Integrand size = 23, antiderivative size = 468 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {2 \left (96 c^2 d^2-64 b c d e+3 b^2 e^2\right ) x \sqrt {b x+c x^2}}{21 e^4 (d+e x)^{5/2}}-\frac {10 c (16 c d-9 b e) x^2 \sqrt {b x+c x^2}}{21 e^3 (d+e x)^{5/2}}-\frac {2 \left (128 c^2 d^2-80 b c d e+3 b^2 e^2\right ) \sqrt {b x+c x^2}}{21 e^5 (d+e x)^{3/2}}+\frac {20 c x \left (b x+c x^2\right )^{3/2}}{21 e^2 (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac {2 (2 c d-b e) \left (128 c^2 d^2-128 b c d e+3 b^2 e^2\right ) \sqrt {b x+c x^2} E\left (\arctan \left (\frac {\sqrt {e} \sqrt {x}}{\sqrt {d}}\right )|1-\frac {c d}{b e}\right )}{21 \sqrt {d} e^{11/2} (c d-b e) \sqrt {x} \sqrt {\frac {d (b+c x)}{b (d+e x)}} \sqrt {d+e x}}-\frac {2 c \sqrt {d} \left (128 c^2 d^2-176 b c d e+51 b^2 e^2\right ) \sqrt {b x+c x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {e} \sqrt {x}}{\sqrt {d}}\right ),1-\frac {c d}{b e}\right )}{21 e^{11/2} (c d-b e) \sqrt {x} \sqrt {\frac {d (b+c x)}{b (d+e x)}} \sqrt {d+e x}} \] Output:
-2/21*(3*b^2*e^2-64*b*c*d*e+96*c^2*d^2)*x*(c*x^2+b*x)^(1/2)/e^4/(e*x+d)^(5 /2)-10/21*c*(-9*b*e+16*c*d)*x^2*(c*x^2+b*x)^(1/2)/e^3/(e*x+d)^(5/2)-2/21*( 3*b^2*e^2-80*b*c*d*e+128*c^2*d^2)*(c*x^2+b*x)^(1/2)/e^5/(e*x+d)^(3/2)+20/2 1*c*x*(c*x^2+b*x)^(3/2)/e^2/(e*x+d)^(5/2)-2/7*(c*x^2+b*x)^(5/2)/e/(e*x+d)^ (7/2)+2/21*(-b*e+2*c*d)*(3*b^2*e^2-128*b*c*d*e+128*c^2*d^2)*(c*x^2+b*x)^(1 /2)*EllipticE(e^(1/2)*x^(1/2)/d^(1/2)/(1+e*x/d)^(1/2),(1-c*d/b/e)^(1/2))/d ^(1/2)/e^(11/2)/(-b*e+c*d)/x^(1/2)/(d*(c*x+b)/b/(e*x+d))^(1/2)/(e*x+d)^(1/ 2)-2/21*c*d^(1/2)*(51*b^2*e^2-176*b*c*d*e+128*c^2*d^2)*(c*x^2+b*x)^(1/2)*I nverseJacobiAM(arctan(e^(1/2)*x^(1/2)/d^(1/2)),(1-c*d/b/e)^(1/2))/e^(11/2) /(-b*e+c*d)/x^(1/2)/(d*(c*x+b)/b/(e*x+d))^(1/2)/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 12.97 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.07 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {2 (x (b+c x))^{5/2} \left (b e x (b+c x) \left (3 b^3 e^6 x^3-b^2 c d e^2 \left (51 d^3+169 d^2 e x+194 d e^2 x^2+85 e^3 x^3\right )-c^3 d^2 \left (128 d^4+416 d^3 e x+464 d^2 e^2 x^2+186 d e^3 x^3+7 e^4 x^4\right )+b c^2 d e \left (176 d^4+576 d^3 e x+649 d^2 e^2 x^2+265 d e^3 x^3+7 e^4 x^4\right )\right )+\sqrt {\frac {b}{c}} c (d+e x)^3 \left (\sqrt {\frac {b}{c}} \left (256 c^3 d^3-384 b c^2 d^2 e+134 b^2 c d e^2-3 b^3 e^3\right ) (b+c x) (d+e x)+i b e \left (256 c^3 d^3-384 b c^2 d^2 e+134 b^2 c d e^2-3 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (128 c^3 d^3-208 b c^2 d^2 e+83 b^2 c d e^2-3 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{21 b d e^6 (c d-b e) x^3 (b+c x)^3 (d+e x)^{7/2}} \] Input:
Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(9/2),x]
Output:
(-2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(3*b^3*e^6*x^3 - b^2*c*d*e^2*(51* d^3 + 169*d^2*e*x + 194*d*e^2*x^2 + 85*e^3*x^3) - c^3*d^2*(128*d^4 + 416*d ^3*e*x + 464*d^2*e^2*x^2 + 186*d*e^3*x^3 + 7*e^4*x^4) + b*c^2*d*e*(176*d^4 + 576*d^3*e*x + 649*d^2*e^2*x^2 + 265*d*e^3*x^3 + 7*e^4*x^4)) + Sqrt[b/c] *c*(d + e*x)^3*(Sqrt[b/c]*(256*c^3*d^3 - 384*b*c^2*d^2*e + 134*b^2*c*d*e^2 - 3*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(256*c^3*d^3 - 384*b*c^2*d^2*e + 134*b^2*c*d*e^2 - 3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)* EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(128*c^3*d^3 - 208*b*c^2*d^2*e + 83*b^2*c*d*e^2 - 3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/ (21*b*d*e^6*(c*d - b*e)*x^3*(b + c*x)^3*(d + e*x)^(7/2))
Time = 1.24 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {1161, 1229, 27, 1230, 27, 1269, 1169, 122, 120, 127, 126}
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.
| \(\displaystyle \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {5 \int \frac {(b+2 c x) \left (c x^2+b x\right )^{3/2}}{(d+e x)^{7/2}}dx}{7 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {5 \left (-\frac {2 \int -\frac {c \left (b d (16 c d-13 b e)+\left (32 c^2 d^2-32 b c e d+3 b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{2 (d+e x)^{3/2}}dx}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {c \int \frac {\left (b d (16 c d-13 b e)+\left (32 c^2 d^2-32 b c e d+3 b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{(d+e x)^{3/2}}dx}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {5 \left (\frac {c \left (\frac {2 \sqrt {b x+c x^2} \left (e x \left (3 b^2 e^2-32 b c d e+32 c^2 d^2\right )+d \left (51 b^2 e^2-176 b c d e+128 c^2 d^2\right )\right )}{3 e^2 \sqrt {d+e x}}-\frac {2 \int \frac {b d \left (128 c^2 d^2-176 b c e d+51 b^2 e^2\right )+(2 c d-b e) \left (128 c^2 d^2-128 b c e d+3 b^2 e^2\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 e^2}\right )}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {c \left (\frac {2 \sqrt {b x+c x^2} \left (e x \left (3 b^2 e^2-32 b c d e+32 c^2 d^2\right )+d \left (51 b^2 e^2-176 b c d e+128 c^2 d^2\right )\right )}{3 e^2 \sqrt {d+e x}}-\frac {\int \frac {b d \left (128 c^2 d^2-176 b c e d+51 b^2 e^2\right )+(2 c d-b e) \left (128 c^2 d^2-128 b c e d+3 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 e^2}\right )}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5 \left (\frac {c \left (\frac {2 \sqrt {b x+c x^2} \left (e x \left (3 b^2 e^2-32 b c d e+32 c^2 d^2\right )+d \left (51 b^2 e^2-176 b c d e+128 c^2 d^2\right )\right )}{3 e^2 \sqrt {d+e x}}-\frac {\frac {(2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {2 d (c d-b e) \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{3 e^2}\right )}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {5 \left (\frac {c \left (\frac {2 \sqrt {b x+c x^2} \left (e x \left (3 b^2 e^2-32 b c d e+32 c^2 d^2\right )+d \left (51 b^2 e^2-176 b c d e+128 c^2 d^2\right )\right )}{3 e^2 \sqrt {d+e x}}-\frac {\frac {\sqrt {x} \sqrt {b+c x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {2 d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e^2}\right )}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {5 \left (\frac {c \left (\frac {2 \sqrt {b x+c x^2} \left (e x \left (3 b^2 e^2-32 b c d e+32 c^2 d^2\right )+d \left (51 b^2 e^2-176 b c d e+128 c^2 d^2\right )\right )}{3 e^2 \sqrt {d+e x}}-\frac {\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e^2}\right )}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {5 \left (\frac {c \left (\frac {2 \sqrt {b x+c x^2} \left (e x \left (3 b^2 e^2-32 b c d e+32 c^2 d^2\right )+d \left (51 b^2 e^2-176 b c d e+128 c^2 d^2\right )\right )}{3 e^2 \sqrt {d+e x}}-\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e^2}\right )}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {5 \left (\frac {c \left (\frac {2 \sqrt {b x+c x^2} \left (e x \left (3 b^2 e^2-32 b c d e+32 c^2 d^2\right )+d \left (51 b^2 e^2-176 b c d e+128 c^2 d^2\right )\right )}{3 e^2 \sqrt {d+e x}}-\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 e^2}\right )}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {5 \left (\frac {c \left (\frac {2 \sqrt {b x+c x^2} \left (e x \left (3 b^2 e^2-32 b c d e+32 c^2 d^2\right )+d \left (51 b^2 e^2-176 b c d e+128 c^2 d^2\right )\right )}{3 e^2 \sqrt {d+e x}}-\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {4 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 e^2}\right )}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-22 b c d e+22 c^2 d^2\right )+c d^2 (16 c d-13 b e)\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
Input:
Int[(b*x + c*x^2)^(5/2)/(d + e*x)^(9/2),x]
Output:
(-2*(b*x + c*x^2)^(5/2))/(7*e*(d + e*x)^(7/2)) + (5*((-2*(c*d^2*(16*c*d - 13*b*e) + e*(22*c^2*d^2 - 22*b*c*d*e + 3*b^2*e^2)*x)*(b*x + c*x^2)^(3/2))/ (15*d*e^2*(c*d - b*e)*(d + e*x)^(5/2)) + (c*((2*(d*(128*c^2*d^2 - 176*b*c* d*e + 51*b^2*e^2) + e*(32*c^2*d^2 - 32*b*c*d*e + 3*b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(3*e^2*Sqrt[d + e*x]) - ((2*Sqrt[-b]*(2*c*d - b*e)*(128*c^2*d^2 - 128*b*c*d*e + 3*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE [ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[1 + (e* x)/d]*Sqrt[b*x + c*x^2]) - (4*Sqrt[-b]*d*(c*d - b*e)*(128*c^2*d^2 - 128*b* c*d*e + 27*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[d + e*x] *Sqrt[b*x + c*x^2]))/(3*e^2)))/(5*d*e^2*(c*d - b*e))))/(7*e)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt Q[e, 0] && !LtQ[-b/d, 0]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) ) Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b , c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x ])) Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
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 + 1))), x] - Si mp[p/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2]) Int[(d + e*x)^m/(Sqrt[x]* Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq Q[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(893\) vs. \(2(411)=822\).
Time = 7.61 (sec) , antiderivative size = 894, normalized size of antiderivative = 1.91
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {\left (c x +b \right ) x \left (e x +d \right )}\, \left (-\frac {2 d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{7 e^{9} \left (x +\frac {d}{e}\right )^{4}}+\frac {6 d \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{7 e^{8} \left (x +\frac {d}{e}\right )^{3}}-\frac {2 \left (9 b^{2} e^{2}-52 b c d e +52 c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{21 e^{7} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+b e x \right ) \left (3 b^{3} e^{3}-85 d \,e^{2} b^{2} c +237 d^{2} e b \,c^{2}-158 d^{3} c^{3}\right )}{21 e^{6} d \left (b e -c d \right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 c^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 e^{5}}+\frac {2 \left (\frac {c \left (3 b^{2} e^{2}-12 b c d e +10 c^{2} d^{2}\right )}{e^{6}}-\frac {c \left (9 b^{2} e^{2}-52 b c d e +52 c^{2} d^{2}\right )}{21 e^{6}}+\frac {3 b^{3} e^{3}-85 d \,e^{2} b^{2} c +237 d^{2} e b \,c^{2}-158 d^{3} c^{3}}{21 e^{6} d}-\frac {b \left (3 b^{3} e^{3}-85 d \,e^{2} b^{2} c +237 d^{2} e b \,c^{2}-158 d^{3} c^{3}\right )}{21 e^{5} d \left (b e -c d \right )}-\frac {c^{2} b d}{3 e^{5}}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {c^{2} \left (3 b e -4 c d \right )}{e^{5}}-\frac {c \left (3 b^{3} e^{3}-85 d \,e^{2} b^{2} c +237 d^{2} e b \,c^{2}-158 d^{3} c^{3}\right )}{21 e^{5} d \left (b e -c d \right )}-\frac {2 c^{2} \left (b e +c d \right )}{3 e^{5}}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}+\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )-\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{c}\right )}{e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\) | \(894\) |
| default | \(\text {Expression too large to display}\) | \(3260\) |
Input:
int((c*x^2+b*x)^(5/2)/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*((c*x+b)*x*(e*x+d))^(1/2)/x/(c*x+b)*(-2/ 7*d^2*(b^2*e^2-2*b*c*d*e+c^2*d^2)/e^9*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2 )/(x+d/e)^4+6/7*d*(b^2*e^2-3*b*c*d*e+2*c^2*d^2)/e^8*(c*e*x^3+b*e*x^2+c*d*x ^2+b*d*x)^(1/2)/(x+d/e)^3-2/21*(9*b^2*e^2-52*b*c*d*e+52*c^2*d^2)/e^7*(c*e* x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(x+d/e)^2+2/21*(c*e*x^2+b*e*x)/e^6/d/(b*e -c*d)*(3*b^3*e^3-85*b^2*c*d*e^2+237*b*c^2*d^2*e-158*c^3*d^3)/((x+d/e)*(c*e *x^2+b*e*x))^(1/2)+2/3*c^2/e^5*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2*(c* (3*b^2*e^2-12*b*c*d*e+10*c^2*d^2)/e^6-1/21*c*(9*b^2*e^2-52*b*c*d*e+52*c^2* d^2)/e^6+1/21/e^6*(3*b^3*e^3-85*b^2*c*d*e^2+237*b*c^2*d^2*e-158*c^3*d^3)/d -1/21*b/e^5/d/(b*e-c*d)*(3*b^3*e^3-85*b^2*c*d*e^2+237*b*c^2*d^2*e-158*c^3* d^3)-1/3*c^2/e^5*b*d)*d/e*((x+d/e)/d*e)^(1/2)*((b/c+x)/(-d/e+b/c))^(1/2)*( -e*x/d)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*EllipticF(((x+d/e)/d*e )^(1/2),(-d/e/(-d/e+b/c))^(1/2))+2*(1/e^5*c^2*(3*b*e-4*c*d)-1/21/e^5*c*(3* b^3*e^3-85*b^2*c*d*e^2+237*b*c^2*d^2*e-158*c^3*d^3)/d/(b*e-c*d)-2/3*c^2/e^ 5*(b*e+c*d))*d/e*((x+d/e)/d*e)^(1/2)*((b/c+x)/(-d/e+b/c))^(1/2)*(-e*x/d)^( 1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-d/e+b/c)*EllipticE(((x+d/e)/ d*e)^(1/2),(-d/e/(-d/e+b/c))^(1/2))-b/c*EllipticF(((x+d/e)/d*e)^(1/2),(-d/ e/(-d/e+b/c))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1184 vs. \(2 (411) = 822\).
Time = 0.34 (sec) , antiderivative size = 1184, normalized size of antiderivative = 2.53 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(9/2),x, algorithm="fricas")
Output:
2/63*((256*c^4*d^8 - 512*b*c^3*d^7*e + 278*b^2*c^2*d^6*e^2 - 22*b^3*c*d^5* e^3 - 3*b^4*d^4*e^4 + (256*c^4*d^4*e^4 - 512*b*c^3*d^3*e^5 + 278*b^2*c^2*d ^2*e^6 - 22*b^3*c*d*e^7 - 3*b^4*e^8)*x^4 + 4*(256*c^4*d^5*e^3 - 512*b*c^3* d^4*e^4 + 278*b^2*c^2*d^3*e^5 - 22*b^3*c*d^2*e^6 - 3*b^4*d*e^7)*x^3 + 6*(2 56*c^4*d^6*e^2 - 512*b*c^3*d^5*e^3 + 278*b^2*c^2*d^4*e^4 - 22*b^3*c*d^3*e^ 5 - 3*b^4*d^2*e^6)*x^2 + 4*(256*c^4*d^7*e - 512*b*c^3*d^6*e^2 + 278*b^2*c^ 2*d^5*e^3 - 22*b^3*c*d^4*e^4 - 3*b^4*d^3*e^5)*x)*sqrt(c*e)*weierstrassPInv erse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c ^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e) /(c*e)) + 3*(256*c^4*d^7*e - 384*b*c^3*d^6*e^2 + 134*b^2*c^2*d^5*e^3 - 3*b ^3*c*d^4*e^4 + (256*c^4*d^3*e^5 - 384*b*c^3*d^2*e^6 + 134*b^2*c^2*d*e^7 - 3*b^3*c*e^8)*x^4 + 4*(256*c^4*d^4*e^4 - 384*b*c^3*d^3*e^5 + 134*b^2*c^2*d^ 2*e^6 - 3*b^3*c*d*e^7)*x^3 + 6*(256*c^4*d^5*e^3 - 384*b*c^3*d^4*e^4 + 134* b^2*c^2*d^3*e^5 - 3*b^3*c*d^2*e^6)*x^2 + 4*(256*c^4*d^6*e^2 - 384*b*c^3*d^ 5*e^3 + 134*b^2*c^2*d^4*e^4 - 3*b^3*c*d^3*e^5)*x)*sqrt(c*e)*weierstrassZet a(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2* d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2 *d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3* b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) + 3* (128*c^4*d^6*e^2 - 176*b*c^3*d^5*e^3 + 51*b^2*c^2*d^4*e^4 + 7*(c^4*d^2*...
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \] Input:
integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(9/2),x)
Output:
Integral((x*(b + c*x))**(5/2)/(d + e*x)**(9/2), x)
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(9/2),x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(9/2), x)
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((c*x^2+b*x)^(5/2)/(e*x+d)^(9/2),x, algorithm="giac")
Output:
integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(9/2), x)
Timed out. \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \] Input:
int((b*x + c*x^2)^(5/2)/(d + e*x)^(9/2),x)
Output:
int((b*x + c*x^2)^(5/2)/(d + e*x)^(9/2), x)
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\text {too large to display} \] Input:
int((c*x^2+b*x)^(5/2)/(e*x+d)^(9/2),x)
Output:
(30*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**4*d*e**3 + 60*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**4*e**4*x - 270*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b **3*c*d**2*e**2 - 560*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**3*c*d*e**3*x - 108*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**3*c*e**4*x**2 + 960*sqrt(x)*s qrt(d + e*x)*sqrt(b + c*x)*b**2*c**2*d**3*e + 2100*sqrt(x)*sqrt(d + e*x)*s qrt(b + c*x)*b**2*c**2*d**2*e**2*x + 816*sqrt(x)*sqrt(d + e*x)*sqrt(b + c* x)*b**2*c**2*d*e**3*x**2 + 84*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**2*c** 2*e**4*x**3 - 960*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c**3*d**4 - 2560*s qrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c**3*d**3*e*x - 1220*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c**3*d**2*e**2*x**2 - 148*sqrt(x)*sqrt(d + e*x)*sqrt (b + c*x)*b*c**3*d*e**3*x**3 + 12*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c* *3*e**4*x**4 + 640*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**4*d**4*x + 320*s qrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**4*d**3*e*x**2 + 40*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**4*d**2*e**2*x**3 - 4*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**4*d*e**3*x**4 - 45*int((sqrt(d + e*x)*sqrt(b + c*x))/(3*sqrt(x)*b* *2*d**5*e + 15*sqrt(x)*b**2*d**4*e**2*x + 30*sqrt(x)*b**2*d**3*e**3*x**2 + 30*sqrt(x)*b**2*d**2*e**4*x**3 + 15*sqrt(x)*b**2*d*e**5*x**4 + 3*sqrt(x)* b**2*e**6*x**5 - sqrt(x)*b*c*d**6 - 2*sqrt(x)*b*c*d**5*e*x + 5*sqrt(x)*b*c *d**4*e**2*x**2 + 20*sqrt(x)*b*c*d**3*e**3*x**3 + 25*sqrt(x)*b*c*d**2*e**4 *x**4 + 14*sqrt(x)*b*c*d*e**5*x**5 + 3*sqrt(x)*b*c*e**6*x**6 - sqrt(x)*...