3.2988 \(\int \frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^2} \, dx\)

Optimal. Leaf size=417 \[ -\frac{3 \sqrt [3]{b} d^{2/3} \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 f^2}-\frac{\sqrt{3} \sqrt [3]{b} d^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{f^2}-\frac{\log (e+f x) (-2 a d f-b c f+3 b d e)}{6 f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac{(-2 a d f-b c f+3 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac{(-2 a d f-b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{f (e+f x)}-\frac{\sqrt [3]{b} d^{2/3} \log (a+b x)}{2 f^2} \]

[Out]

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

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Rubi [A]  time = 0.883703, antiderivative size = 417, 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 \[ -\frac{3 \sqrt [3]{b} d^{2/3} \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 f^2}-\frac{\sqrt{3} \sqrt [3]{b} d^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{f^2}-\frac{\log (e+f x) (-2 a d f-b c f+3 b d e)}{6 f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac{(-2 a d f-b c f+3 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac{(-2 a d f-b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{f (e+f x)}-\frac{\sqrt [3]{b} d^{2/3} \log (a+b x)}{2 f^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 86.202, size = 393, normalized size = 0.94 \[ - \frac{\sqrt [3]{b} d^{\frac{2}{3}} \log{\left (a + b x \right )}}{2 f^{2}} - \frac{3 \sqrt [3]{b} d^{\frac{2}{3}} \log{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x}}{\sqrt [3]{d} \sqrt [3]{a + b x}} - 1 \right )}}{2 f^{2}} - \frac{\sqrt{3} \sqrt [3]{b} d^{\frac{2}{3}} \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{c + d x}}{3 \sqrt [3]{d} \sqrt [3]{a + b x}} + \frac{\sqrt{3}}{3} \right )}}{f^{2}} - \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}}}{f \left (e + f x\right )} - \frac{\left (2 a d f + b c f - 3 b d e\right ) \log{\left (e + f x \right )}}{6 f^{2} \left (a f - b e\right )^{\frac{2}{3}} \sqrt [3]{c f - d e}} + \frac{\left (2 a d f + b c f - 3 b d e\right ) \log{\left (- \sqrt [3]{a + b x} + \frac{\sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{\sqrt [3]{c f - d e}} \right )}}{2 f^{2} \left (a f - b e\right )^{\frac{2}{3}} \sqrt [3]{c f - d e}} + \frac{\sqrt{3} \left (2 a d f + b c f - 3 b d e\right ) \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{3 \sqrt [3]{a + b x} \sqrt [3]{c f - d e}} \right )}}{3 f^{2} \left (a f - b e\right )^{\frac{2}{3}} \sqrt [3]{c f - d e}} \]

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)**2,x)

[Out]

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

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Mathematica [C]  time = 2.6108, size = 743, normalized size = 1.78 \[ \frac{(c+d x)^{2/3} \left (-\frac{4 b \left (-\frac{5 b c f (c+d x) F_1\left (1;\frac{2}{3},1;2;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )}{6 b f (c+d x) F_1\left (1;\frac{2}{3},1;2;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+b (3 c f-3 d e) F_1\left (2;\frac{2}{3},2;3;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+2 f (b c-a d) F_1\left (2;\frac{5}{3},1;3;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )}-\frac{5 a d f (c+d x) F_1\left (1;\frac{2}{3},1;2;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )}{-6 b f (c+d x) F_1\left (1;\frac{2}{3},1;2;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+3 b (d e-c f) F_1\left (2;\frac{2}{3},2;3;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+2 f (a d-b c) F_1\left (2;\frac{5}{3},1;3;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )}-\frac{6 (b c-a d) (c f-d e) F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};\frac{b (c+d x)}{b c-a d},\frac{f (c+d x)}{c f-d e}\right )}{\frac{8 (b c-a d) (c f-d e) F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};\frac{b (c+d x)}{b c-a d},\frac{f (c+d x)}{c f-d e}\right )}{c+d x}+3 f (b c-a d) F_1\left (\frac{8}{3};\frac{2}{3},2;\frac{11}{3};\frac{b (c+d x)}{b c-a d},\frac{f (c+d x)}{c f-d e}\right )+2 b (c f-d e) F_1\left (\frac{8}{3};\frac{5}{3},1;\frac{11}{3};\frac{b (c+d x)}{b c-a d},\frac{f (c+d x)}{c f-d e}\right )}\right )}{d}-5 (a+b x)\right )}{5 f (a+b x)^{2/3} (e+f x)} \]

Warning: Unable to verify antiderivative.

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

[Out]

((c + d*x)^(2/3)*(-5*(a + b*x) - (4*b*((-5*b*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)*App
ellF1[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))]) - (5*a*d*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)*App
ellF1[1, 2/3, 1, 2, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] + 3
*b*(d*e - 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))]) - (6*(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)])/((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))/(5*f*(a + b*x)^(2/3)*(e + f*x))

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Maple [F]  time = 0.088, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( fx+e \right ) ^{2}}\sqrt [3]{bx+a} \left ( dx+c \right ) ^{{\frac{2}{3}}}}\, dx \]

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)^2,x)

[Out]

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

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

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 4.5821, size = 1463, normalized size = 3.51 \[ \text{result too large to display} \]

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)^2,x, algorithm="fricas")

[Out]

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

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

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)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.853726, size = 1, normalized size = 0. \[ \mathit{Done} \]

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)^2,x, algorithm="giac")

[Out]

Done