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

Optimal. Leaf size=344 \[ \frac{2 \sqrt{d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{3 b^4 \sqrt{b x+c x^2} (c d-b e)}-\frac{2 \sqrt{d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (4 c d-3 b e) (4 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{16 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]

[Out]

(-2*Sqrt[d + e*x]*(b*d + (2*c*d - b*e)*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + (2*Sqrt
[d + e*x]*(b*(8*c*d - 5*b*e)*(c*d - b*e) + 8*c*(c*d - b*e)*(2*c*d - b*e)*x))/(3*
b^4*(c*d - b*e)*Sqrt[b*x + c*x^2]) - (16*Sqrt[c]*(2*c*d - b*e)*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)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*(4*c*d - 3*b*e)*(4*c*
d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*S
qrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*Sqrt[c]*Sqrt[d + e*x]*Sqrt[b*x +
c*x^2])

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Rubi [A]  time = 1.22425, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ \frac{2 \sqrt{d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{3 b^4 \sqrt{b x+c x^2} (c d-b e)}-\frac{2 \sqrt{d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (4 c d-3 b e) (4 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{16 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d + e*x]*(b*d + (2*c*d - b*e)*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + (2*Sqrt
[d + e*x]*(b*(8*c*d - 5*b*e)*(c*d - b*e) + 8*c*(c*d - b*e)*(2*c*d - b*e)*x))/(3*
b^4*(c*d - b*e)*Sqrt[b*x + c*x^2]) - (16*Sqrt[c]*(2*c*d - b*e)*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)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*(4*c*d - 3*b*e)*(4*c*
d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*S
qrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*Sqrt[c]*Sqrt[d + e*x]*Sqrt[b*x +
c*x^2])

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Rubi in Sympy [A]  time = 158.682, size = 294, normalized size = 0.85 \[ \frac{16 \sqrt{c} \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e - 2 c d\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 \left (- b\right )^{\frac{7}{2}} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - 4 c d\right ) \left (3 b e - 4 c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 \sqrt{c} \left (- b\right )^{\frac{7}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{2 \sqrt{d + e x} \left (b d - x \left (b e - 2 c d\right )\right )}{3 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{4 \sqrt{d + e x} \left (\frac{b \left (5 b e - 8 c d\right )}{2} + 4 c x \left (b e - 2 c d\right )\right )}{3 b^{4} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16*sqrt(c)*sqrt(x)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(b*e - 2*c*d)*elliptic_e(asin(s
qrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(3*(-b)**(7/2)*sqrt(1 + e*x/d)*sqrt(b*x + c
*x**2)) + 2*sqrt(x)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*(b*e - 4*c*d)*(3*b*e - 4*c*d
)*elliptic_f(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(3*sqrt(c)*(-b)**(7/2)*s
qrt(d + e*x)*sqrt(b*x + c*x**2)) - 2*sqrt(d + e*x)*(b*d - x*(b*e - 2*c*d))/(3*b*
*2*(b*x + c*x**2)**(3/2)) - 4*sqrt(d + e*x)*(b*(5*b*e - 8*c*d)/2 + 4*c*x*(b*e -
2*c*d))/(3*b**4*sqrt(b*x + c*x**2))

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Mathematica [C]  time = 1.6735, size = 290, normalized size = 0.84 \[ \frac{2 \left (\frac{(d+e x) \left (b^3 (-(d+4 e x))+b^2 c x (6 d-13 e x)-8 b c^2 x^2 (e x-3 d)+16 c^3 d x^3\right )}{b+c x}+i c e x^{5/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (8 c d-5 b e) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+8 i c e x^{5/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-8 x (b+c x) (d+e x) (2 c d-b e)\right )}{3 b^4 x \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-8*(2*c*d - b*e)*x*(b + c*x)*(d + e*x) + ((d + e*x)*(16*c^3*d*x^3 + b^2*c*x*
(6*d - 13*e*x) - 8*b*c^2*x^2*(-3*d + e*x) - b^3*(d + 4*e*x)))/(b + c*x) + (8*I)*
Sqrt[b/c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(5/2)*Ellipti
cE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*Sqrt[b/c]*c*e*(8*c*d - 5*b*e)*
Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(5/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x
]], (c*d)/(b*e)]))/(3*b^4*x*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

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Maple [B]  time = 0.045, size = 1099, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(3*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipti
cF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*e^2-16*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^2*c^2*d*e+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^3*d^2-8*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^3*c*e^2+24*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^2*d*e-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^3*d^2+3*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*e^2-16*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^3*c*d*e+
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^2*d^2-8*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^4*e^2+24*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*d*e-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^2*d^2-8*x^4*b*c^3*e^2+16*x^4*c^4*d*e-13*x^3*b^2*c^2*e
^2+16*x^3*b*c^3*d*e+16*x^3*c^4*d^2-4*x^2*b^3*c*e^2-7*x^2*b^2*c^2*d*e+24*x^2*b*c^
3*d^2-5*x*b^3*c*d*e+6*x*b^2*c^2*d^2-b^3*c*d^2)/x^2*(x*(c*x+b))^(1/2)/c/b^4/(c*x+
b)^2/(e*x+d)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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