3.80 \(\int \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^3 \sqrt{d+e x} \, dx\)

Optimal. Leaf size=778 \[ -\frac{4 x (d+e x)^{3/2} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left (8 a^2 d^2+a e (4 b d-7 c e)+3 b^2 e^2\right )}{315 a^2 e^3}+\frac{2 x \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left (19 a^3 d^3-6 a^2 c d e^2+3 a b e^2 (b d-9 c e)+8 b^3 e^3\right )}{315 a^3 e^3}-\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (8 a^4 d^4-a^3 d^2 e (4 b d-9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )-4 a b^2 e^3 (b d+9 c e)+8 b^4 e^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \left (a x^2+b x+c\right ) \sqrt{\frac{a (d+e x)}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (16 a^3 d^3+6 a^2 c d e^2-3 a b e^2 (b d-9 c e)-8 b^3 e^3\right ) \left (a d^2-e (b d-c e)\right ) \sqrt{\frac{a (d+e x)}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \sqrt{d+e x} \left (a x^2+b x+c\right )}+\frac{2 x (d+e x)^{5/2} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (a d+b e)}{63 a e^3}+\frac{2}{9} x^4 \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \]

[Out]

(2*(19*a^3*d^3 - 6*a^2*c*d*e^2 + 8*b^3*e^3 + 3*a*b*e^2*(b*d - 9*c*e))*Sqrt[a + c
/x^2 + b/x]*x*Sqrt[d + e*x])/(315*a^3*e^3) + (2*Sqrt[a + c/x^2 + b/x]*x^4*Sqrt[d
+ e*x])/9 - (4*(8*a^2*d^2 + 3*b^2*e^2 + a*e*(4*b*d - 7*c*e))*Sqrt[a + c/x^2 + b
/x]*x*(d + e*x)^(3/2))/(315*a^2*e^3) + (2*(a*d + b*e)*Sqrt[a + c/x^2 + b/x]*x*(d
+ e*x)^(5/2))/(63*a*e^3) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(8*a^4*d^4 + 8*b^4*e^4
- a^3*d^2*e*(4*b*d - 9*c*e) - 4*a*b^2*e^3*(b*d + 9*c*e) - 3*a^2*e^2*(b^2*d^2 - 5
*b*c*d*e - 7*c^2*e^2))*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x]*Sqrt[-((a*(c + b*x
+ a*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/
Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4
*a*c])*e)])/(315*a^4*e^4*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]
*(c + b*x + a*x^2)) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(16*a^3*d^3 + 6*a^2*c*d*e^2 -
8*b^3*e^3 - 3*a*b*e^2*(b*d - 9*c*e))*(a*d^2 - e*(b*d - c*e))*Sqrt[a + c/x^2 + b
/x]*x*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((a*(c + b*x
+ a*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)
/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 -
4*a*c])*e)])/(315*a^4*e^4*Sqrt[d + e*x]*(c + b*x + a*x^2))

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Rubi [A]  time = 4.66489, antiderivative size = 778, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241 \[ -\frac{4 x (d+e x)^{3/2} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left (8 a^2 d^2+a e (4 b d-7 c e)+3 b^2 e^2\right )}{315 a^2 e^3}+\frac{2 x \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left (19 a^3 d^3-6 a^2 c d e^2+3 a b e^2 (b d-9 c e)+8 b^3 e^3\right )}{315 a^3 e^3}-\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (8 a^4 d^4-a^3 d^2 e (4 b d-9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )-4 a b^2 e^3 (b d+9 c e)+8 b^4 e^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \left (a x^2+b x+c\right ) \sqrt{\frac{a (d+e x)}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (16 a^3 d^3+6 a^2 c d e^2-3 a b e^2 (b d-9 c e)-8 b^3 e^3\right ) \left (a d^2-e (b d-c e)\right ) \sqrt{\frac{a (d+e x)}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \sqrt{d+e x} \left (a x^2+b x+c\right )}+\frac{2 x (d+e x)^{5/2} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (a d+b e)}{63 a e^3}+\frac{2}{9} x^4 \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + c/x^2 + b/x]*x^3*Sqrt[d + e*x],x]

[Out]

(2*(19*a^3*d^3 - 6*a^2*c*d*e^2 + 8*b^3*e^3 + 3*a*b*e^2*(b*d - 9*c*e))*Sqrt[a + c
/x^2 + b/x]*x*Sqrt[d + e*x])/(315*a^3*e^3) + (2*Sqrt[a + c/x^2 + b/x]*x^4*Sqrt[d
+ e*x])/9 - (4*(8*a^2*d^2 + 3*b^2*e^2 + a*e*(4*b*d - 7*c*e))*Sqrt[a + c/x^2 + b
/x]*x*(d + e*x)^(3/2))/(315*a^2*e^3) + (2*(a*d + b*e)*Sqrt[a + c/x^2 + b/x]*x*(d
+ e*x)^(5/2))/(63*a*e^3) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(8*a^4*d^4 + 8*b^4*e^4
- a^3*d^2*e*(4*b*d - 9*c*e) - 4*a*b^2*e^3*(b*d + 9*c*e) - 3*a^2*e^2*(b^2*d^2 - 5
*b*c*d*e - 7*c^2*e^2))*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x]*Sqrt[-((a*(c + b*x
+ a*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/
Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4
*a*c])*e)])/(315*a^4*e^4*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]
*(c + b*x + a*x^2)) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(16*a^3*d^3 + 6*a^2*c*d*e^2 -
8*b^3*e^3 - 3*a*b*e^2*(b*d - 9*c*e))*(a*d^2 - e*(b*d - c*e))*Sqrt[a + c/x^2 + b
/x]*x*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((a*(c + b*x
+ a*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)
/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 -
4*a*c])*e)])/(315*a^4*e^4*Sqrt[d + e*x]*(c + b*x + a*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(x**3*(a+c/x**2+b/x)**(1/2)*(e*x+d)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 14.2211, size = 7531, normalized size = 9.68 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[a + c/x^2 + b/x]*x^3*Sqrt[d + e*x],x]

[Out]

Result too large to show

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Maple [B]  time = 0.074, size = 9182, normalized size = 11.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{e x + d} \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out