3.2537 \(\int \frac{1}{(d+e x)^{3/2} \sqrt [4]{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=239 \[ \frac{2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{\sqrt{d+e x} \sqrt [4]{a+b x+c x^2} \left (e \sqrt{b^2-4 a c}-b e+2 c d\right )} \]

[Out]

(2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(((2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(b + Sq
rt[b^2 - 4*a*c] + 2*c*x))/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(b - Sqrt[b^2 - 4
*a*c] + 2*c*x)))^(1/4)*Hypergeometric2F1[-1/2, 1/4, 1/2, (-4*c*Sqrt[b^2 - 4*a*c]
*(d + e*x))/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)
)])/((2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*Sqrt[d + e*x]*(a + b*x + c*x^2)^(1/4))

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Rubi [A]  time = 0.418002, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{\sqrt{d+e x} \sqrt [4]{a+b x+c x^2} \left (e \sqrt{b^2-4 a c}-b e+2 c d\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/4)),x]

[Out]

(2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(((2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(b + Sq
rt[b^2 - 4*a*c] + 2*c*x))/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(b - Sqrt[b^2 - 4
*a*c] + 2*c*x)))^(1/4)*Hypergeometric2F1[-1/2, 1/4, 1/2, (-4*c*Sqrt[b^2 - 4*a*c]
*(d + e*x))/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)
)])/((2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*Sqrt[d + e*x]*(a + b*x + c*x^2)^(1/4))

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Rubi in Sympy [A]  time = 20.687, size = 219, normalized size = 0.92 \[ - \frac{2 \sqrt [4]{\frac{\left (b + 2 c x + \sqrt{- 4 a c + b^{2}}\right ) \left (b e - 2 c d - e \sqrt{- 4 a c + b^{2}}\right )}{\left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right ) \left (b e - 2 c d + e \sqrt{- 4 a c + b^{2}}\right )}} \left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{4 c \left (d + e x\right ) \sqrt{- 4 a c + b^{2}}}{\left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right ) \left (b e - 2 c d + e \sqrt{- 4 a c + b^{2}}\right )}} \right )}}{\sqrt{d + e x} \sqrt [4]{a + b x + c x^{2}} \left (b e - 2 c d - e \sqrt{- 4 a c + b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(1/4),x)

[Out]

-2*((b + 2*c*x + sqrt(-4*a*c + b**2))*(b*e - 2*c*d - e*sqrt(-4*a*c + b**2))/((b
+ 2*c*x - sqrt(-4*a*c + b**2))*(b*e - 2*c*d + e*sqrt(-4*a*c + b**2))))**(1/4)*(b
+ 2*c*x - sqrt(-4*a*c + b**2))*hyper((-1/2, 1/4), (1/2,), 4*c*(d + e*x)*sqrt(-4
*a*c + b**2)/((b + 2*c*x - sqrt(-4*a*c + b**2))*(b*e - 2*c*d + e*sqrt(-4*a*c + b
**2))))/(sqrt(d + e*x)*(a + b*x + c*x**2)**(1/4)*(b*e - 2*c*d - e*sqrt(-4*a*c +
b**2)))

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Mathematica [A]  time = 4.75965, size = 306, normalized size = 1.28 \[ -\frac{2^{3/4} \left (-d \sqrt{e^2 \left (b^2-4 a c\right )}-e x \sqrt{e^2 \left (b^2-4 a c\right )}+2 a e^2+b e (e x-d)-2 c d e x\right ) \sqrt [4]{\frac{b (e x-d) \sqrt{e^2 \left (b^2-4 a c\right )}-2 c d x \sqrt{e^2 \left (b^2-4 a c\right )}+2 a e \left (\sqrt{e^2 \left (b^2-4 a c\right )}-2 c (d+e x)\right )+b^2 e (d+e x)}{e \left (b^2-4 a c\right ) (d+e x)}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{-2 a e^2+2 c d x e+b (d-e x) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{2 \sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}\right )}{3 e \sqrt{d+e x} \sqrt [4]{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/4)),x]

[Out]

-(2^(3/4)*(2*a*e^2 - d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x - e*Sqrt[(b^2 - 4*a*c
)*e^2]*x + b*e*(-d + e*x))*((-2*c*d*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*Sqrt[(b^2 - 4*
a*c)*e^2]*(-d + e*x) + b^2*e*(d + e*x) + 2*a*e*(Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*(d
+ e*x)))/((b^2 - 4*a*c)*e*(d + e*x)))^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (-
2*a*e^2 + 2*c*d*e*x + b*e*(d - e*x) + Sqrt[(b^2 - 4*a*c)*e^2]*(d + e*x))/(2*Sqrt
[(b^2 - 4*a*c)*e^2]*(d + e*x))])/(3*e*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[d + e*x]*(
a + x*(b + c*x))^(1/4))

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Maple [F]  time = 0.164, size = 0, normalized size = 0. \[ \int{1 \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{c{x}^{2}+bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/4),x)

[Out]

int(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(1/4)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x + a)^(1/4)*(e*x + d)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(1/4)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

integral(1/((c*x^2 + b*x + a)^(1/4)*(e*x + d)^(3/2)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right )^{\frac{3}{2}} \sqrt [4]{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(1/4),x)

[Out]

Integral(1/((d + e*x)**(3/2)*(a + b*x + c*x**2)**(1/4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(1/4)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + b*x + a)^(1/4)*(e*x + d)^(3/2)), x)