3.108 \(\int \frac{1}{x^2 \left (a+b (c+d x)^3\right )} \, dx\)

Optimal. Leaf size=314 \[ -\frac{\sqrt [3]{b} d \left (\sqrt [3]{a} \left (a-2 b c^3\right )-\sqrt [3]{b} c \left (2 a-b c^3\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )^2}+\frac{\sqrt [3]{b} d \left (\sqrt [3]{a} \left (a-2 b c^3\right )-\sqrt [3]{b} c \left (2 a-b c^3\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )^2}+\frac{\sqrt [3]{b} d \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \left (a+b c^3\right )^2}-\frac{1}{x \left (a+b c^3\right )}-\frac{3 b c^2 d \log (x)}{\left (a+b c^3\right )^2}+\frac{b c^2 d \log \left (a+b (c+d x)^3\right )}{\left (a+b c^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 1.09058, antiderivative size = 312, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ \frac{b^{2/3} d \left (-\frac{\sqrt [3]{a} \left (a-2 b c^3\right )}{\sqrt [3]{b}}+2 a c-b c^4\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )^2}+\frac{\sqrt [3]{b} d \left (\sqrt [3]{a} \left (a-2 b c^3\right )-\sqrt [3]{b} c \left (2 a-b c^3\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )^2}+\frac{\sqrt [3]{b} d \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \left (a+b c^3\right )^2}-\frac{1}{x \left (a+b c^3\right )}-\frac{3 b c^2 d \log (x)}{\left (a+b c^3\right )^2}+\frac{b c^2 d \log \left (a+b (c+d x)^3\right )}{\left (a+b c^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*(c + d*x)^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 133.834, size = 314, normalized size = 1. \[ - \frac{3 b c^{2} d \log{\left (- d x \right )}}{\left (a + b c^{3}\right )^{2}} + \frac{b c^{2} d \log{\left (a + b \left (c + d x\right )^{3} \right )}}{\left (a + b c^{3}\right )^{2}} - \frac{1}{x \left (a + b c^{3}\right )} + \frac{\sqrt [3]{b} d \left (\sqrt [3]{a} \left (a - 2 b c^{3}\right ) - \sqrt [3]{b} c \left (2 a - b c^{3}\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 a^{\frac{2}{3}} \left (a + b c^{3}\right )^{2}} - \frac{\sqrt [3]{b} d \left (\sqrt [3]{a} \left (a - 2 b c^{3}\right ) - \sqrt [3]{b} c \left (2 a - b c^{3}\right )\right ) \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 a^{\frac{2}{3}} \left (a + b c^{3}\right )^{2}} + \frac{\sqrt{3} \sqrt [3]{b} d \left (a^{\frac{4}{3}} - 2 \sqrt [3]{a} b c^{3} + 2 a \sqrt [3]{b} c - b^{\frac{4}{3}} c^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} \left (a + b c^{3}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(a+b*(d*x+c)**3),x)

[Out]

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

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Mathematica [C]  time = 0.141238, size = 173, normalized size = 0.55 \[ \frac{d x \text{RootSum}\left [\text{$\#$1}^3 b d^3+3 \text{$\#$1}^2 b c d^2+3 \text{$\#$1} b c^2 d+a+b c^3\&,\frac{3 \text{$\#$1}^2 b c^2 d^2 \log (x-\text{$\#$1})-3 a c \log (x-\text{$\#$1})-\text{$\#$1} a d \log (x-\text{$\#$1})+6 b c^4 \log (x-\text{$\#$1})+8 \text{$\#$1} b c^3 d \log (x-\text{$\#$1})}{\text{$\#$1}^2 d^2+2 \text{$\#$1} c d+c^2}\&\right ]-3 \left (a+b c^3+3 b c^2 d x \log (x)\right )}{3 x \left (a+b c^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*(c + d*x)^3)),x]

[Out]

(-3*(a + b*c^3 + 3*b*c^2*d*x*Log[x]) + d*x*RootSum[a + b*c^3 + 3*b*c^2*d*#1 + 3*
b*c*d^2*#1^2 + b*d^3*#1^3 & , (-3*a*c*Log[x - #1] + 6*b*c^4*Log[x - #1] - a*d*Lo
g[x - #1]*#1 + 8*b*c^3*d*Log[x - #1]*#1 + 3*b*c^2*d^2*Log[x - #1]*#1^2)/(c^2 + 2
*c*d*#1 + d^2*#1^2) & ])/(3*(a + b*c^3)^2*x)

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Maple [C]  time = 0.014, size = 144, normalized size = 0.5 \[ -{\frac{1}{ \left ( b{c}^{3}+a \right ) x}}-3\,{\frac{{c}^{2}db\ln \left ( x \right ) }{ \left ( b{c}^{3}+a \right ) ^{2}}}+{\frac{d}{3\, \left ( b{c}^{3}+a \right ) ^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ( 3\,{{\it \_R}}^{2}b{c}^{2}{d}^{2}+8\,{\it \_R}\,b{c}^{3}d+6\,b{c}^{4}-{\it \_R}\,ad-3\,ac \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b*(d*x+c)^3),x)

[Out]

-1/(b*c^3+a)/x-3*b*c^2*d*ln(x)/(b*c^3+a)^2+1/3*d/(b*c^3+a)^2*sum((3*_R^2*b*c^2*d
^2+8*_R*b*c^3*d+6*b*c^4-_R*a*d-3*a*c)/(_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. \[ -\frac{3 \, b c^{2} d \log \left (x\right )}{b^{2} c^{6} + 2 \, a b c^{3} + a^{2}} + \frac{b d^{2} \int \frac{3 \, b c^{2} d^{2} x^{2} + 6 \, b c^{4} +{\left (8 \, b c^{3} - a\right )} d x - 3 \, a c}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{b^{2} c^{6} + 2 \, a b c^{3} + a^{2}} - \frac{1}{{\left (b c^{3} + a\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)*x^2),x, algorithm="maxima")

[Out]

-3*b*c^2*d*log(x)/(b^2*c^6 + 2*a*b*c^3 + a^2) + b*d^2*integrate((3*b*c^2*d^2*x^2
+ 6*b*c^4 + (8*b*c^3 - a)*d*x - 3*a*c)/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x
+ b*c^3 + a), x)/(b^2*c^6 + 2*a*b*c^3 + a^2) - 1/((b*c^3 + a)*x)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)*x^2),x, algorithm="giac")

[Out]

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