3.1578 \(\int \frac{\sqrt [3]{c+d x}}{(a+b x)^{4/3}} \, dx\)

Optimal. Leaf size=149 \[ -\frac{3 \sqrt [3]{d} \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3}}-\frac{\sqrt{3} \sqrt [3]{d} \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{b^{4/3}}-\frac{3 \sqrt [3]{c+d x}}{b \sqrt [3]{a+b x}}-\frac{\sqrt [3]{d} \log (c+d x)}{2 b^{4/3}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.110484, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{3 \sqrt [3]{d} \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3}}-\frac{\sqrt{3} \sqrt [3]{d} \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{b^{4/3}}-\frac{3 \sqrt [3]{c+d x}}{b \sqrt [3]{a+b x}}-\frac{\sqrt [3]{d} \log (c+d x)}{2 b^{4/3}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(1/3)/(a + b*x)^(4/3),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.3233, size = 143, normalized size = 0.96 \[ - \frac{3 \sqrt [3]{c + d x}}{b \sqrt [3]{a + b x}} - \frac{3 \sqrt [3]{d} \log{\left (-1 + \frac{\sqrt [3]{d} \sqrt [3]{a + b x}}{\sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{2 b^{\frac{4}{3}}} - \frac{\sqrt [3]{d} \log{\left (c + d x \right )}}{2 b^{\frac{4}{3}}} - \frac{\sqrt{3} \sqrt [3]{d} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{d} \sqrt [3]{a + b x}}{3 \sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(1/3)/(b*x+a)**(4/3),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.0920463, size = 74, normalized size = 0.5 \[ \frac{3 \sqrt [3]{c+d x} \left (\sqrt [3]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )-1\right )}{b \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(1/3)/(a + b*x)^(4/3),x]

[Out]

(3*(c + d*x)^(1/3)*(-1 + ((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)]))/(b*(a + b*x)^(1/3))

_______________________________________________________________________________________

Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{1\sqrt [3]{dx+c} \left ( bx+a \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(1/3)/(b*x+a)^(4/3),x)

[Out]

int((d*x+c)^(1/3)/(b*x+a)^(4/3),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/3)/(b*x + a)^(4/3),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(1/3)/(b*x + a)^(4/3), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.220327, size = 313, normalized size = 2.1 \[ \frac{2 \, \sqrt{3}{\left (b x + a\right )} \left (-\frac{d}{b}\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3}{\left ({\left (b x + a\right )} \left (-\frac{d}{b}\right )^{\frac{1}{3}} - 2 \,{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}\right )}}{3 \,{\left (b x + a\right )} \left (-\frac{d}{b}\right )^{\frac{1}{3}}}\right ) -{\left (b x + a\right )} \left (-\frac{d}{b}\right )^{\frac{1}{3}} \log \left (\frac{{\left (b x + a\right )} \left (-\frac{d}{b}\right )^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} \left (-\frac{d}{b}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{b x + a}\right ) + 2 \,{\left (b x + a\right )} \left (-\frac{d}{b}\right )^{\frac{1}{3}} \log \left (\frac{{\left (b x + a\right )} \left (-\frac{d}{b}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{b x + a}\right ) - 6 \,{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{2 \,{\left (b^{2} x + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/3)/(b*x + a)^(4/3),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(1/3)/(b*x+a)**(4/3),x)

[Out]

Integral((c + d*x)**(1/3)/(a + b*x)**(4/3), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/3)/(b*x + a)^(4/3),x, algorithm="giac")

[Out]

integrate((d*x + c)^(1/3)/(b*x + a)^(4/3), x)