3.1582 \(\int \frac{\sqrt [3]{c+d x}}{(a+b x)^{16/3}} \, dx\)

Optimal. Leaf size=136 \[ \frac{243 d^3 (c+d x)^{4/3}}{1820 (a+b x)^{4/3} (b c-a d)^4}-\frac{81 d^2 (c+d x)^{4/3}}{455 (a+b x)^{7/3} (b c-a d)^3}+\frac{27 d (c+d x)^{4/3}}{130 (a+b x)^{10/3} (b c-a d)^2}-\frac{3 (c+d x)^{4/3}}{13 (a+b x)^{13/3} (b c-a d)} \]

[Out]

(-3*(c + d*x)^(4/3))/(13*(b*c - a*d)*(a + b*x)^(13/3)) + (27*d*(c + d*x)^(4/3))/
(130*(b*c - a*d)^2*(a + b*x)^(10/3)) - (81*d^2*(c + d*x)^(4/3))/(455*(b*c - a*d)
^3*(a + b*x)^(7/3)) + (243*d^3*(c + d*x)^(4/3))/(1820*(b*c - a*d)^4*(a + b*x)^(4
/3))

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Rubi [A]  time = 0.126472, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{243 d^3 (c+d x)^{4/3}}{1820 (a+b x)^{4/3} (b c-a d)^4}-\frac{81 d^2 (c+d x)^{4/3}}{455 (a+b x)^{7/3} (b c-a d)^3}+\frac{27 d (c+d x)^{4/3}}{130 (a+b x)^{10/3} (b c-a d)^2}-\frac{3 (c+d x)^{4/3}}{13 (a+b x)^{13/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(1/3)/(a + b*x)^(16/3),x]

[Out]

(-3*(c + d*x)^(4/3))/(13*(b*c - a*d)*(a + b*x)^(13/3)) + (27*d*(c + d*x)^(4/3))/
(130*(b*c - a*d)^2*(a + b*x)^(10/3)) - (81*d^2*(c + d*x)^(4/3))/(455*(b*c - a*d)
^3*(a + b*x)^(7/3)) + (243*d^3*(c + d*x)^(4/3))/(1820*(b*c - a*d)^4*(a + b*x)^(4
/3))

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Rubi in Sympy [A]  time = 20.9597, size = 121, normalized size = 0.89 \[ \frac{243 d^{3} \left (c + d x\right )^{\frac{4}{3}}}{1820 \left (a + b x\right )^{\frac{4}{3}} \left (a d - b c\right )^{4}} + \frac{81 d^{2} \left (c + d x\right )^{\frac{4}{3}}}{455 \left (a + b x\right )^{\frac{7}{3}} \left (a d - b c\right )^{3}} + \frac{27 d \left (c + d x\right )^{\frac{4}{3}}}{130 \left (a + b x\right )^{\frac{10}{3}} \left (a d - b c\right )^{2}} + \frac{3 \left (c + d x\right )^{\frac{4}{3}}}{13 \left (a + b x\right )^{\frac{13}{3}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(1/3)/(b*x+a)**(16/3),x)

[Out]

243*d**3*(c + d*x)**(4/3)/(1820*(a + b*x)**(4/3)*(a*d - b*c)**4) + 81*d**2*(c +
d*x)**(4/3)/(455*(a + b*x)**(7/3)*(a*d - b*c)**3) + 27*d*(c + d*x)**(4/3)/(130*(
a + b*x)**(10/3)*(a*d - b*c)**2) + 3*(c + d*x)**(4/3)/(13*(a + b*x)**(13/3)*(a*d
- b*c))

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Mathematica [A]  time = 0.180244, size = 118, normalized size = 0.87 \[ \frac{3 (c+d x)^{4/3} \left (455 a^3 d^3+195 a^2 b d^2 (3 d x-4 c)+39 a b^2 d \left (14 c^2-12 c d x+9 d^2 x^2\right )+b^3 \left (-140 c^3+126 c^2 d x-108 c d^2 x^2+81 d^3 x^3\right )\right )}{1820 (a+b x)^{13/3} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(1/3)/(a + b*x)^(16/3),x]

[Out]

(3*(c + d*x)^(4/3)*(455*a^3*d^3 + 195*a^2*b*d^2*(-4*c + 3*d*x) + 39*a*b^2*d*(14*
c^2 - 12*c*d*x + 9*d^2*x^2) + b^3*(-140*c^3 + 126*c^2*d*x - 108*c*d^2*x^2 + 81*d
^3*x^3)))/(1820*(b*c - a*d)^4*(a + b*x)^(13/3))

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Maple [A]  time = 0.012, size = 171, normalized size = 1.3 \[{\frac{243\,{b}^{3}{d}^{3}{x}^{3}+1053\,a{b}^{2}{d}^{3}{x}^{2}-324\,{b}^{3}c{d}^{2}{x}^{2}+1755\,{a}^{2}b{d}^{3}x-1404\,a{b}^{2}c{d}^{2}x+378\,{b}^{3}{c}^{2}dx+1365\,{a}^{3}{d}^{3}-2340\,{a}^{2}cb{d}^{2}+1638\,a{b}^{2}{c}^{2}d-420\,{b}^{3}{c}^{3}}{1820\,{d}^{4}{a}^{4}-7280\,b{d}^{3}c{a}^{3}+10920\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-7280\,{b}^{3}d{c}^{3}a+1820\,{b}^{4}{c}^{4}} \left ( dx+c \right ) ^{{\frac{4}{3}}} \left ( bx+a \right ) ^{-{\frac{13}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(1/3)/(b*x+a)^(16/3),x)

[Out]

3/1820*(d*x+c)^(4/3)*(81*b^3*d^3*x^3+351*a*b^2*d^3*x^2-108*b^3*c*d^2*x^2+585*a^2
*b*d^3*x-468*a*b^2*c*d^2*x+126*b^3*c^2*d*x+455*a^3*d^3-780*a^2*b*c*d^2+546*a*b^2
*c^2*d-140*b^3*c^3)/(b*x+a)^(13/3)/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*
b^3*c^3*d+b^4*c^4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{16}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/3)/(b*x + a)^(16/3),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(1/3)/(b*x + a)^(16/3), x)

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Fricas [A]  time = 0.211152, size = 720, normalized size = 5.29 \[ \frac{3 \,{\left (81 \, b^{3} d^{4} x^{4} - 140 \, b^{3} c^{4} + 546 \, a b^{2} c^{3} d - 780 \, a^{2} b c^{2} d^{2} + 455 \, a^{3} c d^{3} - 27 \,{\left (b^{3} c d^{3} - 13 \, a b^{2} d^{4}\right )} x^{3} + 9 \,{\left (2 \, b^{3} c^{2} d^{2} - 13 \, a b^{2} c d^{3} + 65 \, a^{2} b d^{4}\right )} x^{2} -{\left (14 \, b^{3} c^{3} d - 78 \, a b^{2} c^{2} d^{2} + 195 \, a^{2} b c d^{3} - 455 \, a^{3} d^{4}\right )} x\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{1820 \,{\left (a^{5} b^{4} c^{4} - 4 \, a^{6} b^{3} c^{3} d + 6 \, a^{7} b^{2} c^{2} d^{2} - 4 \, a^{8} b c d^{3} + a^{9} d^{4} +{\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} x^{5} + 5 \,{\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} x^{4} + 10 \,{\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} x^{3} + 10 \,{\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} x^{2} + 5 \,{\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/3)/(b*x + a)^(16/3),x, algorithm="fricas")

[Out]

3/1820*(81*b^3*d^4*x^4 - 140*b^3*c^4 + 546*a*b^2*c^3*d - 780*a^2*b*c^2*d^2 + 455
*a^3*c*d^3 - 27*(b^3*c*d^3 - 13*a*b^2*d^4)*x^3 + 9*(2*b^3*c^2*d^2 - 13*a*b^2*c*d
^3 + 65*a^2*b*d^4)*x^2 - (14*b^3*c^3*d - 78*a*b^2*c^2*d^2 + 195*a^2*b*c*d^3 - 45
5*a^3*d^4)*x)*(b*x + a)^(2/3)*(d*x + c)^(1/3)/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6
*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4 + (b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^
7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*x^5 + 5*(a*b^8*c^4 - 4*a^2*b^7*c^3*d
+ 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*x^4 + 10*(a^2*b^7*c^4 - 4*a
^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*x^3 + 10*(a^3*
b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d^4)*x
^2 + 5*(a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^
8*b*d^4)*x)

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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)**(16/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{16}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/3)/(b*x + a)^(16/3),x, algorithm="giac")

[Out]

integrate((d*x + c)^(1/3)/(b*x + a)^(16/3), x)