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

Optimal. Leaf size=666 \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-c^2 f \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 e \sqrt{a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )}{315 c g^4}-\frac{4 \sqrt{a+c x^2} \sqrt{f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (-35 d^3 g^3+168 d^2 e f g^2-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{315 c g^4}-\frac{4 e^2 \sqrt{a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{63 g^4}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 g} \]

[Out]

(-4*(9*a*e^2*g^2*(2*e*f - 5*d*g) + c*(76*e^3*f^3 - 204*d*e^2*f^2*g + 168*d^2*e*f
*g^2 - 35*d^3*g^3))*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(315*c*g^4) + (2*(d + e*x)^3*
Sqrt[f + g*x]*Sqrt[a + c*x^2])/(9*g) + (4*e*(7*a*e^2*g^2 + c*(64*e^2*f^2 - 111*d
*e*f*g + 42*d^2*g^2))*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(315*c*g^4) - (4*e^2*(4*e
*f - 3*d*g)*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/(63*g^4) + (4*Sqrt[-a]*(21*a^2*e^3*
g^4 - 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e*f*g + 63*d^2*g^2) - c^2*f*(64*e^3*f^3 - 2
16*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*Sqrt[f + g*x]*Sqrt[1 + (c*x^2)/
a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*
Sqrt[c]*f - a*g)])/(315*c^(3/2)*g^5*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-
a]*g)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*(c*f^2 + a*g^2)*(9*a*e^2*g^2*(2*e*f - 5*d*
g) - c*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*Sqrt[(Sqr
t[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[S
qrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3
15*c^(3/2)*g^5*Sqrt[f + g*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 3.01197, antiderivative size = 666, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-c^2 f \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 e \sqrt{a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )}{315 c g^4}-\frac{4 \sqrt{a+c x^2} \sqrt{f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (-35 d^3 g^3+168 d^2 e f g^2-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{315 c g^4}-\frac{4 e^2 \sqrt{a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{63 g^4}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 g} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(9*a*e^2*g^2*(2*e*f - 5*d*g) + c*(76*e^3*f^3 - 204*d*e^2*f^2*g + 168*d^2*e*f
*g^2 - 35*d^3*g^3))*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(315*c*g^4) + (2*(d + e*x)^3*
Sqrt[f + g*x]*Sqrt[a + c*x^2])/(9*g) + (4*e*(7*a*e^2*g^2 + c*(64*e^2*f^2 - 111*d
*e*f*g + 42*d^2*g^2))*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(315*c*g^4) - (4*e^2*(4*e
*f - 3*d*g)*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/(63*g^4) + (4*Sqrt[-a]*(21*a^2*e^3*
g^4 - 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e*f*g + 63*d^2*g^2) - c^2*f*(64*e^3*f^3 - 2
16*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*Sqrt[f + g*x]*Sqrt[1 + (c*x^2)/
a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*
Sqrt[c]*f - a*g)])/(315*c^(3/2)*g^5*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-
a]*g)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*(c*f^2 + a*g^2)*(9*a*e^2*g^2*(2*e*f - 5*d*
g) - c*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*Sqrt[(Sqr
t[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[S
qrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(3
15*c^(3/2)*g^5*Sqrt[f + g*x]*Sqrt[a + 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*(c*x**2+a)**(1/2)/(g*x+f)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 13.5582, size = 5379, normalized size = 8.08 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [B]  time = 0.063, size = 5079, normalized size = 7.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError