3.2437 \(\int \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=712 \[ -\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c^2 e^2 \left (-28 a^2 e^2-20 a b d e+3 b^2 d^2\right )+b^2 c e^3 (57 a e+7 b d)-4 c^3 d^2 e (8 b d-15 a e)-8 b^4 e^4+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 c^3 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 c^3 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-6 c e x \left (-c e (7 a e+b d)+2 b^2 e^2+c^2 d^2\right )-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (3 a e+b d)-4 b^3 e^3+8 c^3 d^3\right )}{315 c^2 e^3}+\frac{2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{21 c e} \]

[Out]

(2*Sqrt[d + e*x]*(8*c^3*d^3 - 4*b^3*e^3 - 3*c^2*d*e*(5*b*d - 8*a*e) + 3*b*c*e^2*
(b*d + 3*a*e) - 6*c*e*(c^2*d^2 + 2*b^2*e^2 - c*e*(b*d + 7*a*e))*x)*Sqrt[a + b*x
+ c*x^2])/(315*c^2*e^3) - (2*(2*c*d - b*e)*Sqrt[d + e*x]*(a + b*x + c*x^2)^(3/2)
)/(21*c*e) + (2*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(9*e) - (Sqrt[2]*Sqrt[b
^2 - 4*a*c]*(16*c^4*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*(8*b*d - 15*a*e) + b^2*c*e^3*(
7*b*d + 57*a*e) + 3*c^2*e^2*(3*b^2*d^2 - 20*a*b*d*e - 28*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*Sqrt[b^2 - 4*a*c]*e)/(2*c*d
- (b + Sqrt[b^2 - 4*a*c])*e)])/(315*c^3*e^4*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sq
rt[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)*(c*d^2 - b*d*e + a*e^2)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*Sqr
t[(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
)])/(315*c^3*e^4*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 3.33469, antiderivative size = 712, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c^2 e^2 \left (-28 a^2 e^2-20 a b d e+3 b^2 d^2\right )+b^2 c e^3 (57 a e+7 b d)-4 c^3 d^2 e (8 b d-15 a e)-8 b^4 e^4+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 c^3 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 c^3 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-6 c e x \left (-c e (7 a e+b d)+2 b^2 e^2+c^2 d^2\right )-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (3 a e+b d)-4 b^3 e^3+8 c^3 d^3\right )}{315 c^2 e^3}+\frac{2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{21 c e} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(8*c^3*d^3 - 4*b^3*e^3 - 3*c^2*d*e*(5*b*d - 8*a*e) + 3*b*c*e^2*
(b*d + 3*a*e) - 6*c*e*(c^2*d^2 + 2*b^2*e^2 - c*e*(b*d + 7*a*e))*x)*Sqrt[a + b*x
+ c*x^2])/(315*c^2*e^3) - (2*(2*c*d - b*e)*Sqrt[d + e*x]*(a + b*x + c*x^2)^(3/2)
)/(21*c*e) + (2*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(9*e) - (Sqrt[2]*Sqrt[b
^2 - 4*a*c]*(16*c^4*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*(8*b*d - 15*a*e) + b^2*c*e^3*(
7*b*d + 57*a*e) + 3*c^2*e^2*(3*b^2*d^2 - 20*a*b*d*e - 28*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*Sqrt[b^2 - 4*a*c]*e)/(2*c*d
- (b + Sqrt[b^2 - 4*a*c])*e)])/(315*c^3*e^4*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sq
rt[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)*(c*d^2 - b*d*e + a*e^2)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*Sqr
t[(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
)])/(315*c^3*e^4*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)**(1/2)*(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 14.7979, size = 7541, normalized size = 10.59 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [B]  time = 0.049, size = 9177, normalized size = 12.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out