3.2638 \(\int \frac{A+B x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=705 \[ \frac{2 e \sqrt{a+b x+c x^2} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} (-2 a B e+A b e-2 A c d+b B d) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]

[Out]

(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e
- 2*a*B*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*
x + c*x^2]) + (2*e*(b^2*e*(B*d - 2*A*e) - 2*c*(A*c*d^2 + 4*a*B*d*e - 3*a*A*e^2)
+ b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(c*d^
2 - b*d*e + a*e^2)^2*Sqrt[d + e*x]) - (Sqrt[2]*(b^2*e*(B*d - 2*A*e) - 2*c*(A*c*d
^2 + 4*a*B*d*e - 3*a*A*e^2) + b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[d + e*x]*S
qrt[-((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)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[
(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (2*S
qrt[2]*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*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)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*
d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 2.33523, antiderivative size = 705, 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 \[ \frac{2 e \sqrt{a+b x+c x^2} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} (-2 a B e+A b e-2 A c d+b B d) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e
- 2*a*B*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*
x + c*x^2]) + (2*e*(b^2*e*(B*d - 2*A*e) - 2*c*(A*c*d^2 + 4*a*B*d*e - 3*a*A*e^2)
+ b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(c*d^
2 - b*d*e + a*e^2)^2*Sqrt[d + e*x]) - (Sqrt[2]*(b^2*e*(B*d - 2*A*e) - 2*c*(A*c*d
^2 + 4*a*B*d*e - 3*a*A*e^2) + b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[d + e*x]*S
qrt[-((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)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[
(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (2*S
qrt[2]*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*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)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*
d*e + a*e^2)*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)**(3/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 14.419, size = 6669, normalized size = 9.46 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [B]  time = 0.134, size = 8357, normalized size = 11.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^(3/2)/(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. \[ \int \frac{B x + A}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out