Integrand size = 23, antiderivative size = 570 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {2 \left (c d^2 \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3\right )+e \left (160 c^4 d^4-320 b c^3 d^3 e+171 b^2 c^2 d^2 e^2-11 b^3 c d e^3-2 b^4 e^4\right ) x\right ) \sqrt {b x+c x^2}}{63 d^2 e^5 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 \left (d \left (16 c^2 d^2-11 b c d e-2 b^2 e^2\right )+e \left (26 c^2 d^2-26 b c d e+3 b^2 e^2\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{63 d e^3 (c d-b e) (d+e x)^{7/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {4 \sqrt {-b} \sqrt {c} \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 d^2 e^6 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} \sqrt {c} (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{63 d e^6 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]
-2/63*(d*(-2*b^2*e^2-11*b*c*d*e+16*c^2*d^2)+e*(3*b^2*e^2-26*b*c*d*e+26*c^2 *d^2)*x)*(c*x^2+b*x)^(3/2)/d/e^3/(-b*e+c*d)/(e*x+d)^(7/2)-2/9*(c*x^2+b*x)^ (5/2)/e/(e*x+d)^(9/2)+4/63*(-b^4*e^4-7*b^3*c*d*e^3+135*b^2*c^2*d^2*e^2-256 *b*c^3*d^3*e+128*c^4*d^4)*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^( 1/2))*(-b)^(1/2)*c^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/d^2/e^6/(-b *e+c*d)^2/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-2/63*(-b*e+2*c*d)*(-b^2*e^2-12 8*b*c*d*e+128*c^2*d^2)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2 ))*(-b)^(1/2)*c^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/d/e^6/(-b*e+ c*d)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-2/63*(c*d^2*(-b^3*e^3+111*b^2*c*d*e^2 -240*b*c^2*d^2*e+128*c^3*d^3)+e*(-2*b^4*e^4-11*b^3*c*d*e^3+171*b^2*c^2*d^2 *e^2-320*b*c^3*d^3*e+160*c^4*d^4)*x)*(c*x^2+b*x)^(1/2)/d^2/e^5/(-b*e+c*d)^ 2/(e*x+d)^(3/2)
Result contains complex when optimal does not.
Time = 25.80 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.07 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {2 (x (b+c x))^{5/2} \left (b e x (b+c x) \left (7 d^4 (c d-b e)^4-19 d^3 (c d-b e)^2 \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) (d+e x)+d^2 (c d-b e)^2 \left (88 c^2 d^2-88 b c d e+15 b^2 e^2\right ) (d+e x)^2-d (c d-b e) \left (122 c^3 d^3-183 b c^2 d^2 e+63 b^2 c d e^2-b^3 e^3\right ) (d+e x)^3+\left (193 c^4 d^4-386 b c^3 d^3 e+207 b^2 c^2 d^2 e^2-14 b^3 c d e^3-2 b^4 e^4\right ) (d+e x)^4\right )-\sqrt {\frac {b}{c}} c (d+e x)^4 \left (-2 \sqrt {\frac {b}{c}} \left (-128 c^4 d^4+256 b c^3 d^3 e-135 b^2 c^2 d^2 e^2+7 b^3 c d e^3+b^4 e^4\right ) (b+c x) (d+e x)+2 i b e \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (128 c^4 d^4-272 b c^3 d^3 e+159 b^2 c^2 d^2 e^2-13 b^3 c d e^3-2 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{63 b d^2 e^6 (c d-b e)^2 x^3 (b+c x)^3 (d+e x)^{9/2}} \]
(-2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(7*d^4*(c*d - b*e)^4 - 19*d^3*(c* d - b*e)^2*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*(d + e*x) + d^2*(c*d - b*e)^2 *(88*c^2*d^2 - 88*b*c*d*e + 15*b^2*e^2)*(d + e*x)^2 - d*(c*d - b*e)*(122*c ^3*d^3 - 183*b*c^2*d^2*e + 63*b^2*c*d*e^2 - b^3*e^3)*(d + e*x)^3 + (193*c^ 4*d^4 - 386*b*c^3*d^3*e + 207*b^2*c^2*d^2*e^2 - 14*b^3*c*d*e^3 - 2*b^4*e^4 )*(d + e*x)^4) - Sqrt[b/c]*c*(d + e*x)^4*(-2*Sqrt[b/c]*(-128*c^4*d^4 + 256 *b*c^3*d^3*e - 135*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 + b^4*e^4)*(b + c*x)*(d + e*x) + (2*I)*b*e*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Elli pticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(128*c^4*d^4 - 27 2*b*c^3*d^3*e + 159*b^2*c^2*d^2*e^2 - 13*b^3*c*d*e^3 - 2*b^4*e^4)*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)])))/(63*b*d^2*e^6*(c*d - b*e)^2*x^3*(b + c*x)^3*(d + e*x)^(9 /2))
Time = 0.85 (sec) , antiderivative size = 600, 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, 1229, 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.
| \(\displaystyle \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {5 \int \frac {(b+2 c x) \left (c x^2+b x\right )^{3/2}}{(d+e x)^{9/2}}dx}{9 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {5 \left (-\frac {6 \int -\frac {\left (b \left (16 c^2 d^2-11 b c e d-2 b^2 e^2\right )+c \left (32 c^2 d^2-32 b c e d+b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{2 (d+e x)^{5/2}}dx}{35 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{35 d e^2 (d+e x)^{7/2} (c d-b e)}\right )}{9 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {3 \int \frac {\left (b \left (16 c^2 d^2-11 b c e d-2 b^2 e^2\right )+c \left (32 c^2 d^2-32 b c e d+b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{(d+e x)^{5/2}}dx}{35 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{35 d e^2 (d+e x)^{7/2} (c d-b e)}\right )}{9 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (-\frac {2 \int -\frac {c \left (b d \left (128 c^3 d^3-240 b c^2 e d^2+111 b^2 c e^2 d-b^3 e^3\right )+2 \left (128 c^4 d^4-256 b c^3 e d^3+135 b^2 c^2 e^2 d^2-7 b^3 c e^3 d-b^4 e^4\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (c d^2 \left (-b^3 e^3+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )+e x \left (-2 b^4 e^4-11 b^3 c d e^3+171 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+160 c^4 d^4\right )\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\right )}{35 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{35 d e^2 (d+e x)^{7/2} (c d-b e)}\right )}{9 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {c \int \frac {b d \left (128 c^3 d^3-240 b c^2 e d^2+111 b^2 c e^2 d-b^3 e^3\right )+2 \left (128 c^4 d^4-256 b c^3 e d^3+135 b^2 c^2 e^2 d^2-7 b^3 c e^3 d-b^4 e^4\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (c d^2 \left (-b^3 e^3+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )+e x \left (-2 b^4 e^4-11 b^3 c d e^3+171 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+160 c^4 d^4\right )\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\right )}{35 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{35 d e^2 (d+e x)^{7/2} (c d-b e)}\right )}{9 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {c \left (\frac {2 \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {d (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}\right )}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (c d^2 \left (-b^3 e^3+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )+e x \left (-2 b^4 e^4-11 b^3 c d e^3+171 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+160 c^4 d^4\right )\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\right )}{35 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{35 d e^2 (d+e x)^{7/2} (c d-b e)}\right )}{9 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {c \left (\frac {2 \sqrt {x} \sqrt {b+c x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (c d^2 \left (-b^3 e^3+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )+e x \left (-2 b^4 e^4-11 b^3 c d e^3+171 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+160 c^4 d^4\right )\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\right )}{35 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{35 d e^2 (d+e x)^{7/2} (c d-b e)}\right )}{9 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {c \left (\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (c d^2 \left (-b^3 e^3+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )+e x \left (-2 b^4 e^4-11 b^3 c d e^3+171 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+160 c^4 d^4\right )\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\right )}{35 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{35 d e^2 (d+e x)^{7/2} (c d-b e)}\right )}{9 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {c \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (c d^2 \left (-b^3 e^3+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )+e x \left (-2 b^4 e^4-11 b^3 c d e^3+171 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+160 c^4 d^4\right )\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\right )}{35 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{35 d e^2 (d+e x)^{7/2} (c d-b e)}\right )}{9 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {c \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (c d^2 \left (-b^3 e^3+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )+e x \left (-2 b^4 e^4-11 b^3 c d e^3+171 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+160 c^4 d^4\right )\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\right )}{35 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{35 d e^2 (d+e x)^{7/2} (c d-b e)}\right )}{9 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {c \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{3 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (c d^2 \left (-b^3 e^3+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )+e x \left (-2 b^4 e^4-11 b^3 c d e^3+171 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+160 c^4 d^4\right )\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}\right )}{35 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{35 d e^2 (d+e x)^{7/2} (c d-b e)}\right )}{9 e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
(-2*(b*x + c*x^2)^(5/2))/(9*e*(d + e*x)^(9/2)) + (5*((-2*(d*(16*c^2*d^2 - 11*b*c*d*e - 2*b^2*e^2) + e*(26*c^2*d^2 - 26*b*c*d*e + 3*b^2*e^2)*x)*(b*x + c*x^2)^(3/2))/(35*d*e^2*(c*d - b*e)*(d + e*x)^(7/2)) + (3*((-2*(c*d^2*(1 28*c^3*d^3 - 240*b*c^2*d^2*e + 111*b^2*c*d*e^2 - b^3*e^3) + e*(160*c^4*d^4 - 320*b*c^3*d^3*e + 171*b^2*c^2*d^2*e^2 - 11*b^3*c*d*e^3 - 2*b^4*e^4)*x)* Sqrt[b*x + c*x^2])/(3*d*e^2*(c*d - b*e)*(d + e*x)^(3/2)) + (c*((4*Sqrt[-b] *(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^ 4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*S qrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c *x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(2*c*d - b*e)*(128*c^2*d^2 - 128*b*c*d* e - 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*d*e^2*(c*d - b*e))))/(35*d*e^2*(c*d - b*e))))/(9*e)
3.5.5.3.1 Defintions of rubi rules used
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[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]
Time = 4.20 (sec) , antiderivative size = 995, normalized size of antiderivative = 1.75
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {x \left (e x +d \right ) \left (c x +b \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}}{9 e^{10} \left (x +\frac {d}{e}\right )^{5}}+\frac {38 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}}{63 e^{9} \left (x +\frac {d}{e}\right )^{4}}-\frac {2 \left (15 b^{2} e^{2}-88 b c d e +88 c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{63 e^{8} \left (x +\frac {d}{e}\right )^{3}}+\frac {2 \left (b^{3} e^{3}-63 b^{2} d \,e^{2} c +183 b \,c^{2} d^{2} e -122 c^{3} d^{3}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{63 d \left (b e -c d \right ) e^{7} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+b e x \right ) \left (2 b^{4} e^{4}+14 b^{3} c d \,e^{3}-207 b^{2} c^{2} d^{2} e^{2}+386 b \,c^{3} d^{3} e -193 c^{4} d^{4}\right )}{63 d^{2} \left (b e -c d \right )^{2} e^{6} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {c^{2} \left (3 b e -5 c d \right )}{e^{6}}+\frac {c \left (b^{3} e^{3}-63 b^{2} d \,e^{2} c +183 b \,c^{2} d^{2} e -122 c^{3} d^{3}\right )}{63 d \left (b e -c d \right ) e^{6}}+\frac {2 b^{4} e^{4}+14 b^{3} c d \,e^{3}-207 b^{2} c^{2} d^{2} e^{2}+386 b \,c^{3} d^{3} e -193 c^{4} d^{4}}{63 e^{6} \left (b e -c d \right ) d^{2}}-\frac {b \left (2 b^{4} e^{4}+14 b^{3} c d \,e^{3}-207 b^{2} c^{2} d^{2} e^{2}+386 b \,c^{3} d^{3} e -193 c^{4} d^{4}\right )}{63 e^{5} d^{2} \left (b e -c d \right )^{2}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {c^{3}}{e^{5}}-\frac {c \left (2 b^{4} e^{4}+14 b^{3} c d \,e^{3}-207 b^{2} c^{2} d^{2} e^{2}+386 b \,c^{3} d^{3} e -193 c^{4} d^{4}\right )}{63 e^{5} d^{2} \left (b e -c d \right )^{2}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \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 )}\) | \(995\) |
| default | \(\text {Expression too large to display}\) | \(5005\) |
1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(x*(e*x+d)*(c*x+b))^(1/2)/x/(c*x+b)*(-2/ 9*d^2*(b^2*e^2-2*b*c*d*e+c^2*d^2)/e^10*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/ 2)/(x+d/e)^5+38/63*d*(b^2*e^2-3*b*c*d*e+2*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)^4-2/63*(15*b^2*e^2-88*b*c*d*e+88*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+2/63*(b^3*e^3-63*b^2*c*d*e^ 2+183*b*c^2*d^2*e-122*c^3*d^3)/d/(b*e-c*d)/e^7*(c*e*x^3+b*e*x^2+c*d*x^2+b* d*x)^(1/2)/(x+d/e)^2+2/63*(c*e*x^2+b*e*x)/d^2/(b*e-c*d)^2/e^6*(2*b^4*e^4+1 4*b^3*c*d*e^3-207*b^2*c^2*d^2*e^2+386*b*c^3*d^3*e-193*c^4*d^4)/((x+d/e)*(c *e*x^2+b*e*x))^(1/2)+2*(c^2*(3*b*e-5*c*d)/e^6+1/63*c*(b^3*e^3-63*b^2*c*d*e ^2+183*b*c^2*d^2*e-122*c^3*d^3)/d/(b*e-c*d)/e^6+1/63/e^6/(b*e-c*d)*(2*b^4* e^4+14*b^3*c*d*e^3-207*b^2*c^2*d^2*e^2+386*b*c^3*d^3*e-193*c^4*d^4)/d^2-1/ 63*b/e^5/d^2/(b*e-c*d)^2*(2*b^4*e^4+14*b^3*c*d*e^3-207*b^2*c^2*d^2*e^2+386 *b*c^3*d^3*e-193*c^4*d^4))/c*b*((1/c*b+x)*c/b)^(1/2)*((x+d/e)/(-1/c*b+d/e) )^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*EllipticF(((1 /c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))+2*(c^3/e^5-1/63/e^5*c*(2*b ^4*e^4+14*b^3*c*d*e^3-207*b^2*c^2*d^2*e^2+386*b*c^3*d^3*e-193*c^4*d^4)/d^2 /(b*e-c*d)^2)/c*b*((1/c*b+x)*c/b)^(1/2)*((x+d/e)/(-1/c*b+d/e))^(1/2)*(-c*x /b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-1/c*b+d/e)*EllipticE((( 1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))-d/e*EllipticF(((1/c*b+x)* c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.26 (sec) , antiderivative size = 1675, normalized size of antiderivative = 2.94 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Too large to display} \]
-2/189*((256*c^5*d^10 - 640*b*c^4*d^9*e + 478*b^2*c^3*d^8*e^2 - 77*b^3*c^2 *d^7*e^3 - 13*b^4*c*d^6*e^4 - 2*b^5*d^5*e^5 + (256*c^5*d^5*e^5 - 640*b*c^4 *d^4*e^6 + 478*b^2*c^3*d^3*e^7 - 77*b^3*c^2*d^2*e^8 - 13*b^4*c*d*e^9 - 2*b ^5*e^10)*x^5 + 5*(256*c^5*d^6*e^4 - 640*b*c^4*d^5*e^5 + 478*b^2*c^3*d^4*e^ 6 - 77*b^3*c^2*d^3*e^7 - 13*b^4*c*d^2*e^8 - 2*b^5*d*e^9)*x^4 + 10*(256*c^5 *d^7*e^3 - 640*b*c^4*d^6*e^4 + 478*b^2*c^3*d^5*e^5 - 77*b^3*c^2*d^4*e^6 - 13*b^4*c*d^3*e^7 - 2*b^5*d^2*e^8)*x^3 + 10*(256*c^5*d^8*e^2 - 640*b*c^4*d^ 7*e^3 + 478*b^2*c^3*d^6*e^4 - 77*b^3*c^2*d^5*e^5 - 13*b^4*c*d^4*e^6 - 2*b^ 5*d^3*e^7)*x^2 + 5*(256*c^5*d^9*e - 640*b*c^4*d^8*e^2 + 478*b^2*c^3*d^7*e^ 3 - 77*b^3*c^2*d^6*e^4 - 13*b^4*c*d^5*e^5 - 2*b^5*d^4*e^6)*x)*sqrt(c*e)*we ierstrassPInverse(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)) + 6*(128*c^5*d^9*e - 256*b*c^4*d^8*e^2 + 135*b^2*c^3* d^7*e^3 - 7*b^3*c^2*d^6*e^4 - b^4*c*d^5*e^5 + (128*c^5*d^4*e^6 - 256*b*c^4 *d^3*e^7 + 135*b^2*c^3*d^2*e^8 - 7*b^3*c^2*d*e^9 - b^4*c*e^10)*x^5 + 5*(12 8*c^5*d^5*e^5 - 256*b*c^4*d^4*e^6 + 135*b^2*c^3*d^3*e^7 - 7*b^3*c^2*d^2*e^ 8 - b^4*c*d*e^9)*x^4 + 10*(128*c^5*d^6*e^4 - 256*b*c^4*d^5*e^5 + 135*b^2*c ^3*d^4*e^6 - 7*b^3*c^2*d^3*e^7 - b^4*c*d^2*e^8)*x^3 + 10*(128*c^5*d^7*e^3 - 256*b*c^4*d^6*e^4 + 135*b^2*c^3*d^5*e^5 - 7*b^3*c^2*d^4*e^6 - b^4*c*d^3* e^7)*x^2 + 5*(128*c^5*d^8*e^2 - 256*b*c^4*d^7*e^3 + 135*b^2*c^3*d^6*e^4...
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \]
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]