3.2868 \(\int \frac{1}{(c+d x)^3 \left (a+b (c+d x)^3\right )^2} \, dx\)

Optimal. Leaf size=189 \[ -\frac{5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{8/3} d}+\frac{5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{8/3} d}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3} d}-\frac{5}{6 a^2 d (c+d x)^2}+\frac{1}{3 a d (c+d x)^2 \left (a+b (c+d x)^3\right )} \]

[Out]

-5/(6*a^2*d*(c + d*x)^2) + 1/(3*a*d*(c + d*x)^2*(a + b*(c + d*x)^3)) + (5*b^(2/3
)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(8/3)*
d) - (5*b^(2/3)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(9*a^(8/3)*d) + (5*b^(2/3)*Log
[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/(18*a^(8/3)*d)

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Rubi [A]  time = 0.368759, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ -\frac{5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{8/3} d}+\frac{5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{8/3} d}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3} d}-\frac{5}{6 a^2 d (c+d x)^2}+\frac{1}{3 a d (c+d x)^2 \left (a+b (c+d x)^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-5/(6*a^2*d*(c + d*x)^2) + 1/(3*a*d*(c + d*x)^2*(a + b*(c + d*x)^3)) + (5*b^(2/3
)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(8/3)*
d) - (5*b^(2/3)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(9*a^(8/3)*d) + (5*b^(2/3)*Log
[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/(18*a^(8/3)*d)

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Rubi in Sympy [A]  time = 45.1173, size = 178, normalized size = 0.94 \[ \frac{1}{3 a d \left (a + b \left (c + d x\right )^{3}\right ) \left (c + d x\right )^{2}} - \frac{5}{6 a^{2} d \left (c + d x\right )^{2}} - \frac{5 b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{9 a^{\frac{8}{3}} d} + \frac{5 b^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{18 a^{\frac{8}{3}} d} + \frac{5 \sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{8}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(3*a*d*(a + b*(c + d*x)**3)*(c + d*x)**2) - 5/(6*a**2*d*(c + d*x)**2) - 5*b**(
2/3)*log(a**(1/3) + b**(1/3)*(c + d*x))/(9*a**(8/3)*d) + 5*b**(2/3)*log(a**(2/3)
+ a**(1/3)*b**(1/3)*(-c - d*x) + b**(2/3)*(c + d*x)**2)/(18*a**(8/3)*d) + 5*sqr
t(3)*b**(2/3)*atan(sqrt(3)*(a**(1/3)/3 + b**(1/3)*(-2*c/3 - 2*d*x/3))/a**(1/3))/
(9*a**(8/3)*d)

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Mathematica [A]  time = 0.14356, size = 166, normalized size = 0.88 \[ \frac{5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-\frac{6 a^{2/3} b (c+d x)}{a+b (c+d x)^3}-\frac{9 a^{2/3}}{(c+d x)^2}-10 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-10 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{18 a^{8/3} d} \]

Antiderivative was successfully verified.

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

[Out]

((-9*a^(2/3))/(c + d*x)^2 - (6*a^(2/3)*b*(c + d*x))/(a + b*(c + d*x)^3) - 10*Sqr
t[3]*b^(2/3)*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))] - 10*b^(
2/3)*Log[a^(1/3) + b^(1/3)*(c + d*x)] + 5*b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*
(c + d*x) + b^(2/3)*(c + d*x)^2])/(18*a^(8/3)*d)

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Maple [C]  time = 0.02, size = 174, normalized size = 0.9 \[ -{\frac{bx}{3\,{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }}-{\frac{bc}{3\,{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) d}}-{\frac{5}{9\,{a}^{2}d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}}-{\frac{1}{2\,{a}^{2}d \left ( dx+c \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*b/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-1/3*b/a^2/(b*d^3*x^3+
3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)*c/d-5/9/a^2/d*sum(1/(_R^2*d^2+2*_R*c*d+c^2)*l
n(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^2/d/(d*
x+c)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{5 \, b d^{3} x^{3} + 15 \, b c d^{2} x^{2} + 15 \, b c^{2} d x + 5 \, b c^{3} + 3 \, a}{6 \,{\left (a^{2} b d^{6} x^{5} + 5 \, a^{2} b c d^{5} x^{4} + 10 \, a^{2} b c^{2} d^{4} x^{3} +{\left (10 \, a^{2} b c^{3} + a^{3}\right )} d^{3} x^{2} +{\left (5 \, a^{2} b c^{4} + 2 \, a^{3} c\right )} d^{2} x +{\left (a^{2} b c^{5} + a^{3} c^{2}\right )} d\right )}} - \frac{5 \, b \int \frac{1}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{3 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(5*b*d^3*x^3 + 15*b*c*d^2*x^2 + 15*b*c^2*d*x + 5*b*c^3 + 3*a)/(a^2*b*d^6*x^
5 + 5*a^2*b*c*d^5*x^4 + 10*a^2*b*c^2*d^4*x^3 + (10*a^2*b*c^3 + a^3)*d^3*x^2 + (5
*a^2*b*c^4 + 2*a^3*c)*d^2*x + (a^2*b*c^5 + a^3*c^2)*d) - 5/3*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^2

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Fricas [A]  time = 0.245212, size = 716, normalized size = 3.79 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + b c^{5} +{\left (10 \, b c^{3} + a\right )} d^{2} x^{2} + a c^{2} +{\left (5 \, b c^{4} + 2 \, a c\right )} d x\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} +{\left (a b d x + a b c\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 10 \, \sqrt{3}{\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + b c^{5} +{\left (10 \, b c^{3} + a\right )} d^{2} x^{2} + a c^{2} +{\left (5 \, b c^{4} + 2 \, a c\right )} d x\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b d x + b c - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 30 \,{\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + b c^{5} +{\left (10 \, b c^{3} + a\right )} d^{2} x^{2} + a c^{2} +{\left (5 \, b c^{4} + 2 \, a c\right )} d x\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + 2 \, \sqrt{3}{\left (b d x + b c\right )}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (5 \, b d^{3} x^{3} + 15 \, b c d^{2} x^{2} + 15 \, b c^{2} d x + 5 \, b c^{3} + 3 \, a\right )}\right )}}{54 \,{\left (a^{2} b d^{6} x^{5} + 5 \, a^{2} b c d^{5} x^{4} + 10 \, a^{2} b c^{2} d^{4} x^{3} +{\left (10 \, a^{2} b c^{3} + a^{3}\right )} d^{3} x^{2} +{\left (5 \, a^{2} b c^{4} + 2 \, a^{3} c\right )} d^{2} x +{\left (a^{2} b c^{5} + a^{3} c^{2}\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(5*sqrt(3)*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3*x^3 + b*c^5 +
(10*b*c^3 + a)*d^2*x^2 + a*c^2 + (5*b*c^4 + 2*a*c)*d*x)*(-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)) - 10*sqrt(3)*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3*x^3 + b*c^
5 + (10*b*c^3 + a)*d^2*x^2 + a*c^2 + (5*b*c^4 + 2*a*c)*d*x)*(-b^2/a^2)^(1/3)*log
(b*d*x + b*c - a*(-b^2/a^2)^(1/3)) + 30*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^
3*x^3 + b*c^5 + (10*b*c^3 + a)*d^2*x^2 + a*c^2 + (5*b*c^4 + 2*a*c)*d*x)*(-b^2/a^
2)^(1/3)*arctan(1/3*(sqrt(3)*a*(-b^2/a^2)^(1/3) + 2*sqrt(3)*(b*d*x + b*c))/(a*(-
b^2/a^2)^(1/3))) + 3*sqrt(3)*(5*b*d^3*x^3 + 15*b*c*d^2*x^2 + 15*b*c^2*d*x + 5*b*
c^3 + 3*a))/(a^2*b*d^6*x^5 + 5*a^2*b*c*d^5*x^4 + 10*a^2*b*c^2*d^4*x^3 + (10*a^2*
b*c^3 + a^3)*d^3*x^2 + (5*a^2*b*c^4 + 2*a^3*c)*d^2*x + (a^2*b*c^5 + a^3*c^2)*d)

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Sympy [A]  time = 167.694, size = 197, normalized size = 1.04 \[ - \frac{3 a + 5 b c^{3} + 15 b c^{2} d x + 15 b c d^{2} x^{2} + 5 b d^{3} x^{3}}{6 a^{3} c^{2} d + 6 a^{2} b c^{5} d + 60 a^{2} b c^{2} d^{4} x^{3} + 30 a^{2} b c d^{5} x^{4} + 6 a^{2} b d^{6} x^{5} + x^{2} \left (6 a^{3} d^{3} + 60 a^{2} b c^{3} d^{3}\right ) + x \left (12 a^{3} c d^{2} + 30 a^{2} b c^{4} d^{2}\right )} + \frac{\operatorname{RootSum}{\left (729 t^{3} a^{8} + 125 b^{2}, \left ( t \mapsto t \log{\left (x + \frac{- 9 t a^{3} + 5 b c}{5 b d} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(3*a + 5*b*c**3 + 15*b*c**2*d*x + 15*b*c*d**2*x**2 + 5*b*d**3*x**3)/(6*a**3*c**
2*d + 6*a**2*b*c**5*d + 60*a**2*b*c**2*d**4*x**3 + 30*a**2*b*c*d**5*x**4 + 6*a**
2*b*d**6*x**5 + x**2*(6*a**3*d**3 + 60*a**2*b*c**3*d**3) + x*(12*a**3*c*d**2 + 3
0*a**2*b*c**4*d**2)) + RootSum(729*_t**3*a**8 + 125*b**2, Lambda(_t, _t*log(x +
(-9*_t*a**3 + 5*b*c)/(5*b*d))))/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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