3.2634 \(\int \frac{(A+B x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=678 \[ -\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 e \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right )}{3 c^2 \left (b^2-4 a c\right )}+\frac{2 (d+e x)^{3/2} \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b c \left (-29 a B e^2+6 A c d e+3 B c d^2\right )-2 c^2 \left (-9 a A e^2-20 a B d e+3 A c d^2\right )-b^2 c e (6 A e+13 B d)+8 b^3 B e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]

[Out]

(2*(d + e*x)^(3/2)*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d
+ A*e) + 2*c*(A*c*d - a*B*e))*x))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (2*e
*(4*b^2*B*e - 3*b*c*(B*d + A*e) + 2*c*(3*A*c*d - 5*a*B*e))*Sqrt[d + e*x]*Sqrt[a
+ b*x + c*x^2])/(3*c^2*(b^2 - 4*a*c)) - (Sqrt[2]*(8*b^3*B*e^2 - b^2*c*e*(13*B*d
+ 6*A*e) - 2*c^2*(3*A*c*d^2 - 20*a*B*d*e - 9*a*A*e^2) + b*c*(3*B*c*d^2 + 6*A*c*d
*e - 29*a*B*e^2))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ell
ipticE[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)])/(3*c^3*Sqrt[b^2 -
4*a*c]*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c
*x^2]) - (2*Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*(4*b^2*B*e - 3*b*c*(B*d + A*e) + 2*c
*(3*A*c*d - 5*a*B*e))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sq
rt[-((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)])/(3*c^3*Sqrt[b^2 - 4*a*c]*Sqrt[d + e*x]*Sqrt[a + b*x
+ c*x^2])

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Rubi [A]  time = 2.67727, antiderivative size = 678, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 e \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right )}{3 c^2 \left (b^2-4 a c\right )}+\frac{2 (d+e x)^{3/2} \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b c \left (-29 a B e^2+6 A c d e+3 B c d^2\right )-2 c^2 \left (-9 a A e^2-20 a B d e+3 A c d^2\right )-b^2 c e (6 A e+13 B d)+8 b^3 B e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]

Warning: Unable to verify antiderivative.

[In]  Int[((A + B*x)*(d + e*x)^(5/2))/(a + b*x + c*x^2)^(3/2),x]

[Out]

(2*(d + e*x)^(3/2)*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d
+ A*e) + 2*c*(A*c*d - a*B*e))*x))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (2*e
*(4*b^2*B*e - 3*b*c*(B*d + A*e) + 2*c*(3*A*c*d - 5*a*B*e))*Sqrt[d + e*x]*Sqrt[a
+ b*x + c*x^2])/(3*c^2*(b^2 - 4*a*c)) - (Sqrt[2]*(8*b^3*B*e^2 - b^2*c*e*(13*B*d
+ 6*A*e) - 2*c^2*(3*A*c*d^2 - 20*a*B*d*e - 9*a*A*e^2) + b*c*(3*B*c*d^2 + 6*A*c*d
*e - 29*a*B*e^2))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ell
ipticE[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)])/(3*c^3*Sqrt[b^2 -
4*a*c]*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c
*x^2]) - (2*Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*(4*b^2*B*e - 3*b*c*(B*d + A*e) + 2*c
*(3*A*c*d - 5*a*B*e))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sq
rt[-((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)])/(3*c^3*Sqrt[b^2 - 4*a*c]*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((B*x+A)*(e*x+d)**(5/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 15.2485, size = 7589, normalized size = 11.19 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((A + B*x)*(d + e*x)^(5/2))/(a + b*x + c*x^2)^(3/2),x]

[Out]

Result too large to show

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Maple [B]  time = 0.126, size = 10385, normalized size = 15.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 + b*x + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")

[Out]

integral((B*e^2*x^3 + A*d^2 + (2*B*d*e + A*e^2)*x^2 + (B*d^2 + 2*A*d*e)*x)*sqrt(
e*x + d)/(c*x^2 + b*x + a)^(3/2), 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((B*x+A)*(e*x+d)**(5/2)/(c*x**2+b*x+a)**(3/2),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((B*x + A)*(e*x + d)^(5/2)/(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")

[Out]

Timed out