3.2853 \(\int \frac{(c+d x)^3}{a+b (c+d x)^3} \, dx\)

Optimal. Leaf size=144 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3} d}+\frac{x}{b} \]

[Out]

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

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Rubi [A]  time = 0.311739, antiderivative size = 144, 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 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3} d}+\frac{x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.3598, size = 143, normalized size = 0.99 \[ - \frac{\sqrt [3]{a} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 b^{\frac{4}{3}} d} + \frac{\sqrt [3]{a} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 b^{\frac{4}{3}} d} + \frac{\sqrt{3} \sqrt [3]{a} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{4}{3}} d} + \frac{c + d x}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(1/3)*log(a**(1/3) + b**(1/3)*(c + d*x))/(3*b**(4/3)*d) + a**(1/3)*log(a**(2
/3) + a**(1/3)*b**(1/3)*(-c - d*x) + b**(2/3)*(c + d*x)**2)/(6*b**(4/3)*d) + sqr
t(3)*a**(1/3)*atan(sqrt(3)*(a**(1/3)/3 + b**(1/3)*(-2*c/3 - 2*d*x/3))/a**(1/3))/
(3*b**(4/3)*d) + (c + d*x)/(b*d)

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Mathematica [A]  time = 0.0318521, size = 142, normalized size = 0.99 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )+6 \sqrt [3]{b} c+6 \sqrt [3]{b} d x}{6 b^{4/3} d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.006, size = 78, normalized size = 0.5 \[{\frac{x}{b}}-{\frac{a}{3\,{b}^{2}d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/b-1/3*a/b^2/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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*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)/b + x/b

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Fricas [A]  time = 0.217663, size = 194, normalized size = 1.35 \[ \frac{\sqrt{3}{\left (6 \, \sqrt{3} d x - \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} +{\left (d x + c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) + 2 \, \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (d x + c - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 6 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (d x + c\right )} + \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )\right )}}{18 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.71618, size = 27, normalized size = 0.19 \[ \frac{\operatorname{RootSum}{\left (27 t^{3} b^{4} + a, \left ( t \mapsto t \log{\left (x + \frac{- 3 t b + c}{d} \right )} \right )\right )}}{d} + \frac{x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*b**4 + a, Lambda(_t, _t*log(x + (-3*_t*b + c)/d)))/d + x/b

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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