3.1410 \(\int \frac{\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{10/3}} \, dx\)

Optimal. Leaf size=99 \[ \frac{3 \left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2} \, _2F_1\left (-\frac{4}{3},-\frac{7}{6};-\frac{1}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{56 c^2 d \sqrt [3]{1-\frac{(b+2 c x)^2}{b^2-4 a c}} (d (b+2 c x))^{7/3}} \]

[Out]

(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(1/3)*Hypergeometric2F1[-4/3, -7/6, -1/6, (b
+ 2*c*x)^2/(b^2 - 4*a*c)])/(56*c^2*d*(d*(b + 2*c*x))^(7/3)*(1 - (b + 2*c*x)^2/(b
^2 - 4*a*c))^(1/3))

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Rubi [A]  time = 0.29168, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{3 \left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2} \, _2F_1\left (-\frac{4}{3},-\frac{7}{6};-\frac{1}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{56 c^2 d \sqrt [3]{1-\frac{(b+2 c x)^2}{b^2-4 a c}} (d (b+2 c x))^{7/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(10/3),x]

[Out]

(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(1/3)*Hypergeometric2F1[-4/3, -7/6, -1/6, (b
+ 2*c*x)^2/(b^2 - 4*a*c)])/(56*c^2*d*(d*(b + 2*c*x))^(7/3)*(1 - (b + 2*c*x)^2/(b
^2 - 4*a*c))^(1/3))

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Rubi in Sympy [A]  time = 31.1135, size = 105, normalized size = 1.06 \[ \frac{3 \left (- 4 a c + b^{2}\right ) \sqrt [3]{a - \frac{b^{2}}{4 c} + \frac{\left (b + 2 c x\right )^{2}}{4 c}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, - \frac{7}{6} \\ - \frac{1}{6} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{56 c^{2} d \left (b d + 2 c d x\right )^{\frac{7}{3}} \sqrt [3]{\frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(4/3)/(2*c*d*x+b*d)**(10/3),x)

[Out]

3*(-4*a*c + b**2)*(a - b**2/(4*c) + (b + 2*c*x)**2/(4*c))**(1/3)*hyper((-4/3, -7
/6), (-1/6,), -(b + 2*c*x)**2/(4*a*c - b**2))/(56*c**2*d*(b*d + 2*c*d*x)**(7/3)*
((b + 2*c*x)**2/(4*a*c - b**2) + 1)**(1/3))

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Mathematica [A]  time = 0.603119, size = 170, normalized size = 1.72 \[ \frac{6 \sqrt [3]{2} (b+2 c x)^4 \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{5}{6};\frac{11}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )-15 c \left (a^2 c+2 a \left (b^2+5 b c x+5 c^2 x^2\right )+x \left (2 b^3+11 b^2 c x+18 b c^2 x^2+9 c^3 x^3\right )\right )}{70 c^3 d (a+x (b+c x))^{2/3} (d (b+2 c x))^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(10/3),x]

[Out]

(-15*c*(a^2*c + 2*a*(b^2 + 5*b*c*x + 5*c^2*x^2) + x*(2*b^3 + 11*b^2*c*x + 18*b*c
^2*x^2 + 9*c^3*x^3)) + 6*2^(1/3)*(b + 2*c*x)^4*((c*(a + x*(b + c*x)))/(-b^2 + 4*
a*c))^(2/3)*HypergeometricPFQ[{2/3, 5/6}, {11/6}, (b + 2*c*x)^2/(b^2 - 4*a*c)])/
(70*c^3*d*(d*(b + 2*c*x))^(7/3)*(a + x*(b + c*x))^(2/3))

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Maple [F]  time = 0.155, size = 0, normalized size = 0. \[ \int{1 \left ( c{x}^{2}+bx+a \right ) ^{{\frac{4}{3}}} \left ( 2\,cdx+bd \right ) ^{-{\frac{10}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(10/3),x)

[Out]

int((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(10/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac{10}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(10/3),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(10/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}}{{\left (8 \, c^{3} d^{3} x^{3} + 12 \, b c^{2} d^{3} x^{2} + 6 \, b^{2} c d^{3} x + b^{3} d^{3}\right )}{\left (2 \, c d x + b d\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(10/3),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(4/3)/((8*c^3*d^3*x^3 + 12*b*c^2*d^3*x^2 + 6*b^2*c*d^
3*x + b^3*d^3)*(2*c*d*x + b*d)^(1/3)), x)

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Sympy [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)**(4/3)/(2*c*d*x+b*d)**(10/3),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.44285, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(10/3),x, algorithm="giac")

[Out]

Done