3.2987 \(\int \frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x} \, dx\)

Optimal. Leaf size=409 \[ \frac{\log (a+b x) (-a d f-2 b c f+3 b d e)}{6 b^{2/3} \sqrt [3]{d} f^2}+\frac{(-a d f-2 b c f+3 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} \sqrt [3]{d} f^2}+\frac{(-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} \sqrt [3]{d} f^2}+\frac{\sqrt [3]{b e-a f} (d e-c f)^{2/3} \log (e+f x)}{2 f^2}-\frac{3 \sqrt [3]{b e-a f} (d e-c f)^{2/3} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f^2}-\frac{\sqrt{3} \sqrt [3]{b e-a f} (d e-c f)^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{f^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{f} \]

[Out]

((a + b*x)^(1/3)*(c + d*x)^(2/3))/f + ((3*b*d*e - 2*b*c*f - a*d*f)*ArcTan[1/Sqrt
[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*d^(1/3)*(a + b*x)^(1/3))])/(Sqrt[3]*b
^(2/3)*d^(1/3)*f^2) - (Sqrt[3]*(b*e - a*f)^(1/3)*(d*e - c*f)^(2/3)*ArcTan[1/Sqrt
[3] + (2*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*(d*e - c*f)^(1/3)*(a + b*x)
^(1/3))])/f^2 + ((3*b*d*e - 2*b*c*f - a*d*f)*Log[a + b*x])/(6*b^(2/3)*d^(1/3)*f^
2) + ((b*e - a*f)^(1/3)*(d*e - c*f)^(2/3)*Log[e + f*x])/(2*f^2) - (3*(b*e - a*f)
^(1/3)*(d*e - c*f)^(2/3)*Log[-(a + b*x)^(1/3) + ((b*e - a*f)^(1/3)*(c + d*x)^(1/
3))/(d*e - c*f)^(1/3)])/(2*f^2) + ((3*b*d*e - 2*b*c*f - a*d*f)*Log[-1 + (b^(1/3)
*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(2*b^(2/3)*d^(1/3)*f^2)

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Rubi [A]  time = 1.04945, antiderivative size = 409, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\log (a+b x) (-a d f-2 b c f+3 b d e)}{6 b^{2/3} \sqrt [3]{d} f^2}+\frac{(-a d f-2 b c f+3 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} \sqrt [3]{d} f^2}+\frac{(-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} \sqrt [3]{d} f^2}+\frac{\sqrt [3]{b e-a f} (d e-c f)^{2/3} \log (e+f x)}{2 f^2}-\frac{3 \sqrt [3]{b e-a f} (d e-c f)^{2/3} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f^2}-\frac{\sqrt{3} \sqrt [3]{b e-a f} (d e-c f)^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{f^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{f} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/3)*(c + d*x)^(2/3))/f + ((3*b*d*e - 2*b*c*f - a*d*f)*ArcTan[1/Sqrt
[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*d^(1/3)*(a + b*x)^(1/3))])/(Sqrt[3]*b
^(2/3)*d^(1/3)*f^2) - (Sqrt[3]*(b*e - a*f)^(1/3)*(d*e - c*f)^(2/3)*ArcTan[1/Sqrt
[3] + (2*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*(d*e - c*f)^(1/3)*(a + b*x)
^(1/3))])/f^2 + ((3*b*d*e - 2*b*c*f - a*d*f)*Log[a + b*x])/(6*b^(2/3)*d^(1/3)*f^
2) + ((b*e - a*f)^(1/3)*(d*e - c*f)^(2/3)*Log[e + f*x])/(2*f^2) - (3*(b*e - a*f)
^(1/3)*(d*e - c*f)^(2/3)*Log[-(a + b*x)^(1/3) + ((b*e - a*f)^(1/3)*(c + d*x)^(1/
3))/(d*e - c*f)^(1/3)])/(2*f^2) + ((3*b*d*e - 2*b*c*f - a*d*f)*Log[-1 + (b^(1/3)
*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(2*b^(2/3)*d^(1/3)*f^2)

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Rubi in Sympy [A]  time = 99.9272, size = 388, normalized size = 0.95 \[ \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}}}{f} - \frac{\sqrt [3]{a f - b e} \left (c f - d e\right )^{\frac{2}{3}} \log{\left (e + f x \right )}}{2 f^{2}} + \frac{3 \sqrt [3]{a f - b e} \left (c f - d e\right )^{\frac{2}{3}} \log{\left (- \sqrt [3]{a + b x} + \frac{\sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{\sqrt [3]{c f - d e}} \right )}}{2 f^{2}} + \frac{\sqrt{3} \sqrt [3]{a f - b e} \left (c f - d e\right )^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{3 \sqrt [3]{a + b x} \sqrt [3]{c f - d e}} \right )}}{f^{2}} - \frac{\left (a d f + 2 b c f - 3 b d e\right ) \log{\left (a + b x \right )}}{6 b^{\frac{2}{3}} \sqrt [3]{d} f^{2}} - \frac{\left (a d f + 2 b c f - 3 b d e\right ) \log{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x}}{\sqrt [3]{d} \sqrt [3]{a + b x}} - 1 \right )}}{2 b^{\frac{2}{3}} \sqrt [3]{d} f^{2}} - \frac{\sqrt{3} \left (a d f + 2 b c f - 3 b d e\right ) \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{c + d x}}{3 \sqrt [3]{d} \sqrt [3]{a + b x}} + \frac{\sqrt{3}}{3} \right )}}{3 b^{\frac{2}{3}} \sqrt [3]{d} f^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(1/3)*(c + d*x)**(2/3)/f - (a*f - b*e)**(1/3)*(c*f - d*e)**(2/3)*log(
e + f*x)/(2*f**2) + 3*(a*f - b*e)**(1/3)*(c*f - d*e)**(2/3)*log(-(a + b*x)**(1/3
) + (c + d*x)**(1/3)*(a*f - b*e)**(1/3)/(c*f - d*e)**(1/3))/(2*f**2) + sqrt(3)*(
a*f - b*e)**(1/3)*(c*f - d*e)**(2/3)*atan(sqrt(3)/3 + 2*sqrt(3)*(c + d*x)**(1/3)
*(a*f - b*e)**(1/3)/(3*(a + b*x)**(1/3)*(c*f - d*e)**(1/3)))/f**2 - (a*d*f + 2*b
*c*f - 3*b*d*e)*log(a + b*x)/(6*b**(2/3)*d**(1/3)*f**2) - (a*d*f + 2*b*c*f - 3*b
*d*e)*log(b**(1/3)*(c + d*x)**(1/3)/(d**(1/3)*(a + b*x)**(1/3)) - 1)/(2*b**(2/3)
*d**(1/3)*f**2) - sqrt(3)*(a*d*f + 2*b*c*f - 3*b*d*e)*atan(2*sqrt(3)*b**(1/3)*(c
+ d*x)**(1/3)/(3*d**(1/3)*(a + b*x)**(1/3)) + sqrt(3)/3)/(3*b**(2/3)*d**(1/3)*f
**2)

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Mathematica [C]  time = 6.06163, size = 541, normalized size = 1.32 \[ \frac{(c+d x)^{2/3} \left (5 (a+b x)-\frac{4 (b c-a d) \left (-\frac{5 b f (c+d x) (c f-d e) F_1\left (1;\frac{2}{3},1;2;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )}{6 b f (c+d x) F_1\left (1;\frac{2}{3},1;2;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+b (3 c f-3 d e) F_1\left (2;\frac{2}{3},2;3;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+2 f (b c-a d) F_1\left (2;\frac{5}{3},1;3;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )}-\frac{2 (d e-c f) (-a d f-2 b c f+3 b d e) F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};\frac{b (c+d x)}{b c-a d},\frac{f (c+d x)}{c f-d e}\right )}{\frac{8 (b c-a d) (c f-d e) F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};\frac{b (c+d x)}{b c-a d},\frac{f (c+d x)}{c f-d e}\right )}{c+d x}+3 f (b c-a d) F_1\left (\frac{8}{3};\frac{2}{3},2;\frac{11}{3};\frac{b (c+d x)}{b c-a d},\frac{f (c+d x)}{c f-d e}\right )+2 b (c f-d e) F_1\left (\frac{8}{3};\frac{5}{3},1;\frac{11}{3};\frac{b (c+d x)}{b c-a d},\frac{f (c+d x)}{c f-d e}\right )}\right )}{d^2 (e+f x)}\right )}{5 f (a+b x)^{2/3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((c + d*x)^(2/3)*(5*(a + b*x) - (4*(b*c - a*d)*((-5*b*f*(-(d*e) + c*f)*(c + d*x)
*AppellF1[1, 2/3, 1, 2, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))]
)/(6*b*f*(c + d*x)*AppellF1[1, 2/3, 1, 2, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c
*f)/(f*(c + d*x))] + b*(-3*d*e + 3*c*f)*AppellF1[2, 2/3, 2, 3, (b*c - a*d)/(b*c
+ b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] + 2*(b*c - a*d)*f*AppellF1[2, 5/3, 1, 3,
(b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))]) - (2*(d*e - c*f)*(3*b
*d*e - 2*b*c*f - a*d*f)*AppellF1[5/3, 2/3, 1, 8/3, (b*(c + d*x))/(b*c - a*d), (f
*(c + d*x))/(-(d*e) + c*f)])/((8*(b*c - a*d)*(-(d*e) + c*f)*AppellF1[5/3, 2/3, 1
, 8/3, (b*(c + d*x))/(b*c - a*d), (f*(c + d*x))/(-(d*e) + c*f)])/(c + d*x) + 3*(
b*c - a*d)*f*AppellF1[8/3, 2/3, 2, 11/3, (b*(c + d*x))/(b*c - a*d), (f*(c + d*x)
)/(-(d*e) + c*f)] + 2*b*(-(d*e) + c*f)*AppellF1[8/3, 5/3, 1, 11/3, (b*(c + d*x))
/(b*c - a*d), (f*(c + d*x))/(-(d*e) + c*f)])))/(d^2*(e + f*x))))/(5*f*(a + b*x)^
(2/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 4.21405, size = 1206, normalized size = 2.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(6*sqrt(3)*(-b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*f + sqrt(
3)*(3*b*d*e - (2*b*c + a*d)*f)*log((b^2*d*x + b^2*c - (-b^2*d)^(1/3)*(b*x + a)^(
1/3)*(d*x + c)^(2/3)*b + (-b^2*d)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3))/(d*x +
c)) - 2*sqrt(3)*(3*b*d*e - (2*b*c + a*d)*f)*log((b*d*x + b*c + (-b^2*d)^(1/3)*(b
*x + a)^(1/3)*(d*x + c)^(2/3))/(d*x + c)) - 3*sqrt(3)*(-b*d^2*e^3 + a*c^2*f^3 +
(2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(-b^2*d)^(1/3)*log(-((-
b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)
*(d*e - c*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (d^2*e^2 - 2*c*d*e*f + c^2*f^2)*(
b*x + a)^(2/3)*(d*x + c)^(1/3) - (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2
*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*(d*x + c))/(d*x + c)) + 6*sqrt(3)*(-b*d^2*e^
3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(-b^2*d
)^(1/3)*log(-((d*e - c*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (-b*d^2*e^3 + a*c^2*
f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(d*x + c))/(d*x +
c)) - 6*(3*b*d*e - (2*b*c + a*d)*f)*arctan(1/3*(2*sqrt(3)*(-b^2*d)^(1/3)*(b*x +
a)^(1/3)*(d*x + c)^(2/3) - sqrt(3)*(b*d*x + b*c))/(b*d*x + b*c)) - 18*(-b*d^2*e
^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(-b^2*
d)^(1/3)*arctan(-1/3*sqrt(3)*(2*(d*e - c*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (-
b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)
*(d*x + c))/((-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*
d)*e*f^2)^(1/3)*(d*x + c))))/((-b^2*d)^(1/3)*f^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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