3.2986 \(\int \sqrt [3]{a+b x} (c+d x)^{2/3} \, dx\)

Optimal. Leaf size=219 \[ \frac{(b c-a d)^2 \log (a+b x)}{18 b^{5/3} d^{4/3}}+\frac{(b c-a d)^2 \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{6 b^{5/3} d^{4/3}}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3} d^{4/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d)}{3 b d}+\frac{(a+b x)^{4/3} (c+d x)^{2/3}}{2 b} \]

[Out]

((b*c - a*d)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*b*d) + ((a + b*x)^(4/3)*(c + d*
x)^(2/3))/(2*b) + ((b*c - a*d)^2*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^(5/3)*d^(4/3)) + ((b*c - a*d)^2
*Log[a + b*x])/(18*b^(5/3)*d^(4/3)) + ((b*c - a*d)^2*Log[-1 + (b^(1/3)*(c + d*x)
^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(6*b^(5/3)*d^(4/3))

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Rubi [A]  time = 0.239257, antiderivative size = 219, 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 \[ \frac{(b c-a d)^2 \log (a+b x)}{18 b^{5/3} d^{4/3}}+\frac{(b c-a d)^2 \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{6 b^{5/3} d^{4/3}}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3} d^{4/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d)}{3 b d}+\frac{(a+b x)^{4/3} (c+d x)^{2/3}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*b*d) + ((a + b*x)^(4/3)*(c + d*
x)^(2/3))/(2*b) + ((b*c - a*d)^2*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^(5/3)*d^(4/3)) + ((b*c - a*d)^2
*Log[a + b*x])/(18*b^(5/3)*d^(4/3)) + ((b*c - a*d)^2*Log[-1 + (b^(1/3)*(c + d*x)
^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(6*b^(5/3)*d^(4/3))

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Rubi in Sympy [A]  time = 23.8958, size = 196, normalized size = 0.89 \[ \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{5}{3}}}{2 d} + \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}} \left (a d - b c\right )}{6 b d} + \frac{\left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{18 b^{\frac{5}{3}} d^{\frac{4}{3}}} + \frac{\left (a d - b c\right )^{2} \log{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x}}{\sqrt [3]{d} \sqrt [3]{a + b x}} - 1 \right )}}{6 b^{\frac{5}{3}} d^{\frac{4}{3}}} + \frac{\sqrt{3} \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{c + d x}}{3 \sqrt [3]{d} \sqrt [3]{a + b x}} + \frac{\sqrt{3}}{3} \right )}}{9 b^{\frac{5}{3}} d^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/3)*(d*x+c)**(2/3),x)

[Out]

(a + b*x)**(1/3)*(c + d*x)**(5/3)/(2*d) + (a + b*x)**(1/3)*(c + d*x)**(2/3)*(a*d
- b*c)/(6*b*d) + (a*d - b*c)**2*log(a + b*x)/(18*b**(5/3)*d**(4/3)) + (a*d - b*
c)**2*log(b**(1/3)*(c + d*x)**(1/3)/(d**(1/3)*(a + b*x)**(1/3)) - 1)/(6*b**(5/3)
*d**(4/3)) + sqrt(3)*(a*d - b*c)**2*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**(5/3)*d**(4/3))

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Mathematica [C]  time = 0.198991, size = 109, normalized size = 0.5 \[ \frac{(c+d x)^{2/3} \left (d (a+b x) (a d+2 b c+3 b d x)-(b c-a d)^2 \left (\frac{d (a+b x)}{a d-b c}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )\right )}{6 b d^2 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int \sqrt [3]{bx+a} \left ( dx+c \right ) ^{{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*(d*x + c)^(2/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.227112, size = 394, normalized size = 1.8 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3} \left (b^{2} d\right )^{\frac{1}{3}}{\left (3 \, b d x + 2 \, b c + a d\right )}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} - \sqrt{3}{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac{b^{2} d x + b^{2} c + \left (b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} b + \left (b^{2} d\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{d x + c}\right ) + 2 \, \sqrt{3}{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (-\frac{b d x + b c - \left (b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{d x + c}\right ) - 6 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} + \sqrt{3}{\left (b d x + b c\right )}}{3 \,{\left (b d x + b c\right )}}\right )\right )}}{54 \, \left (b^{2} d\right )^{\frac{1}{3}} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*(d*x + c)^(2/3),x, algorithm="fricas")

[Out]

1/54*sqrt(3)*(3*sqrt(3)*(b^2*d)^(1/3)*(3*b*d*x + 2*b*c + a*d)*(b*x + a)^(1/3)*(d
*x + c)^(2/3) - sqrt(3)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*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)) + 2*sqrt(3)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(-(b*d
*x + b*c - (b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(d*x + c)) - 6*(b^2*c^
2 - 2*a*b*c*d + a^2*d^2)*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)))/((b^2*d)^(1/3)*b*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/3)*(d*x+c)**(2/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*(d*x + c)^(2/3),x, algorithm="giac")

[Out]

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