3.104 \(\int \frac{x^2}{a+b (c+d x)^3} \, dx\)

Optimal. Leaf size=210 \[ \frac{c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3} d^3}-\frac{c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{2/3} d^3}+\frac{c \left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{2/3} d^3}+\frac{\log \left (a+b (c+d x)^3\right )}{3 b d^3} \]

[Out]

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

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Rubi [A]  time = 0.4734, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ \frac{c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3} d^3}-\frac{c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{2/3} d^3}+\frac{c \left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{2/3} d^3}+\frac{\log \left (a+b (c+d x)^3\right )}{3 b d^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*(c + d*x)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 52.2893, size = 204, normalized size = 0.97 \[ \frac{\log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 b d^{3}} + \frac{\sqrt{3} c \left (2 \sqrt [3]{a} - \sqrt [3]{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} b^{\frac{2}{3}} d^{3}} + \frac{c \left (2 \sqrt [3]{a} + \sqrt [3]{b} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 a^{\frac{2}{3}} b^{\frac{2}{3}} d^{3}} - \frac{c \left (2 \sqrt [3]{a} + \sqrt [3]{b} c\right ) \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 a^{\frac{2}{3}} b^{\frac{2}{3}} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0452165, size = 81, normalized size = 0.39 \[ \frac{\text{RootSum}\left [\text{$\#$1}^3 b d^3+3 \text{$\#$1}^2 b c d^2+3 \text{$\#$1} b c^2 d+a+b c^3\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{\text{$\#$1}^2 d^2+2 \text{$\#$1} c d+c^2}\&\right ]}{3 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.006, size = 74, normalized size = 0.4 \[{\frac{1}{3\,bd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{{{\it \_R}}^{2}\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*(d*x+c)^3),x)

[Out]

1/3/b/d*sum(_R^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. \[ \int \frac{x^{2}}{{\left (d x + c\right )}^{3} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/((d*x + c)^3*b + 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(x^2/((d*x + c)^3*b + a),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 3.16654, size = 158, normalized size = 0.75 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{3} d^{9} - 27 t^{2} a^{2} b^{2} d^{6} + t \left (9 a^{2} b d^{3} - 18 a b^{2} c^{3} d^{3}\right ) - a^{2} - 2 a b c^{3} - b^{2} c^{6}, \left ( t \mapsto t \log{\left (x + \frac{18 t^{2} a^{2} b^{2} d^{6} - 12 t a^{2} b d^{3} - 3 t a b^{2} c^{3} d^{3} + 2 a^{2} + a b c^{3} - b^{2} c^{6}}{8 a b c^{2} d - b^{2} c^{5} d} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*a**2*b**3*d**9 - 27*_t**2*a**2*b**2*d**6 + _t*(9*a**2*b*d**3 -
18*a*b**2*c**3*d**3) - a**2 - 2*a*b*c**3 - b**2*c**6, Lambda(_t, _t*log(x + (18*
_t**2*a**2*b**2*d**6 - 12*_t*a**2*b*d**3 - 3*_t*a*b**2*c**3*d**3 + 2*a**2 + a*b*
c**3 - b**2*c**6)/(8*a*b*c**2*d - b**2*c**5*d))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/((d*x + c)^3*b + a), x)