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

Optimal. Leaf size=359 \[ -\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 (b+2 c x) \sqrt{d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (c x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+b (c d-b e) (8 c d-b e)\right )}{3 b^4 d \sqrt{b x+c x^2} (c d-b e)}+\frac{16 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 1.17987, antiderivative size = 359, 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 \[ -\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 (b+2 c x) \sqrt{d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (c x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+b (c d-b e) (8 c d-b e)\right )}{3 b^4 d \sqrt{b x+c x^2} (c d-b e)}+\frac{16 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 153.784, size = 335, normalized size = 0.93 \[ \frac{2 \sqrt{c} \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 d \left (- b\right )^{\frac{7}{2}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} - \frac{2 \left (b + 2 c x\right ) \sqrt{d + e x}}{3 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{16 c \sqrt{x} \sqrt{- d} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - 2 c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{3 b^{4} \sqrt{e} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{4 \sqrt{d + e x} \left (\frac{b \left (b e - 8 c d\right ) \left (b e - c d\right )}{2} + \frac{c x \left (b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right )}{2}\right )}{3 b^{4} d \left (b e - c d\right ) \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 2.25592, size = 375, normalized size = 1.04 \[ \frac{2 \left (b (d+e x) \left (b c^2 d x^2 (c d-b e)+c^2 d x^2 (b+c x) (8 c d-7 b e)+x (b+c x)^2 (c d-b e) (8 c d-b e)+b d (b+c x)^2 (b e-c d)\right )-c x \sqrt{\frac{b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 e^2-9 b c d e+8 c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )\right )\right )}{3 b^5 d (x (b+c x))^{3/2} \sqrt{d+e x} (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*(d + e*x)*(b*c^2*d*(c*d - b*e)*x^2 + c^2*d*(8*c*d - 7*b*e)*x^2*(b + c*x) +
b*d*(-(c*d) + b*e)*(b + c*x)^2 + (c*d - b*e)*(8*c*d - b*e)*x*(b + c*x)^2) - Sqr
t[b/c]*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)*EllipticE[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)*Ellip
ticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^5*d*(c*d - b*e)*(x*(b +
c*x))^(3/2)*Sqrt[d + e*x])

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Maple [B]  time = 0.05, size = 1362, 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)^(1/2)/(c*x^2+b*x)^(5/2),x)

[Out]

-2/3/x^2*(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^2*b^4*c*e^3+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^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+2*x*b^4*c*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)*EllipticE(((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)*El
lipticF(((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^2-24*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^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)*Ell
ipticE(((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+b^4*c*
d^2*e-24*x*((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^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)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b
*e-c*d))^(1/2))*b^4*c*d*e^2)*(x*(c*x+b))^(1/2)/c/b^4/d/(c*x+b)^2/(b*e-c*d)/(e*x+
d)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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