3.3016 \(\int \frac{1}{(a+b x) \sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx\)

Optimal. Leaf size=113 \[ \frac{3 (c+d x)^{2/3} \sqrt [3]{-\frac{a d+b c+2 b d x}{b c-a d}} F_1\left (\frac{2}{3};\frac{4}{3},1;\frac{5}{3};\frac{2 b (c+d x)}{b c-a d},\frac{b (c+d x)}{b c-a d}\right )}{2 (b c-a d)^2 \sqrt [3]{a d+b c+2 b d x}} \]

[Out]

(3*(c + d*x)^(2/3)*(-((b*c + a*d + 2*b*d*x)/(b*c - a*d)))^(1/3)*AppellF1[2/3, 4/
3, 1, 5/3, (2*b*(c + d*x))/(b*c - a*d), (b*(c + d*x))/(b*c - a*d)])/(2*(b*c - a*
d)^2*(b*c + a*d + 2*b*d*x)^(1/3))

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Rubi [A]  time = 0.373524, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{3 (c+d x)^{2/3} \sqrt [3]{-\frac{a d+b c+2 b d x}{b c-a d}} F_1\left (\frac{2}{3};\frac{4}{3},1;\frac{5}{3};\frac{2 b (c+d x)}{b c-a d},\frac{b (c+d x)}{b c-a d}\right )}{2 (b c-a d)^2 \sqrt [3]{a d+b c+2 b d x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(c + d*x)^(2/3)*(-((b*c + a*d + 2*b*d*x)/(b*c - a*d)))^(1/3)*AppellF1[2/3, 4/
3, 1, 5/3, (2*b*(c + d*x))/(b*c - a*d), (b*(c + d*x))/(b*c - a*d)])/(2*(b*c - a*
d)^2*(b*c + a*d + 2*b*d*x)^(1/3))

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Rubi in Sympy [A]  time = 31.2635, size = 99, normalized size = 0.88 \[ \frac{3 \left (c + d x\right )^{\frac{2}{3}} \left (a d + b c + 2 b d x\right )^{\frac{2}{3}} \operatorname{appellf_{1}}{\left (\frac{2}{3},1,\frac{4}{3},\frac{5}{3},\frac{b \left (- c - d x\right )}{a d - b c},\frac{b \left (- 2 c - 2 d x\right )}{a d - b c} \right )}}{2 \left (\frac{a d + b c + 2 b d x}{a d - b c}\right )^{\frac{2}{3}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(c + d*x)**(2/3)*(a*d + b*c + 2*b*d*x)**(2/3)*appellf1(2/3, 1, 4/3, 5/3, b*(-c
- d*x)/(a*d - b*c), b*(-2*c - 2*d*x)/(a*d - b*c))/(2*((a*d + b*c + 2*b*d*x)/(a*
d - b*c))**(2/3)*(a*d - b*c)**3)

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Mathematica [B]  time = 1.37733, size = 395, normalized size = 3.5 \[ \frac{6 d (a+b x) \left (3 F_1\left (\frac{5}{3};-\frac{2}{3},2;\frac{8}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )-F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )\right )-15 b (c+d x) F_1\left (\frac{2}{3};-\frac{2}{3},1;\frac{5}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )}{2 b d (a+b x) \sqrt [3]{c+d x} \sqrt [3]{a d+b (c+2 d x)} \left (5 b (c+d x) F_1\left (\frac{2}{3};-\frac{2}{3},1;\frac{5}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )+(b c-a d) \left (3 F_1\left (\frac{5}{3};-\frac{2}{3},2;\frac{8}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )-F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-15*b*(c + d*x)*AppellF1[2/3, -2/3, 1, 5/3, (b*c - a*d)/(2*b*c + 2*b*d*x), (b*c
- a*d)/(b*c + b*d*x)] + 6*d*(a + b*x)*(3*AppellF1[5/3, -2/3, 2, 8/3, (b*c - a*d
)/(2*b*c + 2*b*d*x), (b*c - a*d)/(b*c + b*d*x)] - AppellF1[5/3, 1/3, 1, 8/3, (b*
c - a*d)/(2*b*c + 2*b*d*x), (b*c - a*d)/(b*c + b*d*x)]))/(2*b*d*(a + b*x)*(c + d
*x)^(1/3)*(a*d + b*(c + 2*d*x))^(1/3)*(5*b*(c + d*x)*AppellF1[2/3, -2/3, 1, 5/3,
(b*c - a*d)/(2*b*c + 2*b*d*x), (b*c - a*d)/(b*c + b*d*x)] + (b*c - a*d)*(3*Appe
llF1[5/3, -2/3, 2, 8/3, (b*c - a*d)/(2*b*c + 2*b*d*x), (b*c - a*d)/(b*c + b*d*x)
] - AppellF1[5/3, 1/3, 1, 8/3, (b*c - a*d)/(2*b*c + 2*b*d*x), (b*c - a*d)/(b*c +
b*d*x)])))

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \[ \int{\frac{1}{bx+a}{\frac{1}{\sqrt [3]{dx+c}}} \left ( 2\,bdx+ad+bc \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError