3.1602 \(\int \frac{(a+b x)^{5/3}}{(c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=216 \[ -\frac{5 (b c-a d)^2 \log (c+d x)}{18 \sqrt [3]{b} d^{8/3}}-\frac{5 (b c-a d)^2 \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{6 \sqrt [3]{b} d^{8/3}}-\frac{5 (b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{b} d^{8/3}}-\frac{5 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)}{6 d^2}+\frac{(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 d} \]

[Out]

(-5*(b*c - a*d)*(a + b*x)^(2/3)*(c + d*x)^(1/3))/(6*d^2) + ((a + b*x)^(5/3)*(c +
d*x)^(1/3))/(2*d) - (5*(b*c - a*d)^2*ArcTan[1/Sqrt[3] + (2*d^(1/3)*(a + b*x)^(1
/3))/(Sqrt[3]*b^(1/3)*(c + d*x)^(1/3))])/(3*Sqrt[3]*b^(1/3)*d^(8/3)) - (5*(b*c -
a*d)^2*Log[c + d*x])/(18*b^(1/3)*d^(8/3)) - (5*(b*c - a*d)^2*Log[-1 + (d^(1/3)*
(a + b*x)^(1/3))/(b^(1/3)*(c + d*x)^(1/3))])/(6*b^(1/3)*d^(8/3))

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Rubi [A]  time = 0.258619, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{5 (b c-a d)^2 \log (c+d x)}{18 \sqrt [3]{b} d^{8/3}}-\frac{5 (b c-a d)^2 \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{6 \sqrt [3]{b} d^{8/3}}-\frac{5 (b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{b} d^{8/3}}-\frac{5 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)}{6 d^2}+\frac{(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/3)/(c + d*x)^(2/3),x]

[Out]

(-5*(b*c - a*d)*(a + b*x)^(2/3)*(c + d*x)^(1/3))/(6*d^2) + ((a + b*x)^(5/3)*(c +
d*x)^(1/3))/(2*d) - (5*(b*c - a*d)^2*ArcTan[1/Sqrt[3] + (2*d^(1/3)*(a + b*x)^(1
/3))/(Sqrt[3]*b^(1/3)*(c + d*x)^(1/3))])/(3*Sqrt[3]*b^(1/3)*d^(8/3)) - (5*(b*c -
a*d)^2*Log[c + d*x])/(18*b^(1/3)*d^(8/3)) - (5*(b*c - a*d)^2*Log[-1 + (d^(1/3)*
(a + b*x)^(1/3))/(b^(1/3)*(c + d*x)^(1/3))])/(6*b^(1/3)*d^(8/3))

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Rubi in Sympy [A]  time = 21.557, size = 202, normalized size = 0.94 \[ \frac{\left (a + b x\right )^{\frac{5}{3}} \sqrt [3]{c + d x}}{2 d} + \frac{5 \left (a + b x\right )^{\frac{2}{3}} \sqrt [3]{c + d x} \left (a d - b c\right )}{6 d^{2}} - \frac{5 \left (a d - b c\right )^{2} \log{\left (-1 + \frac{\sqrt [3]{d} \sqrt [3]{a + b x}}{\sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{6 \sqrt [3]{b} d^{\frac{8}{3}}} - \frac{5 \left (a d - b c\right )^{2} \log{\left (c + d x \right )}}{18 \sqrt [3]{b} d^{\frac{8}{3}}} - \frac{5 \sqrt{3} \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{d} \sqrt [3]{a + b x}}{3 \sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{9 \sqrt [3]{b} d^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/3)/(d*x+c)**(2/3),x)

[Out]

(a + b*x)**(5/3)*(c + d*x)**(1/3)/(2*d) + 5*(a + b*x)**(2/3)*(c + d*x)**(1/3)*(a
*d - b*c)/(6*d**2) - 5*(a*d - b*c)**2*log(-1 + d**(1/3)*(a + b*x)**(1/3)/(b**(1/
3)*(c + d*x)**(1/3)))/(6*b**(1/3)*d**(8/3)) - 5*(a*d - b*c)**2*log(c + d*x)/(18*
b**(1/3)*d**(8/3)) - 5*sqrt(3)*(a*d - b*c)**2*atan(sqrt(3)/3 + 2*sqrt(3)*d**(1/3
)*(a + b*x)**(1/3)/(3*b**(1/3)*(c + d*x)**(1/3)))/(9*b**(1/3)*d**(8/3))

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Mathematica [C]  time = 0.200296, size = 107, normalized size = 0.5 \[ \frac{\sqrt [3]{c+d x} \left (10 (b c-a d)^2 \sqrt [3]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x) (8 a d-5 b c+3 b d x)\right )}{6 d^3 \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/3)/(c + d*x)^(2/3),x]

[Out]

((c + d*x)^(1/3)*(d*(a + b*x)*(-5*b*c + 8*a*d + 3*b*d*x) + 10*(b*c - a*d)^2*((d*
(a + b*x))/(-(b*c) + a*d))^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, (b*(c + d*x))/
(b*c - a*d)]))/(6*d^3*(a + b*x)^(1/3))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{5}{3}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/3)/(d*x+c)^(2/3),x)

[Out]

int((b*x+a)^(5/3)/(d*x+c)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{3}}}{{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/3)/(d*x + c)^(2/3),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/3)/(d*x + c)^(2/3), x)

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Fricas [A]  time = 0.233596, size = 400, normalized size = 1.85 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3} \left (-b d^{2}\right )^{\frac{1}{3}}{\left (3 \, b d x - 5 \, b c + 8 \, a d\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} - 5 \, \sqrt{3}{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac{b d^{2} x + a d^{2} - \left (-b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} d + \left (-b d^{2}\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{b x + a}\right ) + 10 \, \sqrt{3}{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac{b d x + a d + \left (-b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{b x + a}\right ) + 30 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} - \sqrt{3}{\left (b d x + a d\right )}}{3 \,{\left (b d x + a d\right )}}\right )\right )}}{54 \, \left (-b d^{2}\right )^{\frac{1}{3}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/3)/(d*x + c)^(2/3),x, algorithm="fricas")

[Out]

1/54*sqrt(3)*(3*sqrt(3)*(-b*d^2)^(1/3)*(3*b*d*x - 5*b*c + 8*a*d)*(b*x + a)^(2/3)
*(d*x + c)^(1/3) - 5*sqrt(3)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log((b*d^2*x + a*d^
2 - (-b*d^2)^(1/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3)*d + (-b*d^2)^(2/3)*(b*x + a)^
(1/3)*(d*x + c)^(2/3))/(b*x + a)) + 10*sqrt(3)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*l
og((b*d*x + a*d + (-b*d^2)^(1/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3))/(b*x + a)) + 3
0*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(1/3*(2*sqrt(3)*(-b*d^2)^(1/3)*(b*x + a)
^(2/3)*(d*x + c)^(1/3) - sqrt(3)*(b*d*x + a*d))/(b*d*x + a*d)))/((-b*d^2)^(1/3)*
d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{5}{3}}}{\left (c + d x\right )^{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/3)/(d*x+c)**(2/3),x)

[Out]

Integral((a + b*x)**(5/3)/(c + d*x)**(2/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{3}}}{{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/3)/(d*x + c)^(2/3),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/3)/(d*x + c)^(2/3), x)