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

Optimal. Leaf size=343 \[ -\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{3/2} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+b d (8 c d-7 b e)\right )}{3 b^4 \sqrt{b x+c x^2}}+\frac{16 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}} \]

[Out]

(-2*(d + e*x)^(3/2)*(b*d + (2*c*d - b*e)*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + (2*Sq
rt[d + e*x]*(b*d*(8*c*d - 7*b*e) + (16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*x))/(3*b^
4*Sqrt[b*x + c*x^2]) - (2*(16*c^2*d^2 - 16*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])/Sqrt[-b]], (b*e)/(c*d)]
)/(3*(-b)^(7/2)*Sqrt[c]*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (16*d*(c*d - b*e)
*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqr
t[c]*Sqrt[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 = 1.09217, antiderivative size = 343, 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 \[ -\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{3/2} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+b d (8 c d-7 b e)\right )}{3 b^4 \sqrt{b x+c x^2}}+\frac{16 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(d + e*x)^(3/2)*(b*d + (2*c*d - b*e)*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + (2*Sq
rt[d + e*x]*(b*d*(8*c*d - 7*b*e) + (16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*x))/(3*b^
4*Sqrt[b*x + c*x^2]) - (2*(16*c^2*d^2 - 16*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])/Sqrt[-b]], (b*e)/(c*d)]
)/(3*(-b)^(7/2)*Sqrt[c]*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (16*d*(c*d - b*e)
*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqr
t[c]*Sqrt[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 = 121.773, size = 323, normalized size = 0.94 \[ \frac{16 d \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - 2 c d\right ) \left (b e - c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 \sqrt{c} \left (- b\right )^{\frac{7}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{2 \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 \sqrt{c} \left (- b\right )^{\frac{7}{2}} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} - \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (b d - x \left (b e - 2 c d\right )\right )}{3 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \sqrt{d + e x} \left (- \frac{b d \left (7 b e - 8 c d\right )}{2} + x \left (\frac{b^{2} e^{2}}{2} - 8 b c d e + 8 c^{2} d^{2}\right )\right )}{3 b^{4} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16*d*sqrt(x)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*(b*e - 2*c*d)*(b*e - c*d)*elliptic_
f(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(3*sqrt(c)*(-b)**(7/2)*sqrt(d + e*x
)*sqrt(b*x + c*x**2)) - 2*sqrt(x)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(b**2*e**2 - 16*
b*c*d*e + 16*c**2*d**2)*elliptic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(3
*sqrt(c)*(-b)**(7/2)*sqrt(1 + e*x/d)*sqrt(b*x + c*x**2)) - 2*(d + e*x)**(3/2)*(b
*d - x*(b*e - 2*c*d))/(3*b**2*(b*x + c*x**2)**(3/2)) + 4*sqrt(d + e*x)*(-b*d*(7*
b*e - 8*c*d)/2 + x*(b**2*e**2/2 - 8*b*c*d*e + 8*c**2*d**2))/(3*b**4*sqrt(b*x + c
*x**2))

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Mathematica [C]  time = 2.19756, size = 353, normalized size = 1.03 \[ \frac{2 \left (b (d+e x) \left (b^3 \left (-\left (d^2+7 d e x-2 e^2 x^2\right )\right )+b^2 c x \left (6 d^2-25 d e x+e^2 x^2\right )+8 b c^2 d x^2 (3 d-2 e x)+16 c^3 d^2 x^3\right )-x \sqrt{\frac{b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 e^2-9 b c d e+8 c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )\right )\right )}{3 b^5 (x (b+c x))^{3/2} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*(d + e*x)*(16*c^3*d^2*x^3 + 8*b*c^2*d*x^2*(3*d - 2*e*x) - b^3*(d^2 + 7*d*e
*x - 2*e^2*x^2) + b^2*c*x*(6*d^2 - 25*d*e*x + e^2*x^2)) - Sqrt[b/c]*x*(b + c*x)*
(Sqrt[b/c]*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*(b + c*x)*(d + e*x) + I*b*e*(16*c
^2*d^2 - 16*b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Ellip
ticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^2*d^2 - 9*b*c*d*e +
b^2*e^2)*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*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])

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Maple [B]  time = 0.048, size = 1318, normalized size = 3.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x^4*b^2*c^3*e^3+16*x^4*c^5*d^2*e+24*x^2*b*c^4*d^3+6*x*b^2*c^3*d^3+2*x^3*b^3
*c^2*e^3+x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipti
cE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*e^3-16*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^3*d^3-24*x^3*b^2*c^3*d*e^2+32*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^3*d^2*e+8*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*d*e^2+32*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^2*d^2*e+16*x^3*c^5*d^3-5*x^2*b^3*c^2*d*e^2-19*x^2*
b^2*c^3*d^2*e+8*x^3*b*c^4*d^2*e-16*x^4*b*c^4*d*e^2-8*x*b^3*c^2*d^2*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^4*d^3+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*e^3-16*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellip
ticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^3+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^3*d^3+8*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^2*d*e^2-2
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^2*c^3*d^2*e-17*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^2*d*e^2-b^3*c^2*d^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^3
*c^2*d^2*e-17*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*El
lipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d*e^2)/x^2*(x*(c*x+b))^(1
/2)/b^4/c^2/(c*x+b)^2/(e*x+d)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e^2*x^2 + 2*d*e*x + d^2)*sqrt(e*x + d)/((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)**(5/2)/(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. \[ \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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