Integrand size = 23, antiderivative size = 476 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\frac {4 (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {b x+c x^2}}{35 d^2 e^3 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 \left (d \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+e \left (14 c^2 d^2-14 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{35 d e^3 (c d-b e) (d+e x)^{5/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac {4 \sqrt {-b} \sqrt {c} (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{35 d^2 e^4 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} \sqrt {c} \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{35 d e^4 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]
-2/7*(c*x^2+b*x)^(3/2)/e/(e*x+d)^(7/2)-4/35*(-b*e+2*c*d)*(-b^2*e^2-4*b*c*d *e+4*c^2*d^2)*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^4/(-b*e+c*d)^2/(1 +e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/35*(-b^2*e^2-16*b*c*d*e+16*c^2*d^2)*Elli pticF(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^4/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x^2+b *x)^(1/2)-2/35*(d*(-2*b^2*e^2-5*b*c*d*e+8*c^2*d^2)+e*(b^2*e^2-14*b*c*d*e+1 4*c^2*d^2)*x)*(c*x^2+b*x)^(1/2)/d/e^3/(-b*e+c*d)/(e*x+d)^(5/2)+4/35*(-b*e+ 2*c*d)*(-b^2*e^2-4*b*c*d*e+4*c^2*d^2)*(c*x^2+b*x)^(1/2)/d^2/e^3/(-b*e+c*d) ^2/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 13.62 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.01 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=-\frac {2 (x (b+c x))^{3/2} \left (b e x (b+c x) \left (5 d^3 (c d-b e)^3-8 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)+d (c d-b e) \left (19 c^2 d^2-19 b c d e+b^2 e^2\right ) (d+e x)^2-2 \left (8 c^3 d^3-12 b c^2 d^2 e+2 b^2 c d e^2+b^3 e^3\right ) (d+e x)^3\right )+\sqrt {\frac {b}{c}} c (d+e x)^3 \left (2 \sqrt {\frac {b}{c}} \left (8 c^3 d^3-12 b c^2 d^2 e+2 b^2 c d e^2+b^3 e^3\right ) (b+c x) (d+e x)+2 i b e \left (8 c^3 d^3-12 b c^2 d^2 e+2 b^2 c d e^2+b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (8 c^3 d^3-13 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{35 b d^2 e^4 (c d-b e)^2 x^2 (b+c x)^2 (d+e x)^{7/2}} \]
(-2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(5*d^3*(c*d - b*e)^3 - 8*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + d*(c*d - b*e)*(19*c^2*d^2 - 19*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 2*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*(d + e*x)^3) + Sqrt[b/c]*c*(d + e*x)^3*(2*Sqrt[b/c]*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*(b + c*x)*(d + e*x) + (2*I)*b*e* (8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*Sqrt[1 + b/(c*x)]*S qrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b* e)] - I*b*e*(8*c^3*d^3 - 13*b*c^2*d^2*e + 3*b^2*c*d*e^2 + 2*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)])))/(35*b*d^2*e^4*(c*d - b*e)^2*x^2*(b + c*x)^2*(d + e*x) ^(7/2))
Time = 0.74 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.04, 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, 1237, 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.
\(\displaystyle \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx\) |
\(\Big \downarrow \) 1161 |
\(\displaystyle \frac {3 \int \frac {(b+2 c x) \sqrt {c x^2+b x}}{(d+e x)^{7/2}}dx}{7 e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}\) |
\(\Big \downarrow \) 1229 |
\(\displaystyle \frac {3 \left (-\frac {2 \int -\frac {b \left (8 c^2 d^2-5 b c e d-2 b^2 e^2\right )+c \left (16 c^2 d^2-16 b c e d-b^2 e^2\right ) x}{2 (d+e x)^{3/2} \sqrt {c x^2+b x}}dx}{15 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (b^2 e^2-14 b c d e+14 c^2 d^2\right )+d \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 \left (\frac {\int \frac {b \left (8 c^2 d^2-5 b c e d-2 b^2 e^2\right )+c \left (16 c^2 d^2-16 b c e d-b^2 e^2\right ) x}{(d+e x)^{3/2} \sqrt {c x^2+b x}}dx}{15 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (b^2 e^2-14 b c d e+14 c^2 d^2\right )+d \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}\) |
\(\Big \downarrow \) 1237 |
\(\displaystyle \frac {3 \left (\frac {\frac {4 \sqrt {b x+c x^2} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {2 \int \frac {c \left (b d \left (8 c^2 d^2-11 b c e d+b^2 e^2\right )+2 (2 c d-b e) \left (4 c^2 d^2-4 b c e d-b^2 e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{d (c d-b e)}}{15 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (b^2 e^2-14 b c d e+14 c^2 d^2\right )+d \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 \left (\frac {\frac {4 \sqrt {b x+c x^2} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \int \frac {b d \left (8 c^2 d^2-11 b c e d+b^2 e^2\right )+2 (2 c d-b e) \left (4 c^2 d^2-4 b c e d-b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{d (c d-b e)}}{15 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (b^2 e^2-14 b c d e+14 c^2 d^2\right )+d \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {3 \left (\frac {\frac {4 \sqrt {b x+c x^2} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \left (\frac {2 (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {d (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}\right )}{d (c d-b e)}}{15 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (b^2 e^2-14 b c d e+14 c^2 d^2\right )+d \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}\) |
\(\Big \downarrow \) 1169 |
\(\displaystyle \frac {3 \left (\frac {\frac {4 \sqrt {b x+c x^2} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \left (\frac {2 \sqrt {x} \sqrt {b+c x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{d (c d-b e)}}{15 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (b^2 e^2-14 b c d e+14 c^2 d^2\right )+d \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}\) |
\(\Big \downarrow \) 122 |
\(\displaystyle \frac {3 \left (\frac {\frac {4 \sqrt {b x+c x^2} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \left (\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{d (c d-b e)}}{15 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (b^2 e^2-14 b c d e+14 c^2 d^2\right )+d \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}\) |
\(\Big \downarrow \) 120 |
\(\displaystyle \frac {3 \left (\frac {\frac {4 \sqrt {b x+c x^2} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{d (c d-b e)}}{15 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (b^2 e^2-14 b c d e+14 c^2 d^2\right )+d \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}\) |
\(\Big \downarrow \) 127 |
\(\displaystyle \frac {3 \left (\frac {\frac {4 \sqrt {b x+c x^2} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{d (c d-b e)}}{15 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (b^2 e^2-14 b c d e+14 c^2 d^2\right )+d \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}\) |
\(\Big \downarrow \) 126 |
\(\displaystyle \frac {3 \left (\frac {\frac {4 \sqrt {b x+c x^2} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {c \left (\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{d (c d-b e)}}{15 d e^2 (c d-b e)}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (b^2 e^2-14 b c d e+14 c^2 d^2\right )+d \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\right )}{7 e}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}\) |
(-2*(b*x + c*x^2)^(3/2))/(7*e*(d + e*x)^(7/2)) + (3*((-2*(d*(8*c^2*d^2 - 5 *b*c*d*e - 2*b^2*e^2) + e*(14*c^2*d^2 - 14*b*c*d*e + b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(15*d*e^2*(c*d - b*e)*(d + e*x)^(5/2)) + ((4*(2*c*d - b*e)*(4*c^ 2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)*Sqrt[d + e* x]) - (c*((4*Sqrt[-b]*(2*c*d - b*e)*(4*c^2*d^2 - 4*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])/Sqr t[-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^2*d^2 - 16*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])))/(d*(c*d - b* e)))/(15*d*e^2*(c*d - b*e))))/(7*e)
3.4.98.3.1 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[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (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]
Time = 2.66 (sec) , antiderivative size = 792, normalized size of antiderivative = 1.66
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 \left (b e -c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{7 e^{7} \left (x +\frac {d}{e}\right )^{4}}-\frac {16 \left (b e -2 c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{35 e^{6} \left (x +\frac {d}{e}\right )^{3}}+\frac {2 \left (b^{2} e^{2}-19 b c d e +19 c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{35 d \left (b e -c d \right ) e^{5} \left (x +\frac {d}{e}\right )^{2}}+\frac {4 \left (c e \,x^{2}+b e x \right ) \left (b^{3} e^{3}+2 b^{2} d \,e^{2} c -12 b \,c^{2} d^{2} e +8 c^{3} d^{3}\right )}{35 d^{2} \left (b e -c d \right )^{2} e^{4} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {c^{2}}{e^{4}}+\frac {c \left (b^{2} e^{2}-19 b c d e +19 c^{2} d^{2}\right )}{35 e^{4} d \left (b e -c d \right )}+\frac {\frac {4}{35} b^{2} d \,e^{2} c -\frac {24}{35} b \,c^{2} d^{2} e +\frac {16}{35} c^{3} d^{3}+\frac {2}{35} b^{3} e^{3}}{e^{4} \left (b e -c d \right ) d^{2}}-\frac {2 b \left (b^{3} e^{3}+2 b^{2} d \,e^{2} c -12 b \,c^{2} d^{2} e +8 c^{3} d^{3}\right )}{35 e^{3} 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 {4 \left (b^{3} e^{3}+2 b^{2} d \,e^{2} c -12 b \,c^{2} d^{2} e +8 c^{3} d^{3}\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 )}{35 e^{3} d^{2} \left (b e -c d \right )^{2} \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 )}\) | \(792\) |
default | \(\text {Expression too large to display}\) | \(3267\) |
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)*(2/7 *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)^4-16/35*(b* e-2*c*d)/e^6*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(x+d/e)^3+2/35*(b^2*e^2 -19*b*c*d*e+19*c^2*d^2)/d/(b*e-c*d)/e^5*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1 /2)/(x+d/e)^2+4/35*(c*e*x^2+b*e*x)/d^2/(b*e-c*d)^2/e^4*(b^3*e^3+2*b^2*c*d* e^2-12*b*c^2*d^2*e+8*c^3*d^3)/((x+d/e)*(c*e*x^2+b*e*x))^(1/2)+2*(c^2/e^4+1 /35*c*(b^2*e^2-19*b*c*d*e+19*c^2*d^2)/e^4/d/(b*e-c*d)+2/35/e^4/(b*e-c*d)*( b^3*e^3+2*b^2*c*d*e^2-12*b*c^2*d^2*e+8*c^3*d^3)/d^2-2/35*b/e^3/d^2/(b*e-c* d)^2*(b^3*e^3+2*b^2*c*d*e^2-12*b*c^2*d^2*e+8*c^3*d^3))/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) )-4/35/e^3*(b^3*e^3+2*b^2*c*d*e^2-12*b*c^2*d^2*e+8*c^3*d^3)/d^2/(b*e-c*d)^ 2*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 = 0.13 (sec) , antiderivative size = 1230, normalized size of antiderivative = 2.58 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\text {Too large to display} \]
2/105*((16*c^4*d^8 - 32*b*c^3*d^7*e + 13*b^2*c^2*d^6*e^2 + 3*b^3*c*d^5*e^3 + 2*b^4*d^4*e^4 + (16*c^4*d^4*e^4 - 32*b*c^3*d^3*e^5 + 13*b^2*c^2*d^2*e^6 + 3*b^3*c*d*e^7 + 2*b^4*e^8)*x^4 + 4*(16*c^4*d^5*e^3 - 32*b*c^3*d^4*e^4 + 13*b^2*c^2*d^3*e^5 + 3*b^3*c*d^2*e^6 + 2*b^4*d*e^7)*x^3 + 6*(16*c^4*d^6*e ^2 - 32*b*c^3*d^5*e^3 + 13*b^2*c^2*d^4*e^4 + 3*b^3*c*d^3*e^5 + 2*b^4*d^2*e ^6)*x^2 + 4*(16*c^4*d^7*e - 32*b*c^3*d^6*e^2 + 13*b^2*c^2*d^5*e^3 + 3*b^3* c*d^4*e^4 + 2*b^4*d^3*e^5)*x)*sqrt(c*e)*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)) + 6*(8*c^4* d^7*e - 12*b*c^3*d^6*e^2 + 2*b^2*c^2*d^5*e^3 + b^3*c*d^4*e^4 + (8*c^4*d^3* e^5 - 12*b*c^3*d^2*e^6 + 2*b^2*c^2*d*e^7 + b^3*c*e^8)*x^4 + 4*(8*c^4*d^4*e ^4 - 12*b*c^3*d^3*e^5 + 2*b^2*c^2*d^2*e^6 + b^3*c*d*e^7)*x^3 + 6*(8*c^4*d^ 5*e^3 - 12*b*c^3*d^4*e^4 + 2*b^2*c^2*d^3*e^5 + b^3*c*d^2*e^6)*x^2 + 4*(8*c ^4*d^6*e^2 - 12*b*c^3*d^5*e^3 + 2*b^2*c^2*d^4*e^4 + b^3*c*d^3*e^5)*x)*sqrt (c*e)*weierstrassZeta(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), weierstr assPInverse(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*(8*c^4*d^6*e^2 - 11*b*c^3*d^5*e^3 + b^2*c^2*d^4*e^4 + 2*(8*c^4*d^3*e^5 - 12*b*c^3*d^2*e^6 + 2*b^2*c^2*d*e^7 + b^3*c*e^8)*x^3 ...
\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]
\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]