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

Optimal. Leaf size=197 \[ -\frac{(5 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}}+\frac{(5 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac{(5 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} d^{4/3}}+\frac{x (5 a d+b c)}{18 c^2 d \left (c+d x^3\right )}-\frac{x (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]

[Out]

-((b*c - a*d)*x)/(6*c*d*(c + d*x^3)^2) + ((b*c + 5*a*d)*x)/(18*c^2*d*(c + d*x^3)
) - ((b*c + 5*a*d)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(9*Sqrt[3]
*c^(8/3)*d^(4/3)) + ((b*c + 5*a*d)*Log[c^(1/3) + d^(1/3)*x])/(27*c^(8/3)*d^(4/3)
) - ((b*c + 5*a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d
^(4/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.22915, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ -\frac{(5 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}}+\frac{(5 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac{(5 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} d^{4/3}}+\frac{x (5 a d+b c)}{18 c^2 d \left (c+d x^3\right )}-\frac{x (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*x)/(6*c*d*(c + d*x^3)^2) + ((b*c + 5*a*d)*x)/(18*c^2*d*(c + d*x^3)
) - ((b*c + 5*a*d)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(9*Sqrt[3]
*c^(8/3)*d^(4/3)) + ((b*c + 5*a*d)*Log[c^(1/3) + d^(1/3)*x])/(27*c^(8/3)*d^(4/3)
) - ((b*c + 5*a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d
^(4/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 39.1236, size = 180, normalized size = 0.91 \[ \frac{x \left (a d - b c\right )}{6 c d \left (c + d x^{3}\right )^{2}} + \frac{x \left (5 a d + b c\right )}{18 c^{2} d \left (c + d x^{3}\right )} + \frac{\left (5 a d + b c\right ) \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{27 c^{\frac{8}{3}} d^{\frac{4}{3}}} - \frac{\left (5 a d + b c\right ) \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{54 c^{\frac{8}{3}} d^{\frac{4}{3}}} - \frac{\sqrt{3} \left (5 a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{27 c^{\frac{8}{3}} d^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a*d - b*c)/(6*c*d*(c + d*x**3)**2) + x*(5*a*d + b*c)/(18*c**2*d*(c + d*x**3))
+ (5*a*d + b*c)*log(c**(1/3) + d**(1/3)*x)/(27*c**(8/3)*d**(4/3)) - (5*a*d + b*
c)*log(c**(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**2)/(54*c**(8/3)*d**(4/3)) -
sqrt(3)*(5*a*d + b*c)*atan(sqrt(3)*(c**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(27*c
**(8/3)*d**(4/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.225062, size = 175, normalized size = 0.89 \[ \frac{-(5 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )-\frac{9 c^{5/3} \sqrt [3]{d} x (b c-a d)}{\left (c+d x^3\right )^2}+\frac{3 c^{2/3} \sqrt [3]{d} x (5 a d+b c)}{c+d x^3}+2 (5 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-2 \sqrt{3} (5 a d+b c) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{54 c^{8/3} d^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-9*c^(5/3)*d^(1/3)*(b*c - a*d)*x)/(c + d*x^3)^2 + (3*c^(2/3)*d^(1/3)*(b*c + 5*
a*d)*x)/(c + d*x^3) - 2*Sqrt[3]*(b*c + 5*a*d)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))
/Sqrt[3]] + 2*(b*c + 5*a*d)*Log[c^(1/3) + d^(1/3)*x] - (b*c + 5*a*d)*Log[c^(2/3)
- c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d^(4/3))

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 249, normalized size = 1.3 \[{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{2}} \left ({\frac{ \left ( 5\,ad+bc \right ){x}^{4}}{18\,{c}^{2}}}+{\frac{ \left ( 4\,ad-bc \right ) x}{9\,cd}} \right ) }+{\frac{5\,a}{27\,{c}^{2}d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{27\,c{d}^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,a}{54\,{c}^{2}d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{54\,c{d}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,\sqrt{3}a}{27\,{c}^{2}d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}b}{27\,c{d}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)/(d*x^3+c)^3,x)

[Out]

(1/18*(5*a*d+b*c)/c^2*x^4+1/9*(4*a*d-b*c)/c/d*x)/(d*x^3+c)^2+5/27/c^2/d/(c/d)^(2
/3)*ln(x+(c/d)^(1/3))*a+1/27/c/d^2/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*b-5/54/c^2/d/(c
/d)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))*a-1/54/c/d^2/(c/d)^(2/3)*ln(x^2-x*(c
/d)^(1/3)+(c/d)^(2/3))*b+5/27/c^2/d/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c
/d)^(1/3)*x-1))*a+1/27/c/d^2/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/
3)*x-1))*b

_______________________________________________________________________________________

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^3 + a)/(d*x^3 + c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.214195, size = 414, normalized size = 2.1 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (b c d^{2} + 5 \, a d^{3}\right )} x^{6} + b c^{3} + 5 \, a c^{2} d + 2 \,{\left (b c^{2} d + 5 \, a c d^{2}\right )} x^{3}\right )} \log \left (\left (c^{2} d\right )^{\frac{2}{3}} x^{2} - \left (c^{2} d\right )^{\frac{1}{3}} c x + c^{2}\right ) - 2 \, \sqrt{3}{\left ({\left (b c d^{2} + 5 \, a d^{3}\right )} x^{6} + b c^{3} + 5 \, a c^{2} d + 2 \,{\left (b c^{2} d + 5 \, a c d^{2}\right )} x^{3}\right )} \log \left (\left (c^{2} d\right )^{\frac{1}{3}} x + c\right ) - 6 \,{\left ({\left (b c d^{2} + 5 \, a d^{3}\right )} x^{6} + b c^{3} + 5 \, a c^{2} d + 2 \,{\left (b c^{2} d + 5 \, a c d^{2}\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (c^{2} d\right )^{\frac{1}{3}} x - \sqrt{3} c}{3 \, c}\right ) - 3 \, \sqrt{3}{\left ({\left (b c d + 5 \, a d^{2}\right )} x^{4} - 2 \,{\left (b c^{2} - 4 \, a c d\right )} x\right )} \left (c^{2} d\right )^{\frac{1}{3}}\right )}}{162 \,{\left (c^{2} d^{3} x^{6} + 2 \, c^{3} d^{2} x^{3} + c^{4} d\right )} \left (c^{2} d\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(sqrt(3)*((b*c*d^2 + 5*a*d^3)*x^6 + b*c^3 + 5*a*c^2*d + 2*(b*c^2*
d + 5*a*c*d^2)*x^3)*log((c^2*d)^(2/3)*x^2 - (c^2*d)^(1/3)*c*x + c^2) - 2*sqrt(3)
*((b*c*d^2 + 5*a*d^3)*x^6 + b*c^3 + 5*a*c^2*d + 2*(b*c^2*d + 5*a*c*d^2)*x^3)*log
((c^2*d)^(1/3)*x + c) - 6*((b*c*d^2 + 5*a*d^3)*x^6 + b*c^3 + 5*a*c^2*d + 2*(b*c^
2*d + 5*a*c*d^2)*x^3)*arctan(1/3*(2*sqrt(3)*(c^2*d)^(1/3)*x - sqrt(3)*c)/c) - 3*
sqrt(3)*((b*c*d + 5*a*d^2)*x^4 - 2*(b*c^2 - 4*a*c*d)*x)*(c^2*d)^(1/3))/((c^2*d^3
*x^6 + 2*c^3*d^2*x^3 + c^4*d)*(c^2*d)^(1/3))

_______________________________________________________________________________________

Sympy [A]  time = 3.77063, size = 133, normalized size = 0.68 \[ \frac{x^{4} \left (5 a d^{2} + b c d\right ) + x \left (8 a c d - 2 b c^{2}\right )}{18 c^{4} d + 36 c^{3} d^{2} x^{3} + 18 c^{2} d^{3} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} c^{8} d^{4} - 125 a^{3} d^{3} - 75 a^{2} b c d^{2} - 15 a b^{2} c^{2} d - b^{3} c^{3}, \left ( t \mapsto t \log{\left (\frac{27 t c^{3} d}{5 a d + b c} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x**4*(5*a*d**2 + b*c*d) + x*(8*a*c*d - 2*b*c**2))/(18*c**4*d + 36*c**3*d**2*x**
3 + 18*c**2*d**3*x**6) + RootSum(19683*_t**3*c**8*d**4 - 125*a**3*d**3 - 75*a**2
*b*c*d**2 - 15*a*b**2*c**2*d - b**3*c**3, Lambda(_t, _t*log(27*_t*c**3*d/(5*a*d
+ b*c) + x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220801, size = 273, normalized size = 1.39 \[ -\frac{{\left (b c + 5 \, a d\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{27 \, c^{3} d} + \frac{\sqrt{3}{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b c + 5 \, \left (-c d^{2}\right )^{\frac{1}{3}} a d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{27 \, c^{3} d^{2}} + \frac{{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b c + 5 \, \left (-c d^{2}\right )^{\frac{1}{3}} a d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{54 \, c^{3} d^{2}} + \frac{b c d x^{4} + 5 \, a d^{2} x^{4} - 2 \, b c^{2} x + 8 \, a c d x}{18 \,{\left (d x^{3} + c\right )}^{2} c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(b*c + 5*a*d)*(-c/d)^(1/3)*ln(abs(x - (-c/d)^(1/3)))/(c^3*d) + 1/27*sqrt(3
)*((-c*d^2)^(1/3)*b*c + 5*(-c*d^2)^(1/3)*a*d)*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(
1/3))/(-c/d)^(1/3))/(c^3*d^2) + 1/54*((-c*d^2)^(1/3)*b*c + 5*(-c*d^2)^(1/3)*a*d)
*ln(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(c^3*d^2) + 1/18*(b*c*d*x^4 + 5*a*d^2*x
^4 - 2*b*c^2*x + 8*a*c*d*x)/((d*x^3 + c)^2*c^2*d)