3.1620 \(\int \frac{1}{(a+b x)^{5/3} (c+d x)^{4/3}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{9 d \sqrt [3]{a+b x}}{2 \sqrt [3]{c+d x} (b c-a d)^2}-\frac{3}{2 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.047648, antiderivative size = 66, 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 \[ -\frac{9 d \sqrt [3]{a+b x}}{2 \sqrt [3]{c+d x} (b c-a d)^2}-\frac{3}{2 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(5/3)*(c + d*x)^(4/3)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.46661, size = 56, normalized size = 0.85 \[ - \frac{9 d \sqrt [3]{a + b x}}{2 \sqrt [3]{c + d x} \left (a d - b c\right )^{2}} + \frac{3}{2 \left (a + b x\right )^{\frac{2}{3}} \sqrt [3]{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(5/3)/(d*x+c)**(4/3),x)

[Out]

-9*d*(a + b*x)**(1/3)/(2*(c + d*x)**(1/3)*(a*d - b*c)**2) + 3/(2*(a + b*x)**(2/3
)*(c + d*x)**(1/3)*(a*d - b*c))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0699262, size = 45, normalized size = 0.68 \[ -\frac{3 (2 a d+b (c+3 d x))}{2 (a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(5/3)*(c + d*x)^(4/3)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 53, normalized size = 0.8 \[ -{\frac{9\,bdx+6\,ad+3\,bc}{2\,{a}^{2}{d}^{2}-4\,abcd+2\,{b}^{2}{c}^{2}} \left ( bx+a \right ) ^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(5/3)/(d*x+c)^(4/3),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{3}}{\left (d x + c\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/3)*(d*x + c)^(4/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/3)*(d*x + c)^(4/3)), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.210547, size = 70, normalized size = 1.06 \[ -\frac{3 \,{\left (3 \, b d x + b c + 2 \, a d\right )}}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/3)*(d*x + c)^(4/3)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{5}{3}} \left (c + d x\right )^{\frac{4}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(5/3)/(d*x+c)**(4/3),x)

[Out]

Integral(1/((a + b*x)**(5/3)*(c + d*x)**(4/3)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{3}}{\left (d x + c\right )}^{\frac{4}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/3)*(d*x + c)^(4/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(5/3)*(d*x + c)^(4/3)), x)