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

Optimal. Leaf size=478 \[ -\frac{4 e \sqrt{b x+c x^2} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^{3/2} (c d-b e)} \]

[Out]

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

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Rubi [A]  time = 1.66556, antiderivative size = 478, 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 \[ -\frac{4 e \sqrt{b x+c x^2} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^{3/2} (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [C]  time = 4.02151, size = 420, normalized size = 0.88 \[ -\frac{2 \left (b \left (b^2 d e^3 x (b+c x) (c d-b e)-5 b^2 e^3 x (b+c x) (d+e x) (b e-2 c d)+3 (b+c x) (d+e x)^2 (c d-b e)^3+3 c^4 d^3 x (d+e x)^2\right )-c \sqrt{\frac{b}{c}} (d+e x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^3 e^3+23 b^2 c d e^2-18 b c^2 d^2 e+3 c^3 d^3\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right )\right )\right )}{3 b^3 d^3 \sqrt{x (b+c x)} (d+e x)^{3/2} (c d-b e)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*(b^2*d*e^3*(c*d - b*e)*x*(b + c*x) - 5*b^2*e^3*(-2*c*d + b*e)*x*(b + c*x)
*(d + e*x) + 3*c^4*d^3*x*(d + e*x)^2 + 3*(c*d - b*e)^3*(b + c*x)*(d + e*x)^2) -
Sqrt[b/c]*c*(d + e*x)*(Sqrt[b/c]*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8
*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^
2 - 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[S
qrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(3*c^3*d^3 - 18*b*c^2*d^2*e + 23*b^2*c*d
*e^2 - 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSin
h[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^3*d^3*(c*d - b*e)^3*Sqrt[x*(b + c*x)]
*(d + e*x)^(3/2))

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Maple [B]  time = 0.061, size = 1708, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(x*(c*x+b))^(1/2)/x*(-15*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))
*x*b^2*c^3*d^3*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)
+6*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^4*d^4*e*((c*x+b)/b)^
(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*EllipticF(((c*x+b)/b)^(1/2),
(b*e/(b*e-c*d))^(1/2))*x*b^4*c*d*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1
/2)*(-c*x/b)^(1/2)-10*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c
^2*d^2*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+12*Elli
pticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^3*d^3*e^2*((c*x+b)/b)^(1/
2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-6*EllipticF(((c*x+b)/b)^(1/2),(b*
e/(b*e-c*d))^(1/2))*x*b*c^4*d^4*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)
*(-c*x/b)^(1/2)-27*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*c*d*
e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-3*b*c^4*d^5+8*
x^2*b^4*c*e^5-6*x*c^5*d^5+8*x^3*b^3*c^2*e^5-27*EllipticE(((c*x+b)/b)^(1/2),(b*e/
(b*e-c*d))^(1/2))*b^4*c*d^2*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(
-c*x/b)^(1/2)+12*x*b^4*c*d*e^4-6*x^3*c^5*d^3*e^2-12*x^2*c^5*d^4*e+3*b^4*c*d^2*e^
3-9*b^3*c^2*d^3*e^2+9*b^2*c^3*d^4*e+8*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d)
)^(1/2))*x*b^5*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)
+8*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*d*e^4*((c*x+b)/b)^(1/2
)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+6*EllipticE(((c*x+b)/b)^(1/2),(b*e
/(b*e-c*d))^(1/2))*b*c^4*d^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*
x/b)^(1/2)-6*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^5*((c*x+
b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+28*EllipticE(((c*x+b)/b)
^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^3*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)-15*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2)
)*b^2*c^3*d^4*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*
EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d^2*e^3*((c*x+b)/b)^(1/
2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-10*EllipticF(((c*x+b)/b)^(1/2),(b
*e/(b*e-c*d))^(1/2))*b^3*c^2*d^3*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1
/2)*(-c*x/b)^(1/2)+12*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3
*d^4*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+28*Elliptic
E(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c^2*d^2*e^3*((c*x+b)/b)^(1/2)*(
-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-20*x^2*b^2*c^3*d^2*e^3+15*x^2*b*c^4*d
^3*e^2-26*x*b^3*c^2*d^2*e^3+9*x*b^2*c^3*d^3*e^2+3*x*b*c^4*d^4*e-19*x^3*b^2*c^3*d
*e^4+9*x^3*b*c^4*d^2*e^3-7*x^2*b^3*c^2*d*e^4)/b^2/d^3/c/(b*e-c*d)^3/(e*x+d)^(3/2
)/(c*x+b)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError