3.427 \(\int \frac{1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=567 \[ \frac{8 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (c d-b e)}+\frac{2 \left (4 c x \left (b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )+b (c d-b e) \left (-4 b^2 e^2-3 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}+\frac{2 e \sqrt{b x+c x^2} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right )}{3 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3} \]

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*Sqrt[d + e*x]*(b*x
+ c*x^2)^(3/2)) + (2*(b*(c*d - b*e)*(8*c^2*d^2 - 3*b*c*d*e - 4*b^2*e^2) + 4*c*(
4*c^3*d^3 - 6*b*c^2*d^2*e + b^3*e^3)*x))/(3*b^4*d^2*(c*d - b*e)^2*Sqrt[d + e*x]*
Sqrt[b*x + c*x^2]) + (2*e*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b
^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*Sqrt[d + e*x
]) - (2*Sqrt[c]*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3
- 8*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)])/(3*(-b)^(7/2)*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)
/d]*Sqrt[b*x + c*x^2]) + (8*Sqrt[c]*(2*c*d - b*e)*(2*c^2*d^2 - 2*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)])/(3*(-b)^(7/2)*d^2*(c*d - b*e)^2*Sqrt[d + e*x]*Sqrt[b
*x + c*x^2])

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Rubi [A]  time = 1.87998, antiderivative size = 567, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ \frac{8 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (c d-b e)}+\frac{2 \left (4 c x \left (b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )+b (c d-b e) \left (-4 b^2 e^2-3 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}+\frac{2 e \sqrt{b x+c x^2} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right )}{3 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*Sqrt[d + e*x]*(b*x
+ c*x^2)^(3/2)) + (2*(b*(c*d - b*e)*(8*c^2*d^2 - 3*b*c*d*e - 4*b^2*e^2) + 4*c*(
4*c^3*d^3 - 6*b*c^2*d^2*e + b^3*e^3)*x))/(3*b^4*d^2*(c*d - b*e)^2*Sqrt[d + e*x]*
Sqrt[b*x + c*x^2]) + (2*e*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b
^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*Sqrt[d + e*x
]) - (2*Sqrt[c]*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3
- 8*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)])/(3*(-b)^(7/2)*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)
/d]*Sqrt[b*x + c*x^2]) + (8*Sqrt[c]*(2*c*d - b*e)*(2*c^2*d^2 - 2*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)])/(3*(-b)^(7/2)*d^2*(c*d - b*e)^2*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(1/(e*x+d)**(3/2)/(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 5.24352, size = 504, normalized size = 0.89 \[ -\frac{2 \left (b \left (3 b^4 e^5 x^2 (b+c x)^2+b c^4 d^3 x^2 (d+e x) (b e-c d)-c^4 d^3 x^2 (b+c x) (d+e x) (8 c d-13 b e)+b d (b+c x)^2 (d+e x) (c d-b e)^3-x (b+c x)^2 (d+e x) (c d-b e)^3 (5 b e+8 c d)\right )+c x \sqrt{\frac{b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^4 e^4+11 b^3 c d e^3+6 b^2 c^2 d^2 e^2-17 b c^3 d^3 e+8 c^4 d^4\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right )\right )\right )}{3 b^5 d^3 (x (b+c x))^{3/2} \sqrt{d+e x} (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(b*(3*b^4*e^5*x^2*(b + c*x)^2 + b*c^4*d^3*(-(c*d) + b*e)*x^2*(d + e*x) - c^4
*d^3*(8*c*d - 13*b*e)*x^2*(b + c*x)*(d + e*x) + b*d*(c*d - b*e)^3*(b + c*x)^2*(d
+ e*x) - (c*d - b*e)^3*(8*c*d + 5*b*e)*x*(b + c*x)^2*(d + e*x)) + Sqrt[b/c]*c*x
*(b + c*x)*(Sqrt[b/c]*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c
*d*e^3 - 8*b^4*e^4)*(b + c*x)*(d + e*x) + I*b*e*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9
*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*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*(8*c^4*d^
4 - 17*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3 - 8*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)])))/(3*b^5*d^3*(c*d - b*e)^3*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])

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Maple [B]  time = 0.06, size = 2189, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x)

[Out]

2/3*(8*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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^7*e^5-b^6*c*d^2*e^3+3*b^5*c^2*d^3*e^
2-3*b^4*c^3*d^4*e+b^3*c^4*d^5-16*x^3*c^7*d^5+8*x^3*b*c^6*d^4*e+4*x*b^6*c*d*e^4-6
*x*b^5*c^2*d^2*e^3-6*x*b^4*c^3*d^3*e^2+8*x^2*b^6*c*e^5+8*x^2*((c*x+b)/b)^(1/2)*(
-(e*x+d)*c/(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^6*c*e^5+16*x^3*b^5*c^2*e^5-6*x*b^2*c^5*d^5-16*x^4*c^7*d^4*e-24*x
^2*b*c^6*d^5+8*x^4*b^4*c^3*e^5+16*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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^5
-16*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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*c^6*d^5+16*x*((c*x+b)/b)^(1/2)*(-(e*x
+d)*c/(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^2*c^5*d^5-16*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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^5-10*x^3*b^
4*c^3*d*e^4-22*x^3*b^3*c^4*d^2*e^3+40*x^3*b^2*c^5*d^3*e^2+x^2*b^5*c^2*d*e^4-18*x
^2*b^4*c^3*d^2*e^3-2*x^2*b^3*c^4*d^3*e^2+43*x^2*b^2*c^5*d^4*e+14*x*b^3*c^4*d^4*e
+32*x^4*b*c^6*d^3*e^2-7*x^4*b^3*c^4*d*e^4-9*x^4*b^2*c^5*d^2*e^3-15*x^2*((c*x+b)/
b)^(1/2)*(-(e*x+d)*c/(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^5*c^2*d*e^4-2*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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^4
*c^3*d^2*e^3+41*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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^3*c^4*d^3*e^2-48*x^2*((c*
x+b)/b)^(1/2)*(-(e*x+d)*c/(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^2*c^5*d^4*e+4*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(
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*e^4-4*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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^2*e^3-48*x*((c*
x+b)/b)^(1/2)*(-(e*x+d)*c/(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^3*c^4*d^4*e+4*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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^4-4*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*El
lipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c^2*d^2*e^3-24*x*((c*x+b)/b
)^(1/2)*(-(e*x+d)*c/(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^2+40*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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-24*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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^3*e^2+40*x^2*((c*x+
b)/b)^(1/2)*(-(e*x+d)*c/(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^4*e-15*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(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
^6*c*d*e^4-2*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ell
ipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c^2*d^2*e^3+41*x*((c*x+b)/b)
^(1/2)*(-(e*x+d)*c/(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^4*c^3*d^3*e^2)/x^2*(x*(c*x+b))^(1/2)/b^4/d^3/c/(b*e-c*d)
^3/(c*x+b)^2/(e*x+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^(5/2)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x)^(5/2)*(e*x + d)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c^{2} e x^{5} + b^{2} d x^{2} +{\left (c^{2} d + 2 \, b c e\right )} x^{4} +{\left (2 \, b c d + b^{2} e\right )} x^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^(5/2)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

integral(1/((c^2*e*x^5 + b^2*d*x^2 + (c^2*d + 2*b*c*e)*x^4 + (2*b*c*d + b^2*e)*x
^3)*sqrt(c*x^2 + b*x)*sqrt(e*x + d)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x)**(5/2),x)

[Out]

Integral(1/((x*(b + c*x))**(5/2)*(d + e*x)**(3/2)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^(5/2)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError