3.3021 \(\int \frac{(a+b x)^{4/3} (e+f x)}{(c+d x)^{4/3}} \, dx\)

Optimal. Leaf size=328 \[ \frac{(b c-a d) \log (a+b x) (a d f-7 b c f+6 b d e)}{9 b^{2/3} d^{10/3}}+\frac{(b c-a d) (a d f-7 b c f+6 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{3 b^{2/3} d^{10/3}}+\frac{2 (b c-a d) (a d f-7 b c f+6 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{2/3} d^{10/3}}+\frac{2 \sqrt [3]{a+b x} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{3 d^3}-\frac{(a+b x)^{4/3} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{2 d^2 (b c-a d)}+\frac{3 (a+b x)^{7/3} (d e-c f)}{d \sqrt [3]{c+d x} (b c-a d)} \]

[Out]

(3*(d*e - c*f)*(a + b*x)^(7/3))/(d*(b*c - a*d)*(c + d*x)^(1/3)) + (2*(6*b*d*e -
7*b*c*f + a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*d^3) - ((6*b*d*e - 7*b*c*f
+ a*d*f)*(a + b*x)^(4/3)*(c + d*x)^(2/3))/(2*d^2*(b*c - a*d)) + (2*(b*c - a*d)*(
6*b*d*e - 7*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))])/(3*Sqrt[3]*b^(2/3)*d^(10/3)) + ((b*c - a*d)*(6*b*d
*e - 7*b*c*f + a*d*f)*Log[a + b*x])/(9*b^(2/3)*d^(10/3)) + ((b*c - a*d)*(6*b*d*e
- 7*b*c*f + a*d*f)*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))
])/(3*b^(2/3)*d^(10/3))

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Rubi [A]  time = 0.670102, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{(b c-a d) \log (a+b x) (a d f-7 b c f+6 b d e)}{9 b^{2/3} d^{10/3}}+\frac{(b c-a d) (a d f-7 b c f+6 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{3 b^{2/3} d^{10/3}}+\frac{2 (b c-a d) (a d f-7 b c f+6 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{2/3} d^{10/3}}+\frac{2 \sqrt [3]{a+b x} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{3 d^3}-\frac{(a+b x)^{4/3} (c+d x)^{2/3} (a d f-7 b c f+6 b d e)}{2 d^2 (b c-a d)}+\frac{3 (a+b x)^{7/3} (d e-c f)}{d \sqrt [3]{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(3*(d*e - c*f)*(a + b*x)^(7/3))/(d*(b*c - a*d)*(c + d*x)^(1/3)) + (2*(6*b*d*e -
7*b*c*f + a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*d^3) - ((6*b*d*e - 7*b*c*f
+ a*d*f)*(a + b*x)^(4/3)*(c + d*x)^(2/3))/(2*d^2*(b*c - a*d)) + (2*(b*c - a*d)*(
6*b*d*e - 7*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))])/(3*Sqrt[3]*b^(2/3)*d^(10/3)) + ((b*c - a*d)*(6*b*d
*e - 7*b*c*f + a*d*f)*Log[a + b*x])/(9*b^(2/3)*d^(10/3)) + ((b*c - a*d)*(6*b*d*e
- 7*b*c*f + a*d*f)*Log[-1 + (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))
])/(3*b^(2/3)*d^(10/3))

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Rubi in Sympy [A]  time = 50.5985, size = 323, normalized size = 0.98 \[ \frac{3 \left (a + b x\right )^{\frac{7}{3}} \left (c f - d e\right )}{d \sqrt [3]{c + d x} \left (a d - b c\right )} + \frac{\left (a + b x\right )^{\frac{4}{3}} \left (c + d x\right )^{\frac{2}{3}} \left (a d f - 7 b c f + 6 b d e\right )}{2 d^{2} \left (a d - b c\right )} + \frac{2 \sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}} \left (a d f - 7 b c f + 6 b d e\right )}{3 d^{3}} - \frac{\left (a d - b c\right ) \left (a d f - 7 b c f + 6 b d e\right ) \log{\left (a + b x \right )}}{9 b^{\frac{2}{3}} d^{\frac{10}{3}}} - \frac{\left (a d - b c\right ) \left (a d f - 7 b c f + 6 b d e\right ) \log{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x}}{\sqrt [3]{d} \sqrt [3]{a + b x}} - 1 \right )}}{3 b^{\frac{2}{3}} d^{\frac{10}{3}}} - \frac{2 \sqrt{3} \left (a d - b c\right ) \left (a d f - 7 b c f + 6 b d e\right ) \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{c + d x}}{3 \sqrt [3]{d} \sqrt [3]{a + b x}} + \frac{\sqrt{3}}{3} \right )}}{9 b^{\frac{2}{3}} d^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(a + b*x)**(7/3)*(c*f - d*e)/(d*(c + d*x)**(1/3)*(a*d - b*c)) + (a + b*x)**(4/
3)*(c + d*x)**(2/3)*(a*d*f - 7*b*c*f + 6*b*d*e)/(2*d**2*(a*d - b*c)) + 2*(a + b*
x)**(1/3)*(c + d*x)**(2/3)*(a*d*f - 7*b*c*f + 6*b*d*e)/(3*d**3) - (a*d - b*c)*(a
*d*f - 7*b*c*f + 6*b*d*e)*log(a + b*x)/(9*b**(2/3)*d**(10/3)) - (a*d - b*c)*(a*d
*f - 7*b*c*f + 6*b*d*e)*log(b**(1/3)*(c + d*x)**(1/3)/(d**(1/3)*(a + b*x)**(1/3)
) - 1)/(3*b**(2/3)*d**(10/3)) - 2*sqrt(3)*(a*d - b*c)*(a*d*f - 7*b*c*f + 6*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)/(9*b**(2/3)*d**(10/3))

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Mathematica [C]  time = 0.463672, size = 137, normalized size = 0.42 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (\frac{2 (a d f-7 b c f+6 b d e) \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt [3]{\frac{d (a+b x)}{a d-b c}}}-\frac{18 (b c-a d) (c f-d e)}{c+d x}+7 a d f-10 b c f+6 b d e+3 b d f x\right )}{6 d^3} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/3)*(c + d*x)^(2/3)*(6*b*d*e - 10*b*c*f + 7*a*d*f + 3*b*d*f*x - (18
*(b*c - a*d)*(-(d*e) + c*f))/(c + d*x) + (2*(6*b*d*e - 7*b*c*f + a*d*f)*Hypergeo
metric2F1[2/3, 2/3, 5/3, (b*(c + d*x))/(b*c - a*d)])/((d*(a + b*x))/(-(b*c) + a*
d))^(1/3)))/(6*d^3)

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Maple [F]  time = 0.091, size = 0, normalized size = 0. \[ \int{(fx+e) \left ( bx+a \right ) ^{{\frac{4}{3}}} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.253114, size = 798, normalized size = 2.43 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (3 \, b d^{2} f x^{2} + 6 \,{\left (4 \, b c d - 3 \, a d^{2}\right )} e -{\left (28 \, b c^{2} - 25 \, a c d\right )} f +{\left (6 \, b d^{2} e - 7 \,{\left (b c d - a d^{2}\right )} f\right )} x\right )} \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} + 2 \, \sqrt{3}{\left (6 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} e -{\left (7 \, b^{2} c^{3} - 8 \, a b c^{2} d + a^{2} c d^{2}\right )} f +{\left (6 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} e -{\left (7 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} f\right )} x\right )} \log \left (\frac{b^{2} d x + b^{2} c - \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} b + \left (-b^{2} d\right )^{\frac{2}{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 (6 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} e -{\left (7 \, b^{2} c^{3} - 8 \, a b c^{2} d + a^{2} c d^{2}\right )} f +{\left (6 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} e -{\left (7 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} f\right )} x\right )} \log \left (\frac{b d x + b c + \left (-b^{2} 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 ) - 12 \,{\left (6 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} e -{\left (7 \, b^{2} c^{3} - 8 \, a b c^{2} d + a^{2} c d^{2}\right )} f +{\left (6 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} e -{\left (7 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} f\right )} x\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} - \sqrt{3}{\left (b d x + b c\right )}}{3 \,{\left (b d x + b c\right )}}\right )\right )}}{54 \,{\left (d^{4} x + c d^{3}\right )} \left (-b^{2} d\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*sqrt(3)*(3*sqrt(3)*(3*b*d^2*f*x^2 + 6*(4*b*c*d - 3*a*d^2)*e - (28*b*c^2 - 2
5*a*c*d)*f + (6*b*d^2*e - 7*(b*c*d - a*d^2)*f)*x)*(-b^2*d)^(1/3)*(b*x + a)^(1/3)
*(d*x + c)^(2/3) + 2*sqrt(3)*(6*(b^2*c^2*d - a*b*c*d^2)*e - (7*b^2*c^3 - 8*a*b*c
^2*d + a^2*c*d^2)*f + (6*(b^2*c*d^2 - a*b*d^3)*e - (7*b^2*c^2*d - 8*a*b*c*d^2 +
a^2*d^3)*f)*x)*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)) - 4*sqrt(3)*
(6*(b^2*c^2*d - a*b*c*d^2)*e - (7*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2)*f + (6*(b^2
*c*d^2 - a*b*d^3)*e - (7*b^2*c^2*d - 8*a*b*c*d^2 + a^2*d^3)*f)*x)*log((b*d*x + b
*c + (-b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(d*x + c)) - 12*(6*(b^2*c^2
*d - a*b*c*d^2)*e - (7*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2)*f + (6*(b^2*c*d^2 - a*
b*d^3)*e - (7*b^2*c^2*d - 8*a*b*c*d^2 + a^2*d^3)*f)*x)*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)))/((d^4*x + c*d^3)*(-b^2*d)^(1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError