrt {d x} \left (3 \left (6 a c+b^2\right )+7 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 c d}-\frac {\frac {2 \sqrt {d x} \left (-180 a^2 c^2+12 b c x \left (b^2-8 a c\right )-27 a b^2 c+4 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {2 \sqrt {x} \left (\sqrt {a} \left (2 \sqrt {a} \left (180 a^2 c^2-21 a b^2 c+2 b^4\right )+\frac {b \left (372 a^2 c^2-93 a b^2 c+8 b^4\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {\sqrt {a} b \left (372 a^2 c^2-93 a b^2 c+8 b^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 c \sqrt {d x}}}{21 c}\right )+\frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{11} \left (\frac {2 \sqrt {d x} \left (3 \left (6 a c+b^2\right )+7 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 c d}-\frac {\frac {2 \sqrt {d x} \left (-180 a^2 c^2+12 b c x \left (b^2-8 a c\right )-27 a b^2 c+4 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {2 \sqrt {x} \left (\sqrt {a} \left (2 \sqrt {a} \left (180 a^2 c^2-21 a b^2 c+2 b^4\right )+\frac {b \left (372 a^2 c^2-93 a b^2 c+8 b^4\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}d\sqrt {x}-\frac {b \left (372 a^2 c^2-93 a b^2 c+8 b^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 c \sqrt {d x}}}{21 c}\right )+\frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {5}{11} \left (\frac {2 \sqrt {d x} \left (3 \left (6 a c+b^2\right )+7 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 c d}-\frac {\frac {2 \sqrt {d x} \left (-180 a^2 c^2+12 b c x \left (b^2-8 a c\right )-27 a b^2 c+4 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \left (180 a^2 c^2-21 a b^2 c+2 b^4\right )+\frac {b \left (372 a^2 c^2-93 a b^2 c+8 b^4\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {b \left (372 a^2 c^2-93 a b^2 c+8 b^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{15 c \sqrt {d x}}}{21 c}\right )+\frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {5}{11} \left (\frac {2 \sqrt {d x} \left (3 \left (6 a c+b^2\right )+7 b c x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 c d}-\frac {\frac {2 \sqrt {d x} \left (-180 a^2 c^2+12 b c x \left (b^2-8 a c\right )-27 a b^2 c+4 b^4\right ) \sqrt {a+b x+c x^2}}{15 c d}-\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \left (180 a^2 c^2-21 a b^2 c+2 b^4\right )+\frac {b \left (372 a^2 c^2-93 a b^2 c+8 b^4\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {b \left (372 a^2 c^2-93 a b^2 c+8 b^4\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{15 c \sqrt {d x}}}{21 c}\right )+\frac {2 \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{11 d}\) |
Input:
Int[(a + b*x + c*x^2)^(5/2)/Sqrt[d*x],x]
Output:
(2*Sqrt[d*x]*(a + b*x + c*x^2)^(5/2))/(11*d) + (5*((2*Sqrt[d*x]*(3*(b^2 + 6*a*c) + 7*b*c*x)*(a + b*x + c*x^2)^(3/2))/(63*c*d) - ((2*Sqrt[d*x]*(4*b^4 - 27*a*b^2*c - 180*a^2*c^2 + 12*b*c*(b^2 - 8*a*c)*x)*Sqrt[a + b*x + c*x^2 ])/(15*c*d) - (2*Sqrt[x]*(-((b*(8*b^4 - 93*a*b^2*c + 372*a^2*c^2)*(-((Sqrt [x]*Sqrt[a + b*x + c*x^2])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqrt[a] + Sq rt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTa n[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[ a + b*x + c*x^2])))/Sqrt[c]) + (a^(1/4)*(2*Sqrt[a]*(2*b^4 - 21*a*b^2*c + 1 80*a^2*c^2) + (b*(8*b^4 - 93*a*b^2*c + 372*a^2*c^2))/Sqrt[c])*(Sqrt[a] + S qrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcT an[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(1/4)*Sq rt[a + b*x + c*x^2])))/(15*c*Sqrt[d*x]))/(21*c)))/11
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x ] - Simp[p/(e*(m + 2*p + 1)) Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x ] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
Int[((f_) + (g_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_ )^2]), x_Symbol] :> Simp[Sqrt[x]/Sqrt[e*x] Int[(f + g*x)/(Sqrt[x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, e, f, g}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Leaf count of result is larger than twice the leaf count of optimal. \(1154\) vs. \(2(515)=1030\).
Time = 3.45 (sec) , antiderivative size = 1155, normalized size of antiderivative = 1.89
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1155\) |
| risch | \(\text {Expression too large to display}\) | \(1369\) |
| default | \(\text {Expression too large to display}\) | \(2274\) |
Input:
int((c*x^2+b*x+a)^(5/2)/(d*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
(d*x*(c*x^2+b*x+a))^(1/2)/(d*x)^(1/2)/(c*x^2+b*x+a)^(1/2)*(2/11*c^2/d*x^4* (c*d*x^3+b*d*x^2+a*d*x)^(1/2)+46/99*b*c/d*x^3*(c*d*x^3+b*d*x^2+a*d*x)^(1/2 )+2/7*(24/11*a*c^2+113/99*b^2*c)/c/d*x^2*(c*d*x^3+b*d*x^2+a*d*x)^(1/2)+2/5 *(433/99*a*b*c+b^3-6/7*b/c*(24/11*a*c^2+113/99*b^2*c))/c/d*x*(c*d*x^3+b*d* x^2+a*d*x)^(1/2)+2/3*(3*a^2*c+3*a*b^2-4/5*b/c*(433/99*a*b*c+b^3-6/7*b/c*(2 4/11*a*c^2+113/99*b^2*c))-5/7*a/c*(24/11*a*c^2+113/99*b^2*c))/c/d*(c*d*x^3 +b*d*x^2+a*d*x)^(1/2)+(a^3-1/3*a/c*(3*a^2*c+3*a*b^2-4/5*b/c*(433/99*a*b*c+ b^3-6/7*b/c*(24/11*a*c^2+113/99*b^2*c))-5/7*a/c*(24/11*a*c^2+113/99*b^2*c) ))*(b+(-4*a*c+b^2)^(1/2))/c*2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+( -4*a*c+b^2)^(1/2))*c)^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(- 4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(-2*c*x/(b+(-4*a *c+b^2)^(1/2)))^(1/2)/(c*d*x^3+b*d*x^2+a*d*x)^(1/2)*EllipticF(2^(1/2)*((x+ 1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2),1/2*(-2*(b+( -4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2) ^(1/2))))^(1/2))+(3*a^2*b-3/5*a/c*(433/99*a*b*c+b^3-6/7*b/c*(24/11*a*c^2+1 13/99*b^2*c))-2/3*b/c*(3*a^2*c+3*a*b^2-4/5*b/c*(433/99*a*b*c+b^3-6/7*b/c*( 24/11*a*c^2+113/99*b^2*c))-5/7*a/c*(24/11*a*c^2+113/99*b^2*c)))*(b+(-4*a*c +b^2)^(1/2))/c*2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^( 1/2))*c)^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1 /2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(-2*c*x/(b+(-4*a*c+b^2)^(1...
Time = 0.11 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.47 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {d x}} \, dx=-\frac {2 \, {\left ({\left (8 \, b^{6} - 105 \, a b^{4} c + 498 \, a^{2} b^{2} c^{2} - 1080 \, a^{3} c^{3}\right )} \sqrt {c d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 3 \, {\left (8 \, b^{5} c - 93 \, a b^{3} c^{2} + 372 \, a^{2} b c^{3}\right )} \sqrt {c d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) - 3 \, {\left (63 \, c^{6} x^{4} + 161 \, b c^{5} x^{3} - 4 \, b^{4} c^{2} + 42 \, a b^{2} c^{3} + 333 \, a^{2} c^{4} + {\left (113 \, b^{2} c^{4} + 216 \, a c^{5}\right )} x^{2} + {\left (3 \, b^{3} c^{3} + 347 \, a b c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {d x}\right )}}{2079 \, c^{4} d} \] Input:
integrate((c*x^2+b*x+a)^(5/2)/(d*x)^(1/2),x, algorithm="fricas")
Output:
-2/2079*((8*b^6 - 105*a*b^4*c + 498*a^2*b^2*c^2 - 1080*a^3*c^3)*sqrt(c*d)* weierstrassPInverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/ 3*(3*c*x + b)/c) + 3*(8*b^5*c - 93*a*b^3*c^2 + 372*a^2*b*c^3)*sqrt(c*d)*we ierstrassZeta(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, weierstr assPInverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/3*(3*c*x + b)/c)) - 3*(63*c^6*x^4 + 161*b*c^5*x^3 - 4*b^4*c^2 + 42*a*b^2*c^3 + 333 *a^2*c^4 + (113*b^2*c^4 + 216*a*c^5)*x^2 + (3*b^3*c^3 + 347*a*b*c^4)*x)*sq rt(c*x^2 + b*x + a)*sqrt(d*x))/(c^4*d)
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {d x}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\sqrt {d x}}\, dx \] Input:
integrate((c*x**2+b*x+a)**(5/2)/(d*x)**(1/2),x)
Output:
Integral((a + b*x + c*x**2)**(5/2)/sqrt(d*x), x)
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {d x}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{\sqrt {d x}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(5/2)/(d*x)^(1/2),x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x + a)^(5/2)/sqrt(d*x), x)
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {d x}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{\sqrt {d x}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(5/2)/(d*x)^(1/2),x, algorithm="giac")
Output:
integrate((c*x^2 + b*x + a)^(5/2)/sqrt(d*x), x)
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {d x}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{\sqrt {d\,x}} \,d x \] Input:
int((a + b*x + c*x^2)^(5/2)/(d*x)^(1/2),x)
Output:
int((a + b*x + c*x^2)^(5/2)/(d*x)^(1/2), x)
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {d x}} \, dx=\frac {\sqrt {d}\, \left (2076 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a^{2} c -18 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a \,b^{2}+1388 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a b c x +864 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a \,c^{2} x^{2}+12 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, b^{3} x +452 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, b^{2} c \,x^{2}+644 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, b \,c^{2} x^{3}+252 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, c^{3} x^{4}-1116 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x}{c \,x^{2}+b x +a}d x \right ) a^{2} c^{2}+279 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x}{c \,x^{2}+b x +a}d x \right ) a \,b^{2} c -24 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x}{c \,x^{2}+b x +a}d x \right ) b^{4}+348 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{3}+b \,x^{2}+a x}d x \right ) a^{3} c +9 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{3}+b \,x^{2}+a x}d x \right ) a^{2} b^{2}\right )}{1386 c d} \] Input:
int((c*x^2+b*x+a)^(5/2)/(d*x)^(1/2),x)
Output:
(sqrt(d)*(2076*sqrt(x)*sqrt(a + b*x + c*x**2)*a**2*c - 18*sqrt(x)*sqrt(a + b*x + c*x**2)*a*b**2 + 1388*sqrt(x)*sqrt(a + b*x + c*x**2)*a*b*c*x + 864* sqrt(x)*sqrt(a + b*x + c*x**2)*a*c**2*x**2 + 12*sqrt(x)*sqrt(a + b*x + c*x **2)*b**3*x + 452*sqrt(x)*sqrt(a + b*x + c*x**2)*b**2*c*x**2 + 644*sqrt(x) *sqrt(a + b*x + c*x**2)*b*c**2*x**3 + 252*sqrt(x)*sqrt(a + b*x + c*x**2)*c **3*x**4 - 1116*int((sqrt(x)*sqrt(a + b*x + c*x**2)*x)/(a + b*x + c*x**2), x)*a**2*c**2 + 279*int((sqrt(x)*sqrt(a + b*x + c*x**2)*x)/(a + b*x + c*x** 2),x)*a*b**2*c - 24*int((sqrt(x)*sqrt(a + b*x + c*x**2)*x)/(a + b*x + c*x* *2),x)*b**4 + 348*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x + b*x**2 + c*x **3),x)*a**3*c + 9*int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x + b*x**2 + c* x**3),x)*a**2*b**2))/(1386*c*d)