3.2457 \(\int \frac{1}{(d+e x)^{7/2} \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=629 \[ -\frac{2 e \sqrt{a+b x+c x^2} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right )}{15 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^3}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{8 e \sqrt{a+b x+c x^2} (2 c d-b e)}{15 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 e \sqrt{a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )} \]

[Out]

(-2*e*Sqrt[a + b*x + c*x^2])/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2)) - (8*e*
(2*c*d - b*e)*Sqrt[a + b*x + c*x^2])/(15*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/
2)) - (2*e*(23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*e))*Sqrt[a + b*x + c*x^2]
)/(15*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(23*
c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*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)/Sqr
t[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)])/(15*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2
- 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (8*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - 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)/Sq
rt[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)])/(15*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 2.03272, antiderivative size = 629, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 e \sqrt{a+b x+c x^2} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right )}{15 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^3}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{8 e \sqrt{a+b x+c x^2} (2 c d-b e)}{15 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 e \sqrt{a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]

[Out]

(-2*e*Sqrt[a + b*x + c*x^2])/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2)) - (8*e*
(2*c*d - b*e)*Sqrt[a + b*x + c*x^2])/(15*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/
2)) - (2*e*(23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*e))*Sqrt[a + b*x + c*x^2]
)/(15*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(23*
c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*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)/Sqr
t[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)])/(15*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2
- 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (8*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - 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)/Sq
rt[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)])/(15*(c*d^2 - b*d*e + a*e^2)^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(1/(e*x+d)**(7/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 12.0238, size = 983, normalized size = 1.56 \[ \frac{2 \sqrt{c x^2+b x+a} \left (\left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \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 )-\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 (\left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\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 (-30 c^3 d^3-c^2 \left (-34 a e^2-45 b d e+23 d \sqrt{\left (b^2-4 a c\right ) e^2}\right ) d+8 b^2 e^2 \left (b e-\sqrt{\left (b^2-4 a c\right ) e^2}\right )+c e \left (-31 d e b^2-17 a e^2 b+23 d \sqrt{\left (b^2-4 a c\right ) e^2} b+9 a e \sqrt{\left (b^2-4 a c\right ) e^2}\right )\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}}\right ) (d+e x)^{3/2}}{15 e \left (c d^2-b e d+a e^2\right )^3 \sqrt{a+x (b+c x)} \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}}}+\frac{\left (c x^2+b x+a\right ) \left (\frac{2 \left (-23 c^2 d^2+23 b c e d-8 b^2 e^2+9 a c e^2\right ) e}{15 \left (c d^2-b e d+a e^2\right )^3 (d+e x)}+\frac{8 (b e-2 c d) e}{15 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^2}-\frac{2 e}{5 \left (c d^2-b e d+a e^2\right ) (d+e x)^3}\right ) \sqrt{d+e x}}{\sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]

[Out]

(Sqrt[d + e*x]*(a + b*x + c*x^2)*((-2*e)/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3)
+ (8*e*(-2*c*d + b*e))/(15*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + (2*e*(-23*c
^2*d^2 + 23*b*c*d*e - 8*b^2*e^2 + 9*a*c*e^2))/(15*(c*d^2 - b*d*e + a*e^2)^3*(d +
e*x))))/Sqrt[a + x*(b + c*x)] + (2*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]*((23*c
^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*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))]*
((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d +
9*a*e))*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]))] + (-30*c^3*d^3 + 8*b^2*e^2*(b
*e - Sqrt[(b^2 - 4*a*c)*e^2]) - c^2*d*(-45*b*d*e - 34*a*e^2 + 23*d*Sqrt[(b^2 - 4
*a*c)*e^2]) + c*e*(-31*b^2*d*e - 17*a*b*e^2 + 23*b*d*Sqrt[(b^2 - 4*a*c)*e^2] + 9
*a*e*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])]*Sq
rt[d + e*x])))/(15*e*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[a + x*(b + c*x)]*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 = 0.107, size = 14312, normalized size = 22.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{c x^{2} + b x + a} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

integral(1/((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*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(1/(e*x+d)**(7/2)/(c*x**2+b*x+a)**(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(1/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

Timed out