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

Optimal. Leaf size=393 \[ \frac{b^{2/3} d^2 \left (-3 a^{2/3} \sqrt [3]{b} c+a+b c^3\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \left (a+b c^3\right )^3}-\frac{b^{2/3} d^2 \left (6 a^{4/3} b^{2/3} c^2+a^2-3 \sqrt [3]{a} b^{5/3} c^5-7 a b c^3+b^2 c^6\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )^3}+\frac{b^{2/3} d^2 \left (6 a^{4/3} b^{2/3} c^2+a^2-3 \sqrt [3]{a} b^{5/3} c^5-7 a b c^3+b^2 c^6\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )^3}-\frac{3 b c d^2 \log (x) \left (a-2 b c^3\right )}{\left (a+b c^3\right )^3}+\frac{b c d^2 \left (a-2 b c^3\right ) \log \left (a+b (c+d x)^3\right )}{\left (a+b c^3\right )^3}-\frac{1}{2 x^2 \left (a+b c^3\right )}+\frac{3 b c^2 d}{x \left (a+b c^3\right )^2} \]

[Out]

-1/(2*(a + b*c^3)*x^2) + (3*b*c^2*d)/((a + b*c^3)^2*x) + (b^(2/3)*(a^(1/3) + b^(
1/3)*c)^3*(a - 3*a^(2/3)*b^(1/3)*c + b*c^3)*d^2*ArcTan[(a^(1/3) - 2*b^(1/3)*(c +
d*x))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*(a + b*c^3)^3) - (3*b*c*(a - 2*b*c^3
)*d^2*Log[x])/(a + b*c^3)^3 - (b^(2/3)*(a^2 + 6*a^(4/3)*b^(2/3)*c^2 - 7*a*b*c^3
- 3*a^(1/3)*b^(5/3)*c^5 + b^2*c^6)*d^2*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(3*a^(2
/3)*(a + b*c^3)^3) + (b^(2/3)*(a^2 + 6*a^(4/3)*b^(2/3)*c^2 - 7*a*b*c^3 - 3*a^(1/
3)*b^(5/3)*c^5 + b^2*c^6)*d^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*
(c + d*x)^2])/(6*a^(2/3)*(a + b*c^3)^3) + (b*c*(a - 2*b*c^3)*d^2*Log[a + b*(c +
d*x)^3])/(a + b*c^3)^3

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Rubi [A]  time = 1.21127, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ \frac{b^{2/3} d^2 \left (-3 a^{2/3} \sqrt [3]{b} c+a+b c^3\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \left (a+b c^3\right )^3}-\frac{b^{2/3} d^2 \left (6 a^{4/3} b^{2/3} c^2+a^2-3 \sqrt [3]{a} b^{5/3} c^5-7 a b c^3+b^2 c^6\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )^3}+\frac{b^{2/3} d^2 \left (6 a^{4/3} b^{2/3} c^2+a^2-3 \sqrt [3]{a} b^{5/3} c^5-7 a b c^3+b^2 c^6\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )^3}-\frac{3 b c d^2 \log (x) \left (a-2 b c^3\right )}{\left (a+b c^3\right )^3}+\frac{b c d^2 \left (a-2 b c^3\right ) \log \left (a+b (c+d x)^3\right )}{\left (a+b c^3\right )^3}-\frac{1}{2 x^2 \left (a+b c^3\right )}+\frac{3 b c^2 d}{x \left (a+b c^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*(a + b*c^3)*x^2) + (3*b*c^2*d)/((a + b*c^3)^2*x) + (b^(2/3)*(a^(1/3) + b^(
1/3)*c)^3*(a - 3*a^(2/3)*b^(1/3)*c + b*c^3)*d^2*ArcTan[(a^(1/3) - 2*b^(1/3)*(c +
d*x))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*(a + b*c^3)^3) - (3*b*c*(a - 2*b*c^3
)*d^2*Log[x])/(a + b*c^3)^3 - (b^(2/3)*(a^2 + 6*a^(4/3)*b^(2/3)*c^2 - 7*a*b*c^3
- 3*a^(1/3)*b^(5/3)*c^5 + b^2*c^6)*d^2*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(3*a^(2
/3)*(a + b*c^3)^3) + (b^(2/3)*(a^2 + 6*a^(4/3)*b^(2/3)*c^2 - 7*a*b*c^3 - 3*a^(1/
3)*b^(5/3)*c^5 + b^2*c^6)*d^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*
(c + d*x)^2])/(6*a^(2/3)*(a + b*c^3)^3) + (b*c*(a - 2*b*c^3)*d^2*Log[a + b*(c +
d*x)^3])/(a + b*c^3)^3

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.23049, size = 244, normalized size = 0.62 \[ -\frac{2 d^2 x^2 \text{RootSum}\left [\text{$\#$1}^3 b d^3+3 \text{$\#$1}^2 b c d^2+3 \text{$\#$1} b c^2 d+a+b c^3\&,\frac{-3 \text{$\#$1}^2 a b c d^2 \log (x-\text{$\#$1})+6 \text{$\#$1}^2 b^2 c^4 d^2 \log (x-\text{$\#$1})+a^2 \log (x-\text{$\#$1})-16 a b c^3 \log (x-\text{$\#$1})-12 \text{$\#$1} a b c^2 d \log (x-\text{$\#$1})+10 b^2 c^6 \log (x-\text{$\#$1})+15 \text{$\#$1} b^2 c^5 d \log (x-\text{$\#$1})}{\text{$\#$1}^2 d^2+2 \text{$\#$1} c d+c^2}\&\right ]+18 b c d^2 x^2 \log (x) \left (a-2 b c^3\right )+3 \left (a+b c^3\right ) \left (a+b c^2 (c-6 d x)\right )}{6 x^2 \left (a+b c^3\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

-(3*(a + b*c^3)*(a + b*c^2*(c - 6*d*x)) + 18*b*c*(a - 2*b*c^3)*d^2*x^2*Log[x] +
2*d^2*x^2*RootSum[a + b*c^3 + 3*b*c^2*d*#1 + 3*b*c*d^2*#1^2 + b*d^3*#1^3 & , (a^
2*Log[x - #1] - 16*a*b*c^3*Log[x - #1] + 10*b^2*c^6*Log[x - #1] - 12*a*b*c^2*d*L
og[x - #1]*#1 + 15*b^2*c^5*d*Log[x - #1]*#1 - 3*a*b*c*d^2*Log[x - #1]*#1^2 + 6*b
^2*c^4*d^2*Log[x - #1]*#1^2)/(c^2 + 2*c*d*#1 + d^2*#1^2) & ])/(6*(a + b*c^3)^3*x
^2)

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Maple [C]  time = 0.017, size = 217, normalized size = 0.6 \[ -{\frac{1}{ \left ( 2\,b{c}^{3}+2\,a \right ){x}^{2}}}+3\,{\frac{{c}^{2}db}{ \left ( b{c}^{3}+a \right ) ^{2}x}}+6\,{\frac{{c}^{4}{d}^{2}{b}^{2}\ln \left ( x \right ) }{ \left ( b{c}^{3}+a \right ) ^{3}}}-3\,{\frac{bc{d}^{2}\ln \left ( x \right ) a}{ \left ( b{c}^{3}+a \right ) ^{3}}}+{\frac{{d}^{2}}{3\, \left ( b{c}^{3}+a \right ) ^{3}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ( -6\,{{\it \_R}}^{2}{b}^{2}{c}^{4}{d}^{2}-15\,{\it \_R}\,{b}^{2}{c}^{5}d-10\,{b}^{2}{c}^{6}+3\,{{\it \_R}}^{2}abc{d}^{2}+12\,{\it \_R}\,ab{c}^{2}d+16\,ab{c}^{3}-{a}^{2} \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a+b*(d*x+c)^3),x)

[Out]

-1/2/(b*c^3+a)/x^2+3*b*c^2*d/(b*c^3+a)^2/x+6*b^2*c^4*d^2/(b*c^3+a)^3*ln(x)-3*b*c
*d^2/(b*c^3+a)^3*ln(x)*a+1/3/(b*c^3+a)^3*d^2*sum((-6*_R^2*b^2*c^4*d^2-15*_R*b^2*
c^5*d-10*b^2*c^6+3*_R^2*a*b*c*d^2+12*_R*a*b*c^2*d+16*a*b*c^3-a^2)/(_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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{b d^{3} \int \frac{10 \, b^{2} c^{6} - 16 \, a b c^{3} + 3 \,{\left (2 \, b^{2} c^{4} - a b c\right )} d^{2} x^{2} + 3 \,{\left (5 \, b^{2} c^{5} - 4 \, a b c^{2}\right )} d x + a^{2}}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{b^{3} c^{9} + 3 \, a b^{2} c^{6} + 3 \, a^{2} b c^{3} + a^{3}} + \frac{3 \,{\left (2 \, b^{2} c^{4} - a b c\right )} d^{2} \log \left (x\right )}{b^{3} c^{9} + 3 \, a b^{2} c^{6} + 3 \, a^{2} b c^{3} + a^{3}} + \frac{6 \, b c^{2} d x - b c^{3} - a}{2 \,{\left (b^{2} c^{6} + 2 \, a b c^{3} + a^{2}\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*d^3*integrate((10*b^2*c^6 - 16*a*b*c^3 + 3*(2*b^2*c^4 - a*b*c)*d^2*x^2 + 3*(5
*b^2*c^5 - 4*a*b*c^2)*d*x + a^2)/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^
3 + a), x)/(b^3*c^9 + 3*a*b^2*c^6 + 3*a^2*b*c^3 + a^3) + 3*(2*b^2*c^4 - a*b*c)*d
^2*log(x)/(b^3*c^9 + 3*a*b^2*c^6 + 3*a^2*b*c^3 + a^3) + 1/2*(6*b*c^2*d*x - b*c^3
- a)/((b^2*c^6 + 2*a*b*c^3 + a^2)*x^2)

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

[Out]

Exception raised: NotImplementedError

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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