3.1580 \(\int \frac{\sqrt [3]{c+d x}}{(a+b x)^{10/3}} \, dx\)

Optimal. Leaf size=66 \[ \frac{9 d (c+d x)^{4/3}}{28 (a+b x)^{4/3} (b c-a d)^2}-\frac{3 (c+d x)^{4/3}}{7 (a+b x)^{7/3} (b c-a d)} \]

[Out]

(-3*(c + d*x)^(4/3))/(7*(b*c - a*d)*(a + b*x)^(7/3)) + (9*d*(c + d*x)^(4/3))/(28
*(b*c - a*d)^2*(a + b*x)^(4/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.0527406, 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 \[ \frac{9 d (c+d x)^{4/3}}{28 (a+b x)^{4/3} (b c-a d)^2}-\frac{3 (c+d x)^{4/3}}{7 (a+b x)^{7/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(1/3)/(a + b*x)^(10/3),x]

[Out]

(-3*(c + d*x)^(4/3))/(7*(b*c - a*d)*(a + b*x)^(7/3)) + (9*d*(c + d*x)^(4/3))/(28
*(b*c - a*d)^2*(a + b*x)^(4/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.29053, size = 56, normalized size = 0.85 \[ \frac{9 d \left (c + d x\right )^{\frac{4}{3}}}{28 \left (a + b x\right )^{\frac{4}{3}} \left (a d - b c\right )^{2}} + \frac{3 \left (c + d x\right )^{\frac{4}{3}}}{7 \left (a + b x\right )^{\frac{7}{3}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(1/3)/(b*x+a)**(10/3),x)

[Out]

9*d*(c + d*x)**(4/3)/(28*(a + b*x)**(4/3)*(a*d - b*c)**2) + 3*(c + d*x)**(4/3)/(
7*(a + b*x)**(7/3)*(a*d - b*c))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0688351, size = 46, normalized size = 0.7 \[ \frac{3 (c+d x)^{4/3} (7 a d-4 b c+3 b d x)}{28 (a+b x)^{7/3} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(1/3)/(a + b*x)^(10/3),x]

[Out]

(3*(c + d*x)^(4/3)*(-4*b*c + 7*a*d + 3*b*d*x))/(28*(b*c - a*d)^2*(a + b*x)^(7/3)
)

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 54, normalized size = 0.8 \[{\frac{9\,bdx+21\,ad-12\,bc}{28\,{a}^{2}{d}^{2}-56\,abcd+28\,{b}^{2}{c}^{2}} \left ( dx+c \right ) ^{{\frac{4}{3}}} \left ( bx+a \right ) ^{-{\frac{7}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(1/3)/(b*x+a)^(10/3),x)

[Out]

3/28*(d*x+c)^(4/3)*(3*b*d*x+7*a*d-4*b*c)/(b*x+a)^(7/3)/(a^2*d^2-2*a*b*c*d+b^2*c^
2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{10}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/3)/(b*x + a)^(10/3),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(1/3)/(b*x + a)^(10/3), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.210012, size = 236, normalized size = 3.58 \[ \frac{3 \,{\left (3 \, b d^{2} x^{2} - 4 \, b c^{2} + 7 \, a c d -{\left (b c d - 7 \, a d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{28 \,{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2} +{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{3} + 3 \,{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2} + 3 \,{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/3)/(b*x + a)^(10/3),x, algorithm="fricas")

[Out]

3/28*(3*b*d^2*x^2 - 4*b*c^2 + 7*a*c*d - (b*c*d - 7*a*d^2)*x)*(b*x + a)^(2/3)*(d*
x + c)^(1/3)/(a^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2 + (b^5*c^2 - 2*a*b^4*c*d + a^2
*b^3*d^2)*x^3 + 3*(a*b^4*c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*x^2 + 3*(a^2*b^3*c^2
- 2*a^3*b^2*c*d + a^4*b*d^2)*x)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(1/3)/(b*x+a)**(10/3),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{10}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/3)/(b*x + a)^(10/3),x, algorithm="giac")

[Out]

integrate((d*x + c)^(1/3)/(b*x + a)^(10/3), x)