3.1617 \(\int \frac{(a+b x)^{4/3}}{(c+d x)^{4/3}} \, dx\)

Optimal. Leaf size=195 \[ \frac{2 \sqrt [3]{b} (b c-a d) \log (a+b x)}{3 d^{7/3}}+\frac{2 \sqrt [3]{b} (b c-a d) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{d^{7/3}}+\frac{4 \sqrt [3]{b} (b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} d^{7/3}}+\frac{4 b \sqrt [3]{a+b x} (c+d x)^{2/3}}{d^2}-\frac{3 (a+b x)^{4/3}}{d \sqrt [3]{c+d x}} \]

[Out]

(-3*(a + b*x)^(4/3))/(d*(c + d*x)^(1/3)) + (4*b*(a + b*x)^(1/3)*(c + d*x)^(2/3))
/d^2 + (4*b^(1/3)*(b*c - a*d)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sq
rt[3]*d^(1/3)*(a + b*x)^(1/3))])/(Sqrt[3]*d^(7/3)) + (2*b^(1/3)*(b*c - a*d)*Log[
a + b*x])/(3*d^(7/3)) + (2*b^(1/3)*(b*c - a*d)*Log[-1 + (b^(1/3)*(c + d*x)^(1/3)
)/(d^(1/3)*(a + b*x)^(1/3))])/d^(7/3)

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Rubi [A]  time = 0.178237, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 \sqrt [3]{b} (b c-a d) \log (a+b x)}{3 d^{7/3}}+\frac{2 \sqrt [3]{b} (b c-a d) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{d^{7/3}}+\frac{4 \sqrt [3]{b} (b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} d^{7/3}}+\frac{4 b \sqrt [3]{a+b x} (c+d x)^{2/3}}{d^2}-\frac{3 (a+b x)^{4/3}}{d \sqrt [3]{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(a + b*x)^(4/3))/(d*(c + d*x)^(1/3)) + (4*b*(a + b*x)^(1/3)*(c + d*x)^(2/3))
/d^2 + (4*b^(1/3)*(b*c - a*d)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sq
rt[3]*d^(1/3)*(a + b*x)^(1/3))])/(Sqrt[3]*d^(7/3)) + (2*b^(1/3)*(b*c - a*d)*Log[
a + b*x])/(3*d^(7/3)) + (2*b^(1/3)*(b*c - a*d)*Log[-1 + (b^(1/3)*(c + d*x)^(1/3)
)/(d^(1/3)*(a + b*x)^(1/3))])/d^(7/3)

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Rubi in Sympy [A]  time = 19.7735, size = 189, normalized size = 0.97 \[ - \frac{2 \sqrt [3]{b} \left (a d - b c\right ) \log{\left (a + b x \right )}}{3 d^{\frac{7}{3}}} - \frac{2 \sqrt [3]{b} \left (a d - b c\right ) \log{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x}}{\sqrt [3]{d} \sqrt [3]{a + b x}} - 1 \right )}}{d^{\frac{7}{3}}} - \frac{4 \sqrt{3} \sqrt [3]{b} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{c + d x}}{3 \sqrt [3]{d} \sqrt [3]{a + b x}} + \frac{\sqrt{3}}{3} \right )}}{3 d^{\frac{7}{3}}} + \frac{4 b \sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}}}{d^{2}} - \frac{3 \left (a + b x\right )^{\frac{4}{3}}}{d \sqrt [3]{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b**(1/3)*(a*d - b*c)*log(a + b*x)/(3*d**(7/3)) - 2*b**(1/3)*(a*d - b*c)*log(b
**(1/3)*(c + d*x)**(1/3)/(d**(1/3)*(a + b*x)**(1/3)) - 1)/d**(7/3) - 4*sqrt(3)*b
**(1/3)*(a*d - b*c)*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*d**(7/3)) + 4*b*(a + b*x)**(1/3)*(c + d*x)**(2/3)/d**
2 - 3*(a + b*x)**(4/3)/(d*(c + d*x)**(1/3))

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Mathematica [C]  time = 0.372927, size = 95, normalized size = 0.49 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (\frac{2 b \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt [3]{\frac{d (a+b x)}{a d-b c}}}+\frac{-3 a d+4 b c+b d x}{c+d x}\right )}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/3)*(c + d*x)^(2/3)*((4*b*c - 3*a*d + b*d*x)/(c + d*x) + (2*b*Hyper
geometric2F1[2/3, 2/3, 5/3, (b*(c + d*x))/(b*c - a*d)])/((d*(a + b*x))/(-(b*c) +
a*d))^(1/3)))/d^2

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Maple [F]  time = 0.071, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{4}{3}}} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.220856, size = 427, normalized size = 2.19 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} \log \left (\frac{{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} \left (-\frac{b}{d}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{d x + c}\right ) - 4 \, \sqrt{3}{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} \log \left (\frac{{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{d x + c}\right ) + 3 \, \sqrt{3}{\left (b d x + 4 \, b c - 3 \, a d\right )}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} - 12 \,{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3}{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} - 2 \, \sqrt{3}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{3 \,{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}}}\right )\right )}}{9 \,{\left (d^{3} x + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*sqrt(3)*(2*sqrt(3)*(b*c^2 - a*c*d + (b*c*d - a*d^2)*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)) - 4*sqrt(3)*(b*c^2 - a*c*d + (b*c*d - a*d^2)*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)) + 3*sqrt(3)*(b*d*x + 4*b*c - 3*a*d)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - 12
*(b*c^2 - a*c*d + (b*c*d - a*d^2)*x)*(-b/d)^(1/3)*arctan(-1/3*(sqrt(3)*(d*x + c)
*(-b/d)^(1/3) - 2*sqrt(3)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/((d*x + c)*(-b/d)^(1/
3))))/(d^3*x + c*d^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError