3.900 \(\int \frac{(d+e x)^3 \sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=774 \[ -\frac{2 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{2 e \sqrt{f+g x} \sqrt{a+b x+c x^2} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (-\left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )\right )}{105 c^3 g^2}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e g \left (a e g (189 d g+19 e f)-b \left (-210 d^2 g^2-63 d e f g+9 e^2 f^2\right )\right )-8 b c e^2 g^2 (13 a e g+21 b d g+2 b e f)+48 b^3 e^3 g^3+c^3 \left (-\left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e^2 (f+g x)^{3/2} \sqrt{a+b x+c x^2} (-6 b e g+11 c d g+c e f)}{35 c^2 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c} \]

[Out]

(2*e*(24*b^2*e^2*g^2 + c*e*g*(13*b*e*f - 84*b*d*g - 25*a*e*g) - c^2*(7*e^2*f^2 +
12*d*e*f*g - 90*d^2*g^2))*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(105*c^3*g^2) +
(2*e*(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(7*c) + (2*e^2*(c*e*f + 11
*c*d*g - 6*b*e*g)*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(35*c^2*g^2) - (Sqrt[2]
*Sqrt[b^2 - 4*a*c]*(48*b^3*e^3*g^3 - 8*b*c*e^2*g^2*(2*b*e*f + 21*b*d*g + 13*a*e*
g) - c^3*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3*g^3) + c^2*e*g*
(a*e*g*(19*e*f + 189*d*g) - b*(9*e^2*f^2 - 63*d*e*f*g - 210*d^2*g^2)))*Sqrt[f +
g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqr
t[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2
*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(105*c^4*g^3*Sqrt[(c*(f + g*x))/(2*c*f - (b
+ Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*e
*(c*f^2 - b*f*g + a*g^2)*(24*b^2*e^2*g^2 + c*e*g*(13*b*e*f - 84*b*d*g - 25*a*e*g
) + c^2*(8*e^2*f^2 - 42*d*e*f*g + 105*d^2*g^2))*Sqrt[(c*(f + g*x))/(2*c*f - (b +
Sqrt[b^2 - 4*a*c])*g)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[A
rcSin[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]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(105*c^4*g^3*Sqrt[f + g*x
]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 4.46528, antiderivative size = 774, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ -\frac{2 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{2 e \sqrt{f+g x} \sqrt{a+b x+c x^2} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (-\left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )\right )}{105 c^3 g^2}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e g \left (a e g (189 d g+19 e f)-b \left (-210 d^2 g^2-63 d e f g+9 e^2 f^2\right )\right )-8 b c e^2 g^2 (13 a e g+21 b d g+2 b e f)+48 b^3 e^3 g^3+c^3 \left (-\left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e^2 (f+g x)^{3/2} \sqrt{a+b x+c x^2} (-6 b e g+11 c d g+c e f)}{35 c^2 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*(24*b^2*e^2*g^2 + c*e*g*(13*b*e*f - 84*b*d*g - 25*a*e*g) - c^2*(7*e^2*f^2 +
12*d*e*f*g - 90*d^2*g^2))*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(105*c^3*g^2) +
(2*e*(d + e*x)^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(7*c) + (2*e^2*(c*e*f + 11
*c*d*g - 6*b*e*g)*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(35*c^2*g^2) - (Sqrt[2]
*Sqrt[b^2 - 4*a*c]*(48*b^3*e^3*g^3 - 8*b*c*e^2*g^2*(2*b*e*f + 21*b*d*g + 13*a*e*
g) - c^3*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3*g^3) + c^2*e*g*
(a*e*g*(19*e*f + 189*d*g) - b*(9*e^2*f^2 - 63*d*e*f*g - 210*d^2*g^2)))*Sqrt[f +
g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqr
t[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2
*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(105*c^4*g^3*Sqrt[(c*(f + g*x))/(2*c*f - (b
+ Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*e
*(c*f^2 - b*f*g + a*g^2)*(24*b^2*e^2*g^2 + c*e*g*(13*b*e*f - 84*b*d*g - 25*a*e*g
) + c^2*(8*e^2*f^2 - 42*d*e*f*g + 105*d^2*g^2))*Sqrt[(c*(f + g*x))/(2*c*f - (b +
Sqrt[b^2 - 4*a*c])*g)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[A
rcSin[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]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(105*c^4*g^3*Sqrt[f + g*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*(g*x+f)**(1/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 16.2576, size = 10649, normalized size = 13.76 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [B]  time = 0.086, size = 14978, normalized size = 19.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(g*x+f)^(1/2)/(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. \[ \int \frac{{\left (e x + d\right )}^{3} \sqrt{g x + f}}{\sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out