3.2445 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=716 \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c^2 e^2 \left (28 a^2 e^2-76 a b d e+45 b^2 d^2\right )-b^2 c e^3 (7 b d-15 a e)-4 c^3 d^2 e (64 b d-57 a e)-b^4 e^4+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-4 c e (32 b d-33 a e)-b^2 e^2+128 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{63 c e^5}-\frac{10 \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^3}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt{d+e x}} \]

[Out]

(-2*Sqrt[d + e*x]*(128*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(37*b*d - 36*a*e) - 12*c^2*
d*e*(20*b*d - 11*a*e) - 3*c*e*(32*c^2*d^2 + b^2*e^2 - 4*c*e*(8*b*d - 7*a*e))*x)*
Sqrt[a + b*x + c*x^2])/(63*c*e^5) - (10*Sqrt[d + e*x]*(16*c*d - 15*b*e - 14*c*e*
x)*(a + b*x + c*x^2)^(3/2))/(63*e^3) - (2*(a + b*x + c*x^2)^(5/2))/(e*Sqrt[d + e
*x]) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(128*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(64*b*d
- 57*a*e) - b^2*c*e^3*(7*b*d - 15*a*e) + 3*c^2*e^2*(45*b^2*d^2 - 76*a*b*d*e + 2
8*a^2*e^2))*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*Sq
rt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(63*c^2*e^6*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]
*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(128*c^2*d^2 - b^2*e^2
- 4*c*e*(32*b*d - 33*a*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)])/(63*c^2*e^6*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2
])

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Rubi [A]  time = 2.94648, antiderivative size = 716, 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 \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c^2 e^2 \left (28 a^2 e^2-76 a b d e+45 b^2 d^2\right )-b^2 c e^3 (7 b d-15 a e)-4 c^3 d^2 e (64 b d-57 a e)-b^4 e^4+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-4 c e (32 b d-33 a e)-b^2 e^2+128 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{63 c e^5}-\frac{10 \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^3}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d + e*x]*(128*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(37*b*d - 36*a*e) - 12*c^2*
d*e*(20*b*d - 11*a*e) - 3*c*e*(32*c^2*d^2 + b^2*e^2 - 4*c*e*(8*b*d - 7*a*e))*x)*
Sqrt[a + b*x + c*x^2])/(63*c*e^5) - (10*Sqrt[d + e*x]*(16*c*d - 15*b*e - 14*c*e*
x)*(a + b*x + c*x^2)^(3/2))/(63*e^3) - (2*(a + b*x + c*x^2)^(5/2))/(e*Sqrt[d + e
*x]) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(128*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(64*b*d
- 57*a*e) - b^2*c*e^3*(7*b*d - 15*a*e) + 3*c^2*e^2*(45*b^2*d^2 - 76*a*b*d*e + 2
8*a^2*e^2))*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*Sq
rt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(63*c^2*e^6*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]
*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(128*c^2*d^2 - b^2*e^2
- 4*c*e*(32*b*d - 33*a*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)])/(63*c^2*e^6*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((c*x**2+b*x+a)**(5/2)/(e*x+d)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 14.6863, size = 7946, normalized size = 11.1 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [B]  time = 0.076, size = 9187, normalized size = 12.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((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)/(e*x + d)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(5/2)/(e*x+d)**(3/2),x)

[Out]

Integral((a + b*x + c*x**2)**(5/2)/(d + e*x)**(3/2), x)

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

[Out]

Timed out