3.405 \(\int \frac{\left (b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=570 \[ -\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 d e^6 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{63 d e^3 (d+e x)^{7/2} (c d-b e)}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 d^2 e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \sqrt{b x+c x^2} \left (c d^2 \left (-b^3 e^3+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )+e x \left (-2 b^4 e^4-11 b^3 c d e^3+171 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+160 c^4 d^4\right )\right )}{63 d^2 e^5 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}} \]

[Out]

(-2*(c*d^2*(128*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])/(63*d^2*e^5*(c*d - b*e)^2*(d + e*x)^(3/2)) - (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))/(63*d*e^3*(c*d - b*e)*(d + e*x)^(7/2)) - (2*(b*x + c*x^2)^(5/2))/
(9*e*(d + e*x)^(9/2)) + (4*Sqrt[-b]*Sqrt[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)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e
*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*d^2*e^6*(c*d
- b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*Sqrt[c]*(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)])/(63*d*e^6*(c*d
- b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.85418, antiderivative size = 570, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ -\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 d e^6 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{2 \left (b x+c x^2\right )^{3/2} \left (e x \left (3 b^2 e^2-26 b c d e+26 c^2 d^2\right )+d \left (-2 b^2 e^2-11 b c d e+16 c^2 d^2\right )\right )}{63 d e^3 (d+e x)^{7/2} (c d-b e)}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 d^2 e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \sqrt{b x+c x^2} \left (c d^2 \left (-b^3 e^3+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )+e x \left (-2 b^4 e^4-11 b^3 c d e^3+171 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+160 c^4 d^4\right )\right )}{63 d^2 e^5 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(-2*(c*d^2*(128*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])/(63*d^2*e^5*(c*d - b*e)^2*(d + e*x)^(3/2)) - (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))/(63*d*e^3*(c*d - b*e)*(d + e*x)^(7/2)) - (2*(b*x + c*x^2)^(5/2))/
(9*e*(d + e*x)^(9/2)) + (4*Sqrt[-b]*Sqrt[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)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e
*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*d^2*e^6*(c*d
- b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*Sqrt[c]*(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)])/(63*d*e^6*(c*d
- b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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Mathematica [C]  time = 6.54994, size = 610, normalized size = 1.07 \[ -\frac{2 (x (b+c x))^{5/2} \left (b e x (b+c x) \left (d^2 (d+e x)^2 \left (15 b^2 e^2-88 b c d e+88 c^2 d^2\right ) (c d-b e)^2-19 d^3 (d+e x) \left (b^2 e^2-3 b c d e+2 c^2 d^2\right ) (c d-b e)^2-d (d+e x)^3 \left (-b^3 e^3+63 b^2 c d e^2-183 b c^2 d^2 e+122 c^3 d^3\right ) (c d-b e)+(d+e x)^4 \left (-2 b^4 e^4-14 b^3 c d e^3+207 b^2 c^2 d^2 e^2-386 b c^3 d^3 e+193 c^4 d^4\right )+7 d^4 (c d-b e)^4\right )-c \sqrt{\frac{b}{c}} (d+e x)^4 \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-2 b^4 e^4-13 b^3 c d e^3+159 b^2 c^2 d^2 e^2-272 b c^3 d^3 e+128 c^4 d^4\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^4 e^4+7 b^3 c d e^3-135 b^2 c^2 d^2 e^2+256 b c^3 d^3 e-128 c^4 d^4\right )\right )\right )}{63 b d^2 e^6 x^3 (b+c x)^3 (d+e x)^{9/2} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(-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)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b
*e*(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)*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))

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Maple [B]  time = 0.107, size = 5005, normalized size = 8.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(5/2)/(e*x+d)^(11/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(11/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x}}{{\left (e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}\right )} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(11/2),x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + b^2*x^2)*sqrt(c*x^2 + b*x)/((e^5*x^5 + 5*d*e^4*x
^4 + 10*d^2*e^3*x^3 + 10*d^3*e^2*x^2 + 5*d^4*e*x + d^5)*sqrt(e*x + d)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(11/2), x)