3.71 \(\int \frac{(a+b x)^3}{c+d x^3} \, dx\)

Optimal. Leaf size=222 \[ \frac{\left (a^3 (-d)+3 a^2 b \sqrt [3]{c} d^{2/3}+b^3 c\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}-\frac{\left (a^3 (-d)+3 a^2 b \sqrt [3]{c} d^{2/3}+b^3 c\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac{\left (a^3 (-d)-3 a^2 b \sqrt [3]{c} d^{2/3}+b^3 c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{4/3}}+\frac{a b^2 \log \left (c+d x^3\right )}{d}+\frac{b^3 x}{d} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.574891, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ \frac{\left (a^3 (-d)+3 a^2 b \sqrt [3]{c} d^{2/3}+b^3 c\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}-\frac{\left (a^3 (-d)+3 a^2 b \sqrt [3]{c} d^{2/3}+b^3 c\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac{\left (a^3 (-d)-3 a^2 b \sqrt [3]{c} d^{2/3}+b^3 c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{4/3}}+\frac{a b^2 \log \left (c+d x^3\right )}{d}+\frac{b^3 x}{d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/(c + d*x^3),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a b^{2} \log{\left (c + d x^{3} \right )}}{d} + \frac{\int b^{3}\, dx}{d} + \frac{\left (a^{3} d - 3 a^{2} b \sqrt [3]{c} d^{\frac{2}{3}} - b^{3} c\right ) \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{2}{3}} d^{\frac{4}{3}}} - \frac{\left (a^{3} d - 3 a^{2} b \sqrt [3]{c} d^{\frac{2}{3}} - b^{3} c\right ) \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{2}{3}} d^{\frac{4}{3}}} - \frac{\sqrt{3} \left (a^{3} d + 3 a^{2} b \sqrt [3]{c} d^{\frac{2}{3}} - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{2}{3}} d^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*b**2*log(c + d*x**3)/d + Integral(b**3, x)/d + (a**3*d - 3*a**2*b*c**(1/3)*d**
(2/3) - b**3*c)*log(c**(1/3) + d**(1/3)*x)/(3*c**(2/3)*d**(4/3)) - (a**3*d - 3*a
**2*b*c**(1/3)*d**(2/3) - b**3*c)*log(c**(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*
x**2)/(6*c**(2/3)*d**(4/3)) - sqrt(3)*(a**3*d + 3*a**2*b*c**(1/3)*d**(2/3) - b**
3*c)*atan(sqrt(3)*(c**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(3*c**(2/3)*d**(4/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.331785, size = 214, normalized size = 0.96 \[ \frac{\left (a^3 (-d)+3 a^2 b \sqrt [3]{c} d^{2/3}+b^3 c\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )-2 \left (a^3 (-d)+3 a^2 b \sqrt [3]{c} d^{2/3}+b^3 c\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )+2 \sqrt{3} \left (a^3 (-d)-3 a^2 b \sqrt [3]{c} d^{2/3}+b^3 c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )+6 a b^2 c^{2/3} \sqrt [3]{d} \log \left (c+d x^3\right )+6 b^3 c^{2/3} \sqrt [3]{d} x}{6 c^{2/3} d^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

(6*b^3*c^(2/3)*d^(1/3)*x + 2*Sqrt[3]*(b^3*c - 3*a^2*b*c^(1/3)*d^(2/3) - a^3*d)*A
rcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] - 2*(b^3*c + 3*a^2*b*c^(1/3)*d^(2/3)
- a^3*d)*Log[c^(1/3) + d^(1/3)*x] + (b^3*c + 3*a^2*b*c^(1/3)*d^(2/3) - a^3*d)*Lo
g[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2] + 6*a*b^2*c^(2/3)*d^(1/3)*Log[c + d
*x^3])/(6*c^(2/3)*d^(4/3))

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 325, normalized size = 1.5 \[{\frac{{b}^{3}x}{d}}+{\frac{{a}^{3}}{3\,d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{b}^{3}c}{3\,{d}^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{3}}{6\,d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{b}^{3}c}{6\,{d}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}{a}^{3}}{3\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}{b}^{3}c}{3\,{d}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}b}{d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{a}^{2}b}{2\,d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{a}^{2}b\sqrt{3}}{d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{a{b}^{2}\ln \left ( d{x}^{3}+c \right ) }{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/(d*x^3+c),x)

[Out]

b^3*x/d+1/3/d/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*a^3-1/3/d^2/(c/d)^(2/3)*ln(x+(c/d)^(
1/3))*b^3*c-1/6/d/(c/d)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))*a^3+1/6/d^2/(c/d
)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))*b^3*c+1/3/d/(c/d)^(2/3)*3^(1/2)*arctan
(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*a^3-1/3/d^2/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(
1/2)*(2/(c/d)^(1/3)*x-1))*b^3*c-1/d*a^2*b/(c/d)^(1/3)*ln(x+(c/d)^(1/3))+1/2/d*a^
2*b/(c/d)^(1/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))+1/d*a^2*b*3^(1/2)/(c/d)^(1/3)*
arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))+a*b^2*ln(d*x^3+c)/d

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

Sympy [A]  time = 4.73957, size = 245, normalized size = 1.1 \[ \frac{b^{3} x}{d} + \operatorname{RootSum}{\left (27 t^{3} c^{2} d^{4} - 81 t^{2} a b^{2} c^{2} d^{3} + t \left (27 a^{5} b c d^{3} + 54 a^{2} b^{4} c^{2} d^{2}\right ) - a^{9} d^{3} + 3 a^{6} b^{3} c d^{2} - 3 a^{3} b^{6} c^{2} d + b^{9} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{27 t^{2} a^{2} b c^{2} d^{3} + 3 t a^{6} c d^{3} - 60 t a^{3} b^{3} c^{2} d^{2} + 3 t b^{6} c^{3} d + 15 a^{7} b^{2} c d^{2} + 15 a^{4} b^{5} c^{2} d - 3 a b^{8} c^{3}}{a^{9} d^{3} + 24 a^{6} b^{3} c d^{2} + 3 a^{3} b^{6} c^{2} d - b^{9} c^{3}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**3*x/d + RootSum(27*_t**3*c**2*d**4 - 81*_t**2*a*b**2*c**2*d**3 + _t*(27*a**5*
b*c*d**3 + 54*a**2*b**4*c**2*d**2) - a**9*d**3 + 3*a**6*b**3*c*d**2 - 3*a**3*b**
6*c**2*d + b**9*c**3, Lambda(_t, _t*log(x + (27*_t**2*a**2*b*c**2*d**3 + 3*_t*a*
*6*c*d**3 - 60*_t*a**3*b**3*c**2*d**2 + 3*_t*b**6*c**3*d + 15*a**7*b**2*c*d**2 +
15*a**4*b**5*c**2*d - 3*a*b**8*c**3)/(a**9*d**3 + 24*a**6*b**3*c*d**2 + 3*a**3*
b**6*c**2*d - b**9*c**3))))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216398, size = 327, normalized size = 1.47 \[ \frac{b^{3} x}{d} + \frac{a b^{2}{\rm ln}\left ({\left | d x^{3} + c \right |}\right )}{d} - \frac{\sqrt{3}{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-c d^{2}\right )^{\frac{1}{3}} a^{3} d + 3 \, \left (-c d^{2}\right )^{\frac{2}{3}} a^{2} b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{3 \, c d^{2}} - \frac{{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-c d^{2}\right )^{\frac{1}{3}} a^{3} d - 3 \, \left (-c d^{2}\right )^{\frac{2}{3}} a^{2} b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \, c d^{2}} - \frac{{\left (3 \, a^{2} b d^{3} \left (-\frac{c}{d}\right )^{\frac{1}{3}} - b^{3} c d^{2} + a^{3} d^{3}\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \, c d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^3*x/d + a*b^2*ln(abs(d*x^3 + c))/d - 1/3*sqrt(3)*((-c*d^2)^(1/3)*b^3*c - (-c*d
^2)^(1/3)*a^3*d + 3*(-c*d^2)^(2/3)*a^2*b)*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3)
)/(-c/d)^(1/3))/(c*d^2) - 1/6*((-c*d^2)^(1/3)*b^3*c - (-c*d^2)^(1/3)*a^3*d - 3*(
-c*d^2)^(2/3)*a^2*b)*ln(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(c*d^2) - 1/3*(3*a^
2*b*d^3*(-c/d)^(1/3) - b^3*c*d^2 + a^3*d^3)*(-c/d)^(1/3)*ln(abs(x - (-c/d)^(1/3)
))/(c*d^3)