3.2466 \(\int \frac{(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=590 \[ -\frac{2 \sqrt{d+e x} \left (-x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-8 b \left (a e^2+c d^2\right )+4 a c d e+7 b^2 d e\right )}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c \left (b^2-4 a c\right )^{3/2} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{16 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c \left (b^2-4 a c\right )^{3/2} \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(-2*(d + e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x +
c*x^2)^(3/2)) - (2*Sqrt[d + e*x]*(7*b^2*d*e + 4*a*c*d*e - 8*b*(c*d^2 + a*e^2) -
(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*x))/(3*(b^2 - 4*a*c)^2*Sqrt[a +
b*x + c*x^2]) - (Sqrt[2]*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*Sqrt[d +
e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sq
rt[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*(b^2 - 4*a*c)^(3/2)*Sqrt[(c*(d + e*x))
/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (16*Sqrt[2]*(2*c*
d - b*e)*(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[-((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*(b^2 - 4*a*c)^(3/2)*Sqrt[d + e*x]*S
qrt[a + b*x + c*x^2])

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Rubi [A]  time = 1.74982, antiderivative size = 590, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 \sqrt{d+e x} \left (-x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-8 b \left (a e^2+c d^2\right )+4 a c d e+7 b^2 d e\right )}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c \left (b^2-4 a c\right )^{3/2} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{16 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c \left (b^2-4 a c\right )^{3/2} \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*(d + e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x +
c*x^2)^(3/2)) - (2*Sqrt[d + e*x]*(7*b^2*d*e + 4*a*c*d*e - 8*b*(c*d^2 + a*e^2) -
(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*x))/(3*(b^2 - 4*a*c)^2*Sqrt[a +
b*x + c*x^2]) - (Sqrt[2]*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*Sqrt[d +
e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sq
rt[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*(b^2 - 4*a*c)^(3/2)*Sqrt[(c*(d + e*x))
/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (16*Sqrt[2]*(2*c*
d - b*e)*(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[-((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*(b^2 - 4*a*c)^(3/2)*Sqrt[d + e*x]*S
qrt[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((e*x+d)**(5/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 13.4987, size = 3577, normalized size = 6.06 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(a + b*x + c*x^2)^3*((2*(b*c*d^2 - 4*a*c*d*e + a*b*e^2 + 2*c^2*d^
2*x - 2*b*c*d*e*x + b^2*e^2*x - 2*a*c*e^2*x))/(3*c*(-b^2 + 4*a*c)*(a + b*x + c*x
^2)^2) + (2*(8*b*c^2*d^2 - 9*b^2*c*d*e + 4*a*c^2*d*e + b^3*e^2 + 4*a*b*c*e^2 + 1
6*c^3*d^2*x - 16*b*c^2*d*e*x + b^2*c*e^2*x + 12*a*c^2*e^2*x))/(3*c*(-b^2 + 4*a*c
)^2*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(5/2) + ((a + b*x + c*x^2)^(5/2)*((-2
*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2 + 12*a*c*e^2)*(d + e*x)^(3/2)*(c + (c*d^2)/(
d + e*x)^2 - (b*d*e)/(d + e*x)^2 + (a*e^2)/(d + e*x)^2 - (2*c*d)/(d + e*x) + (b*
e)/(d + e*x)))/(c*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]) + (2*(c*d^2 - b*d*e + a*e^2)*(d + e*
x)*Sqrt[c + (c*d^2)/(d + e*x)^2 - (b*d*e)/(d + e*x)^2 + (a*e^2)/(d + e*x)^2 - (2
*c*d)/(d + e*x) + (b*e)/(d + e*x)]*(((4*I)*Sqrt[2]*c^2*d^2*(2*c*d - b*e + Sqrt[b
^2*e^2 - 4*a*c*e^2])*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b
^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d -
b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[
-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d +
e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^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*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e
^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])]))/((c*d^2 - b*d*e +
a*e^2)*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))
]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)]) - ((
4*I)*Sqrt[2]*b*c*d*e*(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*Sqrt[1 - (2*(c*d^
2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[
1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d +
e*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e
- Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4
*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])] - EllipticF[I*ArcSinh[(Sqr
t[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])
/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b
^2*e^2 - 4*a*c*e^2])]))/((c*d^2 - b*d*e + a*e^2)*Sqrt[-((c*d^2 - b*d*e + a*e^2)/
(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d
+ e*x)^2 + (-2*c*d + b*e)/(d + e*x)]) + ((I/2)*b^2*e^2*(2*c*d - b*e + Sqrt[b^2*e
^2 - 4*a*c*e^2])*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e
^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e
+ Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c
*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]
], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^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*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 -
4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])]))/(Sqrt[2]*(c*d^2 - b*d*
e + a*e^2)*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^
2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)])
+ ((3*I)*Sqrt[2]*a*c*e^2*(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*Sqrt[1 - (2*(
c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*S
qrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(
d + e*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d -
b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2
- 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^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*e^2 - 4*a*c*e^2]
))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sq
rt[b^2*e^2 - 4*a*c*e^2])]))/((c*d^2 - b*d*e + a*e^2)*Sqrt[-((c*d^2 - b*d*e + a*e
^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)
/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)]) + ((8*I)*Sqrt[2]*c^2*d*Sqrt[1 - (2*(c*
d^2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Sqr
t[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d
+ e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*
e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 -
4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])])/(Sqrt[-((c*d^2 - b*d*e +
a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e + a*
e^2)/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)]) - ((4*I)*Sqrt[2]*b*c*e*Sqrt[1 - (2
*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]
*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])
*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d
- b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^
2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])])/(Sqrt[-((c*d^2 - b*d
*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e
+ a*e^2)/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)])))/(c*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])
))/(3*(-b^2 + 4*a*c)^2*e*(a + x*(b + c*x))^(5/2))

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Maple [B]  time = 0.144, size = 12990, normalized size = 22. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out