3.412 \(\int \frac{1}{(d+e x)^{5/2} \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=317 \[ -\frac{4 e \sqrt{b x+c x^2} (2 c d-b e)}{3 d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)} \]

[Out]

(-2*e*Sqrt[b*x + c*x^2])/(3*d*(c*d - b*e)*(d + e*x)^(3/2)) - (4*e*(2*c*d - b*e)*
Sqrt[b*x + c*x^2])/(3*d^2*(c*d - b*e)^2*Sqrt[d + e*x]) + (4*Sqrt[-b]*Sqrt[c]*(2*
c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqr
t[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*d^2*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x +
c*x^2]) - (2*Sqrt[-b]*Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Ellip
ticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*d*(c*d - b*e)*Sqrt[d +
e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.976746, antiderivative size = 317, 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 \[ -\frac{4 e \sqrt{b x+c x^2} (2 c d-b e)}{3 d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2}}{3 d (d+e x)^{3/2} (c d-b e)}-\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e*Sqrt[b*x + c*x^2])/(3*d*(c*d - b*e)*(d + e*x)^(3/2)) - (4*e*(2*c*d - b*e)*
Sqrt[b*x + c*x^2])/(3*d^2*(c*d - b*e)^2*Sqrt[d + e*x]) + (4*Sqrt[-b]*Sqrt[c]*(2*
c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqr
t[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*d^2*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x +
c*x^2]) - (2*Sqrt[-b]*Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Ellip
ticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*d*(c*d - b*e)*Sqrt[d +
e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 125.961, size = 280, normalized size = 0.88 \[ - \frac{4 \sqrt{c} \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e - 2 c d\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 d^{2} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right )^{2} \sqrt{b x + c x^{2}}} - \frac{2 c \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{3 \sqrt{e} \sqrt{- d} \sqrt{d + e x} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} + \frac{2 e \sqrt{b x + c x^{2}}}{3 d \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )} + \frac{4 e \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{3 d^{2} \sqrt{d + e x} \left (b e - c d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*sqrt(c)*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(b*e - 2*c*d)*elliptic
_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(3*d**2*sqrt(1 + e*x/d)*(b*e - c*d
)**2*sqrt(b*x + c*x**2)) - 2*c*sqrt(x)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*elliptic_
f(asin(sqrt(e)*sqrt(x)/sqrt(-d)), c*d/(b*e))/(3*sqrt(e)*sqrt(-d)*sqrt(d + e*x)*(
b*e - c*d)*sqrt(b*x + c*x**2)) + 2*e*sqrt(b*x + c*x**2)/(3*d*(d + e*x)**(3/2)*(b
*e - c*d)) + 4*e*(b*e - 2*c*d)*sqrt(b*x + c*x**2)/(3*d**2*sqrt(d + e*x)*(b*e - c
*d)**2)

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Mathematica [C]  time = 1.72278, size = 290, normalized size = 0.91 \[ -\frac{2 \left (-b e x (b+c x) (b e (3 d+2 e x)-c d (5 d+4 e x))-c \sqrt{\frac{b}{c}} (d+e x) \left (i x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 b^2 e^2-5 b c d e+3 c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (2 c d-b e) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) (b e-2 c d)\right )\right )}{3 b d^2 \sqrt{x (b+c x)} (d+e x)^{3/2} (c d-b e)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-(b*e*x*(b + c*x)*(b*e*(3*d + 2*e*x) - c*d*(5*d + 4*e*x))) - Sqrt[b/c]*c*(d
+ e*x)*(-2*Sqrt[b/c]*(-2*c*d + b*e)*(b + c*x)*(d + e*x) + (2*I)*b*e*(2*c*d - b*
e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqr
t[x]], (c*d)/(b*e)] + I*(3*c^2*d^2 - 5*b*c*d*e + 2*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sq
rt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/
(3*b*d^2*(c*d - b*e)^2*Sqrt[x*(b + c*x)]*(d + e*x)^(3/2))

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Maple [B]  time = 0.047, size = 897, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x*(c*x+b))^(1/2)*(EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*
c*d*e^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))*x*b*c^2*d^2*e*((c*x+b)/b)^(1/2)*(-(e*x+
d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+2*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d
))^(1/2))*x*b^3*e^3*((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^2*c*d*e^2*((c*x+b)/b)
^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*EllipticE(((c*x+b)/b)^(1/2)
,(b*e/(b*e-c*d))^(1/2))*x*b*c^2*d^2*e*((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*d^2
*e*((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^2*d^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)+2*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))
*b^3*d*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-6*Ellip
ticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c*d^2*e*((c*x+b)/b)^(1/2)*(-(e
*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-
c*d))^(1/2))*b*c^2*d^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(
1/2)+2*x^3*b*c^2*e^3-4*x^3*c^3*d*e^2+2*x^2*b^2*c*e^3-x^2*b*c^2*d*e^2-5*x^2*c^3*d
^2*e+3*b^2*c*d*e^2*x-5*b*c^2*d^2*e*x)/(c*x+b)/x/(b*e-c*d)^2/(e*x+d)^(3/2)/c/d^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((e^2*x^2 + 2*d*e*x + d^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. \[ \int \frac{1}{\sqrt{x \left (b + c x\right )} \left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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