3.2430 \(\int (d+e x)^{3/2} \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=581 \[ \frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right )}{105 c^2 e}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c} \]

[Out]

(2*Sqrt[d + e*x]*(3*c^2*d^2 - 4*b^2*e^2 + c*e*(9*b*d - 5*a*e) + 12*c*e*(2*c*d -
b*e)*x)*Sqrt[a + b*x + c*x^2])/(105*c^2*e) + (2*e*Sqrt[d + e*x]*(a + b*x + c*x^2
)^(3/2))/(7*c) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(3*c^2*d^2 + 8*b^2*e^2
- c*e*(3*b*d + 29*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)/Sqrt[b^2 - 4*a*c]]/Sqr
t[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(105*c^3*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]
) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(3*c^2*d^2 + 2*b^2*e^2
- c*e*(3*b*d + 5*a*e))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*S
qrt[-((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)])/(105*c^3*e^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 2.22615, antiderivative size = 581, 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 \[ \frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right )}{105 c^2 e}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(3*c^2*d^2 - 4*b^2*e^2 + c*e*(9*b*d - 5*a*e) + 12*c*e*(2*c*d -
b*e)*x)*Sqrt[a + b*x + c*x^2])/(105*c^2*e) + (2*e*Sqrt[d + e*x]*(a + b*x + c*x^2
)^(3/2))/(7*c) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(3*c^2*d^2 + 8*b^2*e^2
- c*e*(3*b*d + 29*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)/Sqrt[b^2 - 4*a*c]]/Sqr
t[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(105*c^3*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]
) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(3*c^2*d^2 + 2*b^2*e^2
- c*e*(3*b*d + 5*a*e))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*S
qrt[-((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)])/(105*c^3*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((e*x+d)**(3/2)*(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 13.6195, size = 5328, normalized size = 9.17 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [B]  time = 0.132, size = 6517, normalized size = 11.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(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. \[ \int \left (d + e x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(3/2)*(c*x**2+b*x+a)**(1/2),x)

[Out]

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

[Out]

Timed out