3.2446 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=622 \[ \frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{21 c e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{21 c e^6 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{21 e^5}+\frac{10 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{21 e^3 \sqrt{d+e x}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}} \]

[Out]

(2*Sqrt[d + e*x]*(128*c^2*d^2 + 51*b^2*e^2 - 4*c*e*(44*b*d - 5*a*e) - 48*c*e*(2*
c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(21*e^5) + (10*(16*c*d - 7*b*e + 2*c*e*x)*(
a + b*x + c*x^2)^(3/2))/(21*e^3*Sqrt[d + e*x]) - (2*(a + b*x + c*x^2)^(5/2))/(3*
e*(d + e*x)^(3/2)) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(128*c^2*d^2 + 3*b
^2*e^2 - 4*c*e*(32*b*d - 29*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^
2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*
a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(
21*c*e^6*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x +
c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(128*c^2*d^2 + 27
*b^2*e^2 - 4*c*e*(32*b*d - 5*a*e))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4
*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b
+ Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]
*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(21*c*e^6*Sqrt[d + e*x]*Sqrt[a + b*x +
c*x^2])

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Rubi [A]  time = 2.22769, antiderivative size = 622, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{21 c e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{21 c e^6 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{21 e^5}+\frac{10 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{21 e^3 \sqrt{d+e x}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(128*c^2*d^2 + 51*b^2*e^2 - 4*c*e*(44*b*d - 5*a*e) - 48*c*e*(2*
c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(21*e^5) + (10*(16*c*d - 7*b*e + 2*c*e*x)*(
a + b*x + c*x^2)^(3/2))/(21*e^3*Sqrt[d + e*x]) - (2*(a + b*x + c*x^2)^(5/2))/(3*
e*(d + e*x)^(3/2)) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(128*c^2*d^2 + 3*b
^2*e^2 - 4*c*e*(32*b*d - 29*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^
2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*
a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(
21*c*e^6*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x +
c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(128*c^2*d^2 + 27
*b^2*e^2 - 4*c*e*(32*b*d - 5*a*e))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4
*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b
+ Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]
*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(21*c*e^6*Sqrt[d + e*x]*Sqrt[a + b*x +
c*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((c*x**2+b*x+a)**(5/2)/(e*x+d)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 14.5479, size = 5407, normalized size = 8.69 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [B]  time = 0.085, size = 12847, normalized size = 20.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(c*x^2 +
b*x + a)/((e^2*x^2 + 2*d*e*x + d^2)*sqrt(e*x + d)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Timed out