3.2859 \(\int \frac{1}{(c+d x)^3 \left (a+b (c+d x)^3\right )} \, dx\)

Optimal. Leaf size=156 \[ -\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{5/3} d}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{5/3} d}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} d}-\frac{1}{2 a d (c+d x)^2} \]

[Out]

-1/(2*a*d*(c + d*x)^2) + (b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3
]*a^(1/3))])/(Sqrt[3]*a^(5/3)*d) - (b^(2/3)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(3
*a^(5/3)*d) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*
x)^2])/(6*a^(5/3)*d)

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Rubi [A]  time = 0.294321, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{5/3} d}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{5/3} d}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} d}-\frac{1}{2 a d (c+d x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*a*d*(c + d*x)^2) + (b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3
]*a^(1/3))])/(Sqrt[3]*a^(5/3)*d) - (b^(2/3)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(3
*a^(5/3)*d) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*
x)^2])/(6*a^(5/3)*d)

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Rubi in Sympy [A]  time = 39.342, size = 148, normalized size = 0.95 \[ - \frac{1}{2 a d \left (c + d x\right )^{2}} - \frac{b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 a^{\frac{5}{3}} d} + \frac{b^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 a^{\frac{5}{3}} d} + \frac{\sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*a*d*(c + d*x)**2) - b**(2/3)*log(a**(1/3) + b**(1/3)*(c + d*x))/(3*a**(5/3
)*d) + b**(2/3)*log(a**(2/3) + a**(1/3)*b**(1/3)*(-c - d*x) + b**(2/3)*(c + d*x)
**2)/(6*a**(5/3)*d) + sqrt(3)*b**(2/3)*atan(sqrt(3)*(a**(1/3)/3 + b**(1/3)*(-2*c
/3 - 2*d*x/3))/a**(1/3))/(3*a**(5/3)*d)

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Mathematica [A]  time = 0.067749, size = 139, normalized size = 0.89 \[ \frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-\frac{3 a^{2/3}}{(c+d x)^2}-2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-2 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 a^{5/3} d} \]

Antiderivative was successfully verified.

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

[Out]

((-3*a^(2/3))/(c + d*x)^2 - 2*Sqrt[3]*b^(2/3)*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c +
d*x))/(Sqrt[3]*a^(1/3))] - 2*b^(2/3)*Log[a^(1/3) + b^(1/3)*(c + d*x)] + b^(2/3)*
Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/(6*a^(5/3)*d)

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Maple [C]  time = 0.007, size = 87, normalized size = 0.6 \[ -{\frac{1}{3\,ad}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}}-{\frac{1}{2\,ad \left ( dx+c \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/a/d*sum(1/(_R^2*d^2+2*_R*c*d+c^2)*ln(x-_R),_R=RootOf(_Z^3*b*d^3+3*_Z^2*b*c*
d^2+3*_Z*b*c^2*d+b*c^3+a))-1/2/a/d/(d*x+c)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{b \int \frac{1}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{a} - \frac{1}{2 \,{\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*integrate(1/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a), x)/a - 1/2
/(a*d^3*x^2 + 2*a*c*d^2*x + a*c^2*d)

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Fricas [A]  time = 0.217644, size = 343, normalized size = 2.2 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} +{\left (a b d x + a b c\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3}{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b d x + b c - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 6 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + 2 \, \sqrt{3}{\left (b d x + b c\right )}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}\right )}}{18 \,{\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(sqrt(3)*(d^2*x^2 + 2*c*d*x + c^2)*(-b^2/a^2)^(1/3)*log(b^2*d^2*x^
2 + 2*b^2*c*d*x + b^2*c^2 + a^2*(-b^2/a^2)^(2/3) + (a*b*d*x + a*b*c)*(-b^2/a^2)^
(1/3)) - 2*sqrt(3)*(d^2*x^2 + 2*c*d*x + c^2)*(-b^2/a^2)^(1/3)*log(b*d*x + b*c -
a*(-b^2/a^2)^(1/3)) + 6*(d^2*x^2 + 2*c*d*x + c^2)*(-b^2/a^2)^(1/3)*arctan(1/3*(s
qrt(3)*a*(-b^2/a^2)^(1/3) + 2*sqrt(3)*(b*d*x + b*c))/(a*(-b^2/a^2)^(1/3))) + 3*s
qrt(3))/(a*d^3*x^2 + 2*a*c*d^2*x + a*c^2*d)

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Sympy [A]  time = 4.37909, size = 61, normalized size = 0.39 \[ - \frac{1}{2 a c^{2} d + 4 a c d^{2} x + 2 a d^{3} x^{2}} + \frac{\operatorname{RootSum}{\left (27 t^{3} a^{5} + b^{2}, \left ( t \mapsto t \log{\left (x + \frac{- 3 t a^{2} + b c}{b d} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*a*c**2*d + 4*a*c*d**2*x + 2*a*d**3*x**2) + RootSum(27*_t**3*a**5 + b**2, L
ambda(_t, _t*log(x + (-3*_t*a**2 + b*c)/(b*d))))/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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