3.3017 \(\int \frac{1}{(a+b x)^2 \sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx\)

Optimal. Leaf size=114 \[ -\frac{3 d (c+d x)^{2/3} \sqrt [3]{-\frac{a d+b c+2 b d x}{b c-a d}} F_1\left (\frac{2}{3};\frac{4}{3},2;\frac{5}{3};\frac{2 b (c+d x)}{b c-a d},\frac{b (c+d x)}{b c-a d}\right )}{2 (b c-a d)^3 \sqrt [3]{a d+b c+2 b d x}} \]

[Out]

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

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Rubi [A]  time = 0.345246, antiderivative size = 114, 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 \[ -\frac{3 d (c+d x)^{2/3} \sqrt [3]{-\frac{a d+b c+2 b d x}{b c-a d}} F_1\left (\frac{2}{3};\frac{4}{3},2;\frac{5}{3};\frac{2 b (c+d x)}{b c-a d},\frac{b (c+d x)}{b c-a d}\right )}{2 (b c-a d)^3 \sqrt [3]{a d+b c+2 b d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.2905, size = 100, normalized size = 0.88 \[ \frac{3 d \left (c + d x\right )^{\frac{2}{3}} \left (a d + b c + 2 b d x\right )^{\frac{2}{3}} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{4}{3},2,\frac{5}{3},\frac{b \left (- 2 c - 2 d x\right )}{a d - b c},\frac{b \left (- c - d x\right )}{a d - b c} \right )}}{2 \left (\frac{a d + b c + 2 b d x}{a d - b c}\right )^{\frac{2}{3}} \left (a d - b c\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 3.66909, size = 605, normalized size = 5.31 \[ \frac{(c+d x)^{2/3} \left (\frac{d \left (7 \left (\frac{16 (b c-a d)^2 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )}{d (a+b x) \left (16 b (c+d x) F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )+(b c-a d) \left (6 F_1\left (\frac{8}{3};\frac{1}{3},2;\frac{11}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )+F_1\left (\frac{8}{3};\frac{4}{3},1;\frac{11}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )\right )\right )}-\frac{5 a d}{b c+b d x}+\frac{5 c}{c+d x}-10\right )-\frac{400 b (c+d x) (b c-a d) F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )}{d (a+b x) \left (10 b (c+d x) F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )+(b c-a d) \left (6 F_1\left (\frac{5}{3};\frac{1}{3},2;\frac{8}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )+F_1\left (\frac{5}{3};\frac{4}{3},1;\frac{8}{3};\frac{b c-a d}{2 b c+2 b d x},\frac{b c-a d}{b c+b d x}\right )\right )\right )}\right )}{(a d-b c)^3}-\frac{5 (13 a d+b (c+14 d x))}{(a+b x) (b c-a d)^3}\right )}{5 \sqrt [3]{a d+b (c+2 d x)}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((c + d*x)^(2/3)*((-5*(13*a*d + b*(c + 14*d*x)))/((b*c - a*d)^3*(a + b*x)) + (d*
((-400*b*(b*c - a*d)*(c + d*x)*AppellF1[2/3, 1/3, 1, 5/3, (b*c - a*d)/(2*b*c + 2
*b*d*x), (b*c - a*d)/(b*c + b*d*x)])/(d*(a + b*x)*(10*b*(c + d*x)*AppellF1[2/3,
1/3, 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)*(6*AppellF1[5/3, 1/3, 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, 4/3, 1, 8/3, (b*c - a*d)/(2*b*c + 2*b*d*x), (b*c -
a*d)/(b*c + b*d*x)]))) + 7*(-10 + (5*c)/(c + d*x) - (5*a*d)/(b*c + b*d*x) + (16*
(b*c - a*d)^2*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)])/(d*(a + b*x)*(16*b*(c + 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)] + (b*c - a*d)*(6*AppellF1[
8/3, 1/3, 2, 11/3, (b*c - a*d)/(2*b*c + 2*b*d*x), (b*c - a*d)/(b*c + b*d*x)] + A
ppellF1[8/3, 4/3, 1, 11/3, (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))/(5*(a*d + b*(c + 2*d*x))^(1/3))

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( bx+a \right ) ^{2}}{\frac{1}{\sqrt [3]{dx+c}}} \left ( 2\,bdx+ad+bc \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/(b*x+a)^2/(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. \[ \int \frac{1}{{\left (2 \, b d x + b c + a d\right )}^{\frac{4}{3}}{\left (b x + a\right )}^{2}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

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)^2*(d*x + c)^(1/3)),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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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)^2*(d*x + c)^(1/3)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError