3.79 \(\int \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^4 \sqrt{d+e x} \, dx\)

Optimal. Leaf size=981 \[ \frac{2}{11} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x^5+\frac{2 (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{7/2} x}{99 a e^4}-\frac{2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{5/2} x}{693 a^2 e^4}+\frac{2 \left (233 a^3 d^3+4 a^2 e (18 b d-37 c e) d+48 b^3 e^3+a b e^2 (67 b d-157 c e)\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{3/2} x}{3465 a^3 e^4}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (128 a^5 d^5-4 a^4 e (14 b d-27 c e) d^3-a^3 e^2 \left (37 b^2 d^2-135 b c e d+156 c^2 e^2\right ) d+128 b^5 e^5-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c e d-771 c^2 e^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3465 a^5 e^5 \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right ) \left (128 a^4 d^4+4 a^3 e (2 b d+3 c e) d^2-64 b^4 e^4-4 a b^2 e^3 (7 b d-69 c e)-3 a^2 e^2 \left (3 b^2 d^2-29 b c e d+50 c^2 e^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3465 a^5 e^5 \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{2 \left (187 a^4 d^4-4 a^3 e (2 b d+3 c e) d^2+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c e d+50 c^2 e^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x}{3465 a^4 e^4} \]

[Out]

(-2*(187*a^4*d^4 + 64*b^4*e^4 + 4*a*b^2*e^3*(7*b*d - 69*c*e) - 4*a^3*d^2*e*(2*b*
d + 3*c*e) + 3*a^2*e^2*(3*b^2*d^2 - 29*b*c*d*e + 50*c^2*e^2))*Sqrt[a + c/x^2 + b
/x]*x*Sqrt[d + e*x])/(3465*a^4*e^4) + (2*Sqrt[a + c/x^2 + b/x]*x^5*Sqrt[d + e*x]
)/11 + (2*(233*a^3*d^3 + 48*b^3*e^3 + a*b*e^2*(67*b*d - 157*c*e) + 4*a^2*d*e*(18
*b*d - 37*c*e))*Sqrt[a + c/x^2 + b/x]*x*(d + e*x)^(3/2))/(3465*a^3*e^4) - (2*(29
*a^2*d^2 + 8*b^2*e^2 + a*e*(19*b*d - 18*c*e))*Sqrt[a + c/x^2 + b/x]*x*(d + e*x)^
(5/2))/(693*a^2*e^4) + (2*(a*d + b*e)*Sqrt[a + c/x^2 + b/x]*x*(d + e*x)^(7/2))/(
99*a*e^4) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(128*a^5*d^5 + 128*b^5*e^5 - 4*a^4*d^3*e*
(14*b*d - 27*c*e) - 8*a*b^3*e^4*(7*b*d + 87*c*e) - a^2*b*e^3*(37*b^2*d^2 - 258*b
*c*d*e - 771*c^2*e^2) - a^3*d*e^2*(37*b^2*d^2 - 135*b*c*d*e + 156*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))]*E
llipticE[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)])/(3465*a^5*e^5*S
qrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*(c + b*x + a*x^2)) - (2*S
qrt[2]*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))*(128*a^4*d^4 - 64*b^4*e^4 - 4*a
*b^2*e^3*(7*b*d - 69*c*e) + 4*a^3*d^2*e*(2*b*d + 3*c*e) - 3*a^2*e^2*(3*b^2*d^2 -
29*b*c*d*e + 50*c^2*e^2))*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))]*Elliptic
F[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*S
qrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3465*a^5*e^5*Sqrt[d +
e*x]*(c + b*x + a*x^2))

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Rubi [A]  time = 10.1348, antiderivative size = 981, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241 \[ \frac{2}{11} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x^5+\frac{2 (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{7/2} x}{99 a e^4}-\frac{2 \left (29 a^2 d^2+8 b^2 e^2+a e (19 b d-18 c e)\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{5/2} x}{693 a^2 e^4}+\frac{2 \left (233 a^3 d^3+4 a^2 e (18 b d-37 c e) d+48 b^3 e^3+a b e^2 (67 b d-157 c e)\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} (d+e x)^{3/2} x}{3465 a^3 e^4}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (128 a^5 d^5-4 a^4 e (14 b d-27 c e) d^3-a^3 e^2 \left (37 b^2 d^2-135 b c e d+156 c^2 e^2\right ) d+128 b^5 e^5-8 a b^3 e^4 (7 b d+87 c e)-a^2 b e^3 \left (37 b^2 d^2-258 b c e d-771 c^2 e^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3465 a^5 e^5 \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right ) \left (128 a^4 d^4+4 a^3 e (2 b d+3 c e) d^2-64 b^4 e^4-4 a b^2 e^3 (7 b d-69 c e)-3 a^2 e^2 \left (3 b^2 d^2-29 b c e d+50 c^2 e^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3465 a^5 e^5 \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{2 \left (187 a^4 d^4-4 a^3 e (2 b d+3 c e) d^2+64 b^4 e^4+4 a b^2 e^3 (7 b d-69 c e)+3 a^2 e^2 \left (3 b^2 d^2-29 b c e d+50 c^2 e^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x}{3465 a^4 e^4} \]

Warning: Unable to verify antiderivative.

[In]  Int[Sqrt[a + c/x^2 + b/x]*x^4*Sqrt[d + e*x],x]

[Out]

(-2*(187*a^4*d^4 + 64*b^4*e^4 + 4*a*b^2*e^3*(7*b*d - 69*c*e) - 4*a^3*d^2*e*(2*b*
d + 3*c*e) + 3*a^2*e^2*(3*b^2*d^2 - 29*b*c*d*e + 50*c^2*e^2))*Sqrt[a + c/x^2 + b
/x]*x*Sqrt[d + e*x])/(3465*a^4*e^4) + (2*Sqrt[a + c/x^2 + b/x]*x^5*Sqrt[d + e*x]
)/11 + (2*(233*a^3*d^3 + 48*b^3*e^3 + a*b*e^2*(67*b*d - 157*c*e) + 4*a^2*d*e*(18
*b*d - 37*c*e))*Sqrt[a + c/x^2 + b/x]*x*(d + e*x)^(3/2))/(3465*a^3*e^4) - (2*(29
*a^2*d^2 + 8*b^2*e^2 + a*e*(19*b*d - 18*c*e))*Sqrt[a + c/x^2 + b/x]*x*(d + e*x)^
(5/2))/(693*a^2*e^4) + (2*(a*d + b*e)*Sqrt[a + c/x^2 + b/x]*x*(d + e*x)^(7/2))/(
99*a*e^4) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(128*a^5*d^5 + 128*b^5*e^5 - 4*a^4*d^3*e*
(14*b*d - 27*c*e) - 8*a*b^3*e^4*(7*b*d + 87*c*e) - a^2*b*e^3*(37*b^2*d^2 - 258*b
*c*d*e - 771*c^2*e^2) - a^3*d*e^2*(37*b^2*d^2 - 135*b*c*d*e + 156*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))]*E
llipticE[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)])/(3465*a^5*e^5*S
qrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*(c + b*x + a*x^2)) - (2*S
qrt[2]*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))*(128*a^4*d^4 - 64*b^4*e^4 - 4*a
*b^2*e^3*(7*b*d - 69*c*e) + 4*a^3*d^2*e*(2*b*d + 3*c*e) - 3*a^2*e^2*(3*b^2*d^2 -
29*b*c*d*e + 50*c^2*e^2))*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))]*Elliptic
F[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*S
qrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3465*a^5*e^5*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**4*(a+c/x**2+b/x)**(1/2)*(e*x+d)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 14.7852, size = 10904, normalized size = 11.12 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[a + c/x^2 + b/x]*x^4*Sqrt[d + e*x],x]

[Out]

Result too large to show

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Maple [B]  time = 0.169, size = 11938, normalized size = 12.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(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. \[ \int \sqrt{e x + d} \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)*sqrt(a + b/x + c/x^2)*x^4,x, algorithm="maxima")

[Out]

integrate(sqrt(e*x + d)*sqrt(a + b/x + c/x^2)*x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)*sqrt(a + b/x + c/x^2)*x^4,x, algorithm="fricas")

[Out]

integral(sqrt(e*x + d)*x^4*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**4*(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^4,x, algorithm="giac")

[Out]

Timed out