3.10 \(\int \frac{c+d x}{\left (a+b x^3\right )^4} \, dx\)

Optimal. Leaf size=240 \[ -\frac{\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac{2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{2 \left (7 \sqrt [3]{a} d+20 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}+\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{x (c+d x)}{9 a \left (a+b x^3\right )^3} \]

[Out]

(x*(c + d*x))/(9*a*(a + b*x^3)^3) + (x*(8*c + 7*d*x))/(54*a^2*(a + b*x^3)^2) + (
2*x*(10*c + 7*d*x))/(81*a^3*(a + b*x^3)) - (2*(20*b^(1/3)*c + 7*a^(1/3)*d)*ArcTa
n[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(11/3)*b^(2/3)) + (2
*(20*b^(1/3)*c - 7*a^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(11/3)*b^(2/3)) -
((20*b^(1/3)*c - 7*a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(
243*a^(11/3)*b^(2/3))

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Rubi [A]  time = 0.42287, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ -\frac{\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac{2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{2 \left (7 \sqrt [3]{a} d+20 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}+\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{x (c+d x)}{9 a \left (a+b x^3\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)/(a + b*x^3)^4,x]

[Out]

(x*(c + d*x))/(9*a*(a + b*x^3)^3) + (x*(8*c + 7*d*x))/(54*a^2*(a + b*x^3)^2) + (
2*x*(10*c + 7*d*x))/(81*a^3*(a + b*x^3)) - (2*(20*b^(1/3)*c + 7*a^(1/3)*d)*ArcTa
n[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(11/3)*b^(2/3)) + (2
*(20*b^(1/3)*c - 7*a^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(11/3)*b^(2/3)) -
((20*b^(1/3)*c - 7*a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(
243*a^(11/3)*b^(2/3))

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Rubi in Sympy [A]  time = 64.5289, size = 228, normalized size = 0.95 \[ \frac{x \left (c + d x\right )}{9 a \left (a + b x^{3}\right )^{3}} + \frac{x \left (8 c + 7 d x\right )}{54 a^{2} \left (a + b x^{3}\right )^{2}} + \frac{x \left (40 c + 28 d x\right )}{162 a^{3} \left (a + b x^{3}\right )} - \frac{2 \left (7 \sqrt [3]{a} d - 20 \sqrt [3]{b} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{243 a^{\frac{11}{3}} b^{\frac{2}{3}}} + \frac{\left (7 \sqrt [3]{a} d - 20 \sqrt [3]{b} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{243 a^{\frac{11}{3}} b^{\frac{2}{3}}} - \frac{2 \sqrt{3} \left (7 \sqrt [3]{a} d + 20 \sqrt [3]{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{243 a^{\frac{11}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c + d*x)/(9*a*(a + b*x**3)**3) + x*(8*c + 7*d*x)/(54*a**2*(a + b*x**3)**2) +
x*(40*c + 28*d*x)/(162*a**3*(a + b*x**3)) - 2*(7*a**(1/3)*d - 20*b**(1/3)*c)*log
(a**(1/3) + b**(1/3)*x)/(243*a**(11/3)*b**(2/3)) + (7*a**(1/3)*d - 20*b**(1/3)*c
)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(243*a**(11/3)*b**(2/3)) -
2*sqrt(3)*(7*a**(1/3)*d + 20*b**(1/3)*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*
x/3)/a**(1/3))/(243*a**(11/3)*b**(2/3))

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Mathematica [A]  time = 0.440452, size = 229, normalized size = 0.95 \[ \frac{\frac{2 \left (7 a^{2/3} d-20 \sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{4 \left (20 \sqrt [3]{a} \sqrt [3]{b} c-7 a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac{54 a^3 x (c+d x)}{\left (a+b x^3\right )^3}+\frac{9 a^2 x (8 c+7 d x)}{\left (a+b x^3\right )^2}-\frac{4 \sqrt{3} \sqrt [3]{a} \left (7 \sqrt [3]{a} d+20 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{12 a x (10 c+7 d x)}{a+b x^3}}{486 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((54*a^3*x*(c + d*x))/(a + b*x^3)^3 + (9*a^2*x*(8*c + 7*d*x))/(a + b*x^3)^2 + (1
2*a*x*(10*c + 7*d*x))/(a + b*x^3) - (4*Sqrt[3]*a^(1/3)*(20*b^(1/3)*c + 7*a^(1/3)
*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (4*(20*a^(1/3)*b^(1/3
)*c - 7*a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (2*(-20*a^(1/3)*b^(1/3)*c
+ 7*a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(486*a^
4)

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Maple [A]  time = 0.007, size = 306, normalized size = 1.3 \[{\frac{cx}{9\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{4\,cx}{27\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{20\,cx}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{40\,c}{243\,{a}^{3}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,c}{243\,{a}^{3}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{40\,c\sqrt{3}}{243\,{a}^{3}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{d{x}^{2}}{9\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{7\,d{x}^{2}}{54\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{14\,d{x}^{2}}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{14\,d}{243\,{a}^{3}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{7\,d}{243\,{a}^{3}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{14\,d\sqrt{3}}{243\,{a}^{3}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(b*x^3+a)^4,x)

[Out]

1/9*c/a*x/(b*x^3+a)^3+4/27*c/a^2*x/(b*x^3+a)^2+20/81*c/a^3*x/(b*x^3+a)+40/243*c/
a^3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-20/243*c/a^3/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1
/3)+(a/b)^(2/3))+40/243*c/a^3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^
(1/3)*x-1))+1/9*d/a*x^2/(b*x^3+a)^3+7/54*d/a^2*x^2/(b*x^3+a)^2+14/81*d/a^3*x^2/(
b*x^3+a)-14/243*d/a^3/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+7/243*d/a^3/b/(a/b)^(1/3)*
ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+14/243*d/a^3*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*
3^(1/2)*(2/(a/b)^(1/3)*x-1))

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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((d*x + c)/(b*x^3 + a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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((d*x + c)/(b*x^3 + a)^4,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 3.93502, size = 185, normalized size = 0.77 \[ \operatorname{RootSum}{\left (14348907 t^{3} a^{11} b^{2} + 408240 t a^{4} b c d + 2744 a d^{3} - 64000 b c^{3}, \left ( t \mapsto t \log{\left (x + \frac{413343 t^{2} a^{8} b d + 194400 t a^{4} b c^{2} + 7840 a c d^{2}}{1372 a d^{3} + 32000 b c^{3}} \right )} \right )\right )} + \frac{82 a^{2} c x + 67 a^{2} d x^{2} + 104 a b c x^{4} + 77 a b d x^{5} + 40 b^{2} c x^{7} + 28 b^{2} d x^{8}}{162 a^{6} + 486 a^{5} b x^{3} + 486 a^{4} b^{2} x^{6} + 162 a^{3} b^{3} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(14348907*_t**3*a**11*b**2 + 408240*_t*a**4*b*c*d + 2744*a*d**3 - 64000*b
*c**3, Lambda(_t, _t*log(x + (413343*_t**2*a**8*b*d + 194400*_t*a**4*b*c**2 + 78
40*a*c*d**2)/(1372*a*d**3 + 32000*b*c**3)))) + (82*a**2*c*x + 67*a**2*d*x**2 + 1
04*a*b*c*x**4 + 77*a*b*d*x**5 + 40*b**2*c*x**7 + 28*b**2*d*x**8)/(162*a**6 + 486
*a**5*b*x**3 + 486*a**4*b**2*x**6 + 162*a**3*b**3*x**9)

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GIAC/XCAS [A]  time = 0.214368, size = 312, normalized size = 1.3 \[ -\frac{2 \,{\left (7 \, d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 20 \, c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{243 \, a^{4}} + \frac{2 \, \sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{243 \, a^{4} b^{2}} + \frac{28 \, b^{2} d x^{8} + 40 \, b^{2} c x^{7} + 77 \, a b d x^{5} + 104 \, a b c x^{4} + 67 \, a^{2} d x^{2} + 82 \, a^{2} c x}{162 \,{\left (b x^{3} + a\right )}^{3} a^{3}} + \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c + 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{243 \, a^{5} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/243*(7*d*(-a/b)^(1/3) + 20*c)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^4 + 2/
243*sqrt(3)*(20*(-a*b^2)^(1/3)*b*c - 7*(-a*b^2)^(2/3)*d)*arctan(1/3*sqrt(3)*(2*x
+ (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^2) + 1/162*(28*b^2*d*x^8 + 40*b^2*c*x^7 +
77*a*b*d*x^5 + 104*a*b*c*x^4 + 67*a^2*d*x^2 + 82*a^2*c*x)/((b*x^3 + a)^3*a^3) +
1/243*(20*(-a*b^2)^(1/3)*a*b^3*c + 7*(-a*b^2)^(2/3)*a*b^2*d)*ln(x^2 + x*(-a/b)^(
1/3) + (-a/b)^(2/3))/(a^5*b^4)