3.3023 \(\int \frac{(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)} \, dx\)

Optimal. Leaf size=380 \[ -\frac{3 b^{4/3} \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 d^{4/3} f}-\frac{\sqrt{3} b^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{d^{4/3} f}-\frac{b^{4/3} \log (a+b x)}{2 d^{4/3} f}+\frac{3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (d e-c f)}-\frac{(b e-a f)^{4/3} \log (e+f x)}{2 f (d e-c f)^{4/3}}+\frac{3 (b e-a f)^{4/3} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f (d e-c f)^{4/3}}+\frac{\sqrt{3} (b e-a f)^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{f (d e-c f)^{4/3}} \]

[Out]

(3*(b*c - a*d)*(a + b*x)^(1/3))/(d*(d*e - c*f)*(c + d*x)^(1/3)) - (Sqrt[3]*b^(4/
3)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*d^(1/3)*(a + b*x)^(1/
3))])/(d^(4/3)*f) + (Sqrt[3]*(b*e - a*f)^(4/3)*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*(d*e -
c*f)^(4/3)) - (b^(4/3)*Log[a + b*x])/(2*d^(4/3)*f) - ((b*e - a*f)^(4/3)*Log[e +
f*x])/(2*f*(d*e - c*f)^(4/3)) + (3*(b*e - a*f)^(4/3)*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*(d*e - c*f)^(4/3)) - (3
*b^(4/3)*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(2*d^(4/
3)*f)

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Rubi [A]  time = 1.08057, antiderivative size = 380, 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 \[ -\frac{3 b^{4/3} \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 d^{4/3} f}-\frac{\sqrt{3} b^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{d^{4/3} f}-\frac{b^{4/3} \log (a+b x)}{2 d^{4/3} f}+\frac{3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (d e-c f)}-\frac{(b e-a f)^{4/3} \log (e+f x)}{2 f (d e-c f)^{4/3}}+\frac{3 (b e-a f)^{4/3} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f (d e-c f)^{4/3}}+\frac{\sqrt{3} (b e-a f)^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{f (d e-c f)^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b*c - a*d)*(a + b*x)^(1/3))/(d*(d*e - c*f)*(c + d*x)^(1/3)) - (Sqrt[3]*b^(4/
3)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*d^(1/3)*(a + b*x)^(1/
3))])/(d^(4/3)*f) + (Sqrt[3]*(b*e - a*f)^(4/3)*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*(d*e -
c*f)^(4/3)) - (b^(4/3)*Log[a + b*x])/(2*d^(4/3)*f) - ((b*e - a*f)^(4/3)*Log[e +
f*x])/(2*f*(d*e - c*f)^(4/3)) + (3*(b*e - a*f)^(4/3)*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*(d*e - c*f)^(4/3)) - (3
*b^(4/3)*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(2*d^(4/
3)*f)

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Rubi in Sympy [A]  time = 100.821, size = 337, normalized size = 0.89 \[ - \frac{b^{\frac{4}{3}} \log{\left (a + b x \right )}}{2 d^{\frac{4}{3}} f} - \frac{3 b^{\frac{4}{3}} \log{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x}}{\sqrt [3]{d} \sqrt [3]{a + b x}} - 1 \right )}}{2 d^{\frac{4}{3}} f} - \frac{\sqrt{3} b^{\frac{4}{3}} \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{c + d x}}{3 \sqrt [3]{d} \sqrt [3]{a + b x}} + \frac{\sqrt{3}}{3} \right )}}{d^{\frac{4}{3}} f} - \frac{\left (a f - b e\right )^{\frac{4}{3}} \log{\left (e + f x \right )}}{2 f \left (c f - d e\right )^{\frac{4}{3}}} + \frac{3 \left (a f - b e\right )^{\frac{4}{3}} \log{\left (- \sqrt [3]{a + b x} + \frac{\sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{\sqrt [3]{c f - d e}} \right )}}{2 f \left (c f - d e\right )^{\frac{4}{3}}} + \frac{\sqrt{3} \left (a f - b e\right )^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{3 \sqrt [3]{a + b x} \sqrt [3]{c f - d e}} \right )}}{f \left (c f - d e\right )^{\frac{4}{3}}} + \frac{3 \sqrt [3]{a + b x} \left (a d - b c\right )}{d \sqrt [3]{c + d x} \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**(4/3)*log(a + b*x)/(2*d**(4/3)*f) - 3*b**(4/3)*log(b**(1/3)*(c + d*x)**(1/3)
/(d**(1/3)*(a + b*x)**(1/3)) - 1)/(2*d**(4/3)*f) - sqrt(3)*b**(4/3)*atan(2*sqrt(
3)*b**(1/3)*(c + d*x)**(1/3)/(3*d**(1/3)*(a + b*x)**(1/3)) + sqrt(3)/3)/(d**(4/3
)*f) - (a*f - b*e)**(4/3)*log(e + f*x)/(2*f*(c*f - d*e)**(4/3)) + 3*(a*f - b*e)*
*(4/3)*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*(c*f - d*e)**(4/3)) + sqrt(3)*(a*f - b*e)**(4/3)*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*(c*f - d*e)**(4/3)) + 3*(a + b*x)**(1/3)*(a*d - b*c)/(d*(c + d*x)**(1/3)
*(c*f - d*e))

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Mathematica [C]  time = 3.42915, size = 559, normalized size = 1.47 \[ \frac{3 \left (-\frac{2 b (c+d x) (b c-a d) \left (\frac{5 f (c+d x) (a d f+b c f-2 b d e) F_1\left (1;\frac{2}{3},1;2;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )}{6 b f (c+d x) F_1\left (1;\frac{2}{3},1;2;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+b (3 c f-3 d e) F_1\left (2;\frac{2}{3},2;3;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+2 f (b c-a d) F_1\left (2;\frac{5}{3},1;3;\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )}-\frac{4 b (d e-c f)^2 F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};\frac{b (c+d x)}{b c-a d},\frac{f (c+d x)}{c f-d e}\right )}{-\frac{8 (b c-a d) (c f-d e) F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};\frac{b (c+d x)}{b c-a d},\frac{f (c+d x)}{c f-d e}\right )}{c+d x}+(3 a d f-3 b c f) F_1\left (\frac{8}{3};\frac{2}{3},2;\frac{11}{3};\frac{b (c+d x)}{b c-a d},\frac{f (c+d x)}{c f-d e}\right )+2 b (d e-c f) F_1\left (\frac{8}{3};\frac{5}{3},1;\frac{11}{3};\frac{b (c+d x)}{b c-a d},\frac{f (c+d x)}{c f-d e}\right )}\right )}{d (e+f x)}-5 d (a+b x) (a d-b c)\right )}{5 d^2 (a+b x)^{2/3} \sqrt [3]{c+d x} (d e-c f)} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \[ \int{\frac{1}{fx+e} \left ( bx+a \right ) ^{{\frac{4}{3}}} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((b*x+a)^(4/3)/(d*x+c)^(4/3)/(f*x+e),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^(4/3)/((d*x + c)^(4/3)*(f*x + e)), x)

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Fricas [A]  time = 0.502872, size = 961, normalized size = 2.53 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)**(4/3)/((c + d*x)**(4/3)*(e + f*x)), 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((b*x + a)^(4/3)/((d*x + c)^(4/3)*(f*x + e)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError