3.2637 \(\int \frac{A+B x}{\sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=528 \[ \frac{2 \sqrt{d+e x} \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (-2 a B e+A b e-2 A c d+b B d) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (b B-2 A c) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 1.27587, antiderivative size = 528, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{2 \sqrt{d+e x} \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (-2 a B e+A b e-2 A c d+b B d) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (b B-2 A c) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(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[a + b*
x + c*x^2]) - (Sqrt[2]*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*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)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^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*Sqrt[2]*(
b*B - 2*A*c)*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)])/(c*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)**(1/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 11.4038, size = 893, normalized size = 1.69 \[ \frac{2 \sqrt{d+e x} \left (-A e b^2+A c d b+a B e b-B c d x b-A c e x b-2 a B c d+2 a A c e+2 A c^2 d x+2 a B c e x\right ) \left (c x^2+b x+a\right )}{\left (4 a c-b^2\right ) \left (c d^2-b e d+a e^2\right ) (a+x (b+c x))^{3/2}}-\frac{2 (d+e x)^{3/2} \left (c x^2+b x+a\right )^{3/2} \left (\frac{i \sqrt{1-\frac{2 \left (c d^2+e (a e-b d)\right )}{\left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt{\frac{2 \left (c d^2+e (a e-b d)\right )}{\left (-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}+1} \left ((b B d-2 A c d+A b e-2 a B e) \left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )-\left (e (B d-A e) b^2+\sqrt{\left (b^2-4 a c\right ) e^2} (B d+A e) b-2 a e \left (2 c d B+\sqrt{\left (b^2-4 a c\right ) e^2} B-2 A c e\right )-2 A c d \sqrt{\left (b^2-4 a c\right ) e^2}\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt{2} \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}} \sqrt{d+e x}}-(b B d-2 A c d+A b e-2 a B e) \left (c \left (\frac{d}{d+e x}-1\right )^2+\frac{e \left (-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right )}{d+e x}\right )\right )}{\left (4 a c-b^2\right ) e \left (c d^2-b e d+a e^2\right ) (a+x (b+c x))^{3/2} \sqrt{\frac{(d+e x)^2 \left (c \left (\frac{d}{d+e x}-1\right )^2+\frac{e \left (-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(A*b*c*d - 2*a*B*c*d - A*b^2*e + a*b*B*e + 2*a*A*c*e - b*B*c*d*
x + 2*A*c^2*d*x - A*b*c*e*x + 2*a*B*c*e*x)*(a + b*x + c*x^2))/((-b^2 + 4*a*c)*(c
*d^2 - b*d*e + a*e^2)*(a + x*(b + c*x))^(3/2)) - (2*(d + e*x)^(3/2)*(a + b*x + c
*x^2)^(3/2)*(-((b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*(c*(-1 + d/(d + e*x))^2 + (e*
(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x))) + ((I/2)*Sqrt[1 - (2*(c*d^2
+ e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[
1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d
+ e*x))]*((b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*
e^2])*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e +
Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*
e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] - (-2*A*c*d*Sqrt[(b^2 - 4*a*c)*e
^2] + b^2*e*(B*d - A*e) + b*Sqrt[(b^2 - 4*a*c)*e^2]*(B*d + A*e) - 2*a*e*(2*B*c*d
- 2*A*c*e + B*Sqrt[(b^2 - 4*a*c)*e^2]))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^
2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -(
(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])
)]))/(Sqrt[2]*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)
*e^2])]*Sqrt[d + e*x])))/((-b^2 + 4*a*c)*e*(c*d^2 - b*d*e + a*e^2)*(a + x*(b + c
*x))^(3/2)*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) +
(a*e)/(d + e*x)))/(d + e*x)))/e^2])

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Maple [B]  time = 0.109, size = 5059, normalized size = 9.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^(1/2)/(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. \[ \int \frac{B x + A}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}} \sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x + A)/((c*x^2 + b*x + a)^(3/2)*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)**(1/2)/(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)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

Timed out