Integrand size = 23, antiderivative size = 521 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-3 c e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac {16 \sqrt {-b} (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} d (c d-b e) \left (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+13 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
2/231*(c^2*d^2+13*b*c*d*e-6*b^2*e^2+14*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x)^(3/ 2)*(e*x+d)^(1/2)/c^2/e+2/11*e*(c*x^2+b*x)^(5/2)*(e*x+d)^(1/2)/c-16/1155*(- 2*b*e+c*d)*(-b*e+2*c*d)*(b*e+c*d)*(b^2*e^2-b*c*d*e+c^2*d^2)*EllipticE(c^(1 /2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(1+c*x/b)^(1/2) *(e*x+d)^(1/2)/c^(7/2)/e^4/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/1155*d*(-b* e+c*d)*(-8*b^4*e^4+13*b^3*c*d*e^3+3*b^2*c^2*d^2*e^2-32*b*c^3*d^3*e+16*c^4* d^4)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1 /2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/c^(7/2)/e^4/(e*x+d)^(1/2)/(c*x^2+b*x)^ (1/2)+2/1155*(8*c^4*d^4-19*b*c^3*d^3*e+6*b^2*c^2*d^2*e^2-19*b^3*c*d*e^3+8* b^4*e^4-3*c*e*(-b*e+2*c*d)*(8*b^2*e^2-b*c*d*e+c^2*d^2)*x)*(e*x+d)^(1/2)*(c *x^2+b*x)^(1/2)/c^3/e^3
Result contains complex when optimal does not.
Time = 24.11 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.07 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (8 b^4 e^4-b^3 c e^3 (19 d+6 e x)+b^2 c^2 e^2 \left (6 d^2+14 d e x+5 e^2 x^2\right )+b c^3 e \left (-19 d^3+14 d^2 e x+205 d e^2 x^2+140 e^3 x^3\right )+c^4 \left (8 d^4-6 d^3 e x+5 d^2 e^2 x^2+140 d e^3 x^3+105 e^4 x^4\right )\right )+\sqrt {\frac {b}{c}} \left (-8 \sqrt {\frac {b}{c}} \left (2 c^5 d^5-5 b c^4 d^4 e+2 b^2 c^3 d^3 e^2+2 b^3 c^2 d^2 e^3-5 b^4 c d e^4+2 b^5 e^5\right ) (b+c x) (d+e x)-8 i b e \left (2 c^5 d^5-5 b c^4 d^4 e+2 b^2 c^3 d^3 e^2+2 b^3 c^2 d^2 e^3-5 b^4 c d e^4+2 b^5 e^5\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e \left (8 c^5 d^5-21 b c^4 d^4 e+10 b^2 c^3 d^3 e^2+35 b^3 c^2 d^2 e^3-48 b^4 c d e^4+16 b^5 e^5\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{1155 b c^3 e^4 x^2 (b+c x)^2 \sqrt {d+e x}} \]
(2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(d + e*x)*(8*b^4*e^4 - b^3*c*e^3*( 19*d + 6*e*x) + b^2*c^2*e^2*(6*d^2 + 14*d*e*x + 5*e^2*x^2) + b*c^3*e*(-19* d^3 + 14*d^2*e*x + 205*d*e^2*x^2 + 140*e^3*x^3) + c^4*(8*d^4 - 6*d^3*e*x + 5*d^2*e^2*x^2 + 140*d*e^3*x^3 + 105*e^4*x^4)) + Sqrt[b/c]*(-8*Sqrt[b/c]*( 2*c^5*d^5 - 5*b*c^4*d^4*e + 2*b^2*c^3*d^3*e^2 + 2*b^3*c^2*d^2*e^3 - 5*b^4* c*d*e^4 + 2*b^5*e^5)*(b + c*x)*(d + e*x) - (8*I)*b*e*(2*c^5*d^5 - 5*b*c^4* d^4*e + 2*b^2*c^3*d^3*e^2 + 2*b^3*c^2*d^2*e^3 - 5*b^4*c*d*e^4 + 2*b^5*e^5) *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*(8*c^5*d^5 - 21*b*c^4*d^4*e + 10*b^2*c^3*d ^3*e^2 + 35*b^3*c^2*d^2*e^3 - 48*b^4*c*d*e^4 + 16*b^5*e^5)*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)])))/(1155*b*c^3*e^4*x^2*(b + c*x)^2*Sqrt[d + e*x])
Time = 0.85 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {1166, 27, 1231, 27, 1231, 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.
\(\displaystyle \int \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2} \, dx\) |
\(\Big \downarrow \) 1166 |
\(\displaystyle \frac {2 \int \frac {(d (11 c d-5 b e)+6 e (2 c d-b e) x) \left (c x^2+b x\right )^{3/2}}{2 \sqrt {d+e x}}dx}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {(d (11 c d-5 b e)+6 e (2 c d-b e) x) \left (c x^2+b x\right )^{3/2}}{\sqrt {d+e x}}dx}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
\(\Big \downarrow \) 1231 |
\(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {2 \int \frac {3 e \left (b d \left (c^2 d^2+13 b c e d-6 b^2 e^2\right )+(2 c d-b e) \left (c^2 d^2-b c e d+8 b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{2 \sqrt {d+e x}}dx}{21 c e^2}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\int \frac {\left (b d \left (c^2 d^2+13 b c e d-6 b^2 e^2\right )+(2 c d-b e) \left (c^2 d^2-b c e d+8 b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{\sqrt {d+e x}}dx}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
\(\Big \downarrow \) 1231 |
\(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {-\frac {2 \int -\frac {b d \left (8 c^4 d^4-19 b c^3 e d^3+6 b^2 c^2 e^2 d^2-19 b^3 c e^3 d+8 b^4 e^4\right )+8 (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c e d+b^2 e^2\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\int \frac {b d \left (8 c^4 d^4-19 b c^3 e d^3+6 b^2 c^2 e^2 d^2-19 b^3 c e^3 d+8 b^4 e^4\right )+8 (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c e d+b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\frac {8 (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {d (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
\(\Big \downarrow \) 1169 |
\(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\frac {8 \sqrt {x} \sqrt {b+c x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
\(\Big \downarrow \) 122 |
\(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\frac {8 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
\(\Big \downarrow \) 120 |
\(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
\(\Big \downarrow \) 127 |
\(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
\(\Big \downarrow \) 126 |
\(\displaystyle \frac {\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{21 c e}-\frac {\frac {\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{15 c e^2}}{7 c e}}{11 c}+\frac {2 e \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
(2*e*Sqrt[d + e*x]*(b*x + c*x^2)^(5/2))/(11*c) + ((2*Sqrt[d + e*x]*(c^2*d^ 2 + 13*b*c*d*e - 6*b^2*e^2 + 14*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/ (21*c*e) - ((-2*Sqrt[d + e*x]*(8*c^4*d^4 - 19*b*c^3*d^3*e + 6*b^2*c^2*d^2* e^2 - 19*b^3*c*d*e^3 + 8*b^4*e^4 - 3*c*e*(2*c*d - b*e)*(c^2*d^2 - b*c*d*e + 8*b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(15*c*e^2) + ((16*Sqrt[-b]*(c*d - 2*b*e )*(2*c*d - b*e)*(c*d + b*e)*(c^2*d^2 - b*c*d*e + 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]) - (2*Sqrt[-b]*d* (c*d - b*e)*(16*c^4*d^4 - 32*b*c^3*d^3*e + 3*b^2*c^2*d^2*e^2 + 13*b^3*c*d* e^3 - 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[Arc Sin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sq rt[b*x + c*x^2]))/(15*c*e^2))/(7*c*e))/(11*c)
3.4.92.3.1 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[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[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)*(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[((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. \(1358\) vs. \(2(461)=922\).
Time = 2.10 (sec) , antiderivative size = 1359, normalized size of antiderivative = 2.61
method | result | size |
default | \(\text {Expression too large to display}\) | \(1359\) |
elliptic | \(\text {Expression too large to display}\) | \(1623\) |
2/1155*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(590*b*c^6*d*e^5*x^5+364*b^2*c^5*d* e^5*x^4+364*b*c^6*d^2*e^4*x^4-6*b^3*c^4*d*e^5*x^3+239*b^2*c^5*d^2*e^4*x^3- 6*b*c^6*d^3*e^3*x^3-17*b^4*c^3*d*e^5*x^2+b^3*c^4*d^2*e^4*x^2+b^2*c^5*d^3*e ^3*x^2-17*b*c^6*d^4*e^2*x^2+8*b^5*c^2*d*e^5*x-19*b^4*c^3*d^2*e^4*x+6*b^3*c ^4*d^3*e^3*x-19*b^2*c^5*d^4*e^2*x+8*b*c^6*d^5*e*x+245*b*c^6*e^6*x^6+245*c^ 7*d*e^5*x^6+145*b^2*c^5*e^6*x^5+145*c^7*d^2*e^4*x^5-b^3*c^4*e^6*x^4-c^7*d^ 3*e^3*x^4+2*b^4*c^3*e^6*x^3+2*c^7*d^4*e^2*x^3+8*b^5*c^2*e^6*x^2+8*c^7*d^5* e*x^2+16*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ell ipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^6*d^6-16*((c*x+b)/b)^( 1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/ 2),(b*e/(b*e-c*d))^(1/2))*b*c^6*d^6+105*c^7*e^6*x^7+8*((c*x+b)/b)^(1/2)*(- c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e /(b*e-c*d))^(1/2))*b^6*c*d*e^5-21*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d)) ^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b ^5*c^2*d^2*e^4+10*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^ (1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^3*d^3*e^3+3 5*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF( ((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^4*d^4*e^2-48*((c*x+b)/b)^(1 /2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2 ),(b*e/(b*e-c*d))^(1/2))*b^2*c^5*d^5*e-56*((c*x+b)/b)^(1/2)*(-c*(e*x+d)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.22 \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left ({\left (16 \, c^{6} d^{6} - 48 \, b c^{5} d^{5} e + 33 \, b^{2} c^{4} d^{4} e^{2} + 14 \, b^{3} c^{3} d^{3} e^{3} + 33 \, b^{4} c^{2} d^{2} e^{4} - 48 \, b^{5} c d e^{5} + 16 \, b^{6} e^{6}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 24 \, {\left (2 \, c^{6} d^{5} e - 5 \, b c^{5} d^{4} e^{2} + 2 \, b^{2} c^{4} d^{3} e^{3} + 2 \, b^{3} c^{3} d^{2} e^{4} - 5 \, b^{4} c^{2} d e^{5} + 2 \, b^{5} c e^{6}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (105 \, c^{6} e^{6} x^{4} + 8 \, c^{6} d^{4} e^{2} - 19 \, b c^{5} d^{3} e^{3} + 6 \, b^{2} c^{4} d^{2} e^{4} - 19 \, b^{3} c^{3} d e^{5} + 8 \, b^{4} c^{2} e^{6} + 140 \, {\left (c^{6} d e^{5} + b c^{5} e^{6}\right )} x^{3} + 5 \, {\left (c^{6} d^{2} e^{4} + 41 \, b c^{5} d e^{5} + b^{2} c^{4} e^{6}\right )} x^{2} - 2 \, {\left (3 \, c^{6} d^{3} e^{3} - 7 \, b c^{5} d^{2} e^{4} - 7 \, b^{2} c^{4} d e^{5} + 3 \, b^{3} c^{3} e^{6}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{3465 \, c^{5} e^{5}} \]
2/3465*((16*c^6*d^6 - 48*b*c^5*d^5*e + 33*b^2*c^4*d^4*e^2 + 14*b^3*c^3*d^3 *e^3 + 33*b^4*c^2*d^2*e^4 - 48*b^5*c*d*e^5 + 16*b^6*e^6)*sqrt(c*e)*weierst rassPInverse(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)) + 24*(2*c^6*d^5*e - 5*b*c^5*d^4*e^2 + 2*b^2*c^4*d^3*e^3 + 2*b^3*c^3*d^2*e^4 - 5*b^4*c^2*d*e^5 + 2*b^5*c*e^6)*sqrt(c*e)*weierstrassZe ta(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 *(105*c^6*e^6*x^4 + 8*c^6*d^4*e^2 - 19*b*c^5*d^3*e^3 + 6*b^2*c^4*d^2*e^4 - 19*b^3*c^3*d*e^5 + 8*b^4*c^2*e^6 + 140*(c^6*d*e^5 + b*c^5*e^6)*x^3 + 5*(c ^6*d^2*e^4 + 41*b*c^5*d*e^5 + b^2*c^4*e^6)*x^2 - 2*(3*c^6*d^3*e^3 - 7*b*c^ 5*d^2*e^4 - 7*b^2*c^4*d*e^5 + 3*b^3*c^3*e^6)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^5*e^5)
\[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\int \left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}\, dx \]
\[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \]
\[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{3/2} \,d x \]