3.2876 \(\int \frac{1}{(c+d x)^2 \left (a+b (c+d x)^3\right )^3} \, dx\)

Optimal. Leaf size=219 \[ -\frac{7 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{10/3} d}+\frac{14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d}+\frac{14 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{10/3} d}-\frac{14}{9 a^3 d (c+d x)}+\frac{7}{18 a^2 d (c+d x) \left (a+b (c+d x)^3\right )}+\frac{1}{6 a d (c+d x) \left (a+b (c+d x)^3\right )^2} \]

[Out]

-14/(9*a^3*d*(c + d*x)) + 1/(6*a*d*(c + d*x)*(a + b*(c + d*x)^3)^2) + 7/(18*a^2*
d*(c + d*x)*(a + b*(c + d*x)^3)) + (14*b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c +
d*x))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*d) + (14*b^(1/3)*Log[a^(1/3) + b^(
1/3)*(c + d*x)])/(27*a^(10/3)*d) - (7*b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c +
d*x) + b^(2/3)*(c + d*x)^2])/(27*a^(10/3)*d)

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Rubi [A]  time = 0.436296, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ -\frac{7 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{10/3} d}+\frac{14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d}+\frac{14 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{10/3} d}-\frac{14}{9 a^3 d (c+d x)}+\frac{7}{18 a^2 d (c+d x) \left (a+b (c+d x)^3\right )}+\frac{1}{6 a d (c+d x) \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-14/(9*a^3*d*(c + d*x)) + 1/(6*a*d*(c + d*x)*(a + b*(c + d*x)^3)^2) + 7/(18*a^2*
d*(c + d*x)*(a + b*(c + d*x)^3)) + (14*b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c +
d*x))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*d) + (14*b^(1/3)*Log[a^(1/3) + b^(
1/3)*(c + d*x)])/(27*a^(10/3)*d) - (7*b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c +
d*x) + b^(2/3)*(c + d*x)^2])/(27*a^(10/3)*d)

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Rubi in Sympy [A]  time = 50.6071, size = 201, normalized size = 0.92 \[ \frac{1}{6 a d \left (a + b \left (c + d x\right )^{3}\right )^{2} \left (c + d x\right )} + \frac{7}{18 a^{2} d \left (a + b \left (c + d x\right )^{3}\right ) \left (c + d x\right )} - \frac{14}{9 a^{3} d \left (c + d x\right )} + \frac{14 \sqrt [3]{b} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{27 a^{\frac{10}{3}} d} - \frac{7 \sqrt [3]{b} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{27 a^{\frac{10}{3}} d} + \frac{14 \sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{10}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(6*a*d*(a + b*(c + d*x)**3)**2*(c + d*x)) + 7/(18*a**2*d*(a + b*(c + d*x)**3)*
(c + d*x)) - 14/(9*a**3*d*(c + d*x)) + 14*b**(1/3)*log(a**(1/3) + b**(1/3)*(c +
d*x))/(27*a**(10/3)*d) - 7*b**(1/3)*log(a**(2/3) + a**(1/3)*b**(1/3)*(-c - d*x)
+ b**(2/3)*(c + d*x)**2)/(27*a**(10/3)*d) + 14*sqrt(3)*b**(1/3)*atan(sqrt(3)*(a*
*(1/3)/3 + b**(1/3)*(-2*c/3 - 2*d*x/3))/a**(1/3))/(27*a**(10/3)*d)

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Mathematica [A]  time = 0.257518, size = 196, normalized size = 0.89 \[ \frac{-14 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-\frac{9 a^{4/3} b (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}-\frac{30 \sqrt [3]{a} b (c+d x)^2}{a+b (c+d x)^3}+28 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-28 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{54 \sqrt [3]{a}}{c+d x}}{54 a^{10/3} d} \]

Antiderivative was successfully verified.

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

[Out]

((-54*a^(1/3))/(c + d*x) - (9*a^(4/3)*b*(c + d*x)^2)/(a + b*(c + d*x)^3)^2 - (30
*a^(1/3)*b*(c + d*x)^2)/(a + b*(c + d*x)^3) - 28*Sqrt[3]*b^(1/3)*ArcTan[(-a^(1/3
) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))] + 28*b^(1/3)*Log[a^(1/3) + b^(1/3)*(
c + d*x)] - 14*b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*
x)^2])/(54*a^(10/3)*d)

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Maple [C]  time = 0.035, size = 524, normalized size = 2.4 \[ -{\frac{5\,{b}^{2}{d}^{4}{x}^{5}}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{25\,{b}^{2}c{d}^{3}{x}^{4}}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{50\,{b}^{2}{c}^{2}{d}^{2}{x}^{3}}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{50\,{b}^{2}{x}^{2}{c}^{3}d}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{13\,bd{x}^{2}}{18\,{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{25\,{b}^{2}x{c}^{4}}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{13\,bxc}{9\,{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{5\,{b}^{2}{c}^{5}}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}-{\frac{13\,b{c}^{2}}{18\,{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}-{\frac{14}{27\,{a}^{3}d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}}-{\frac{1}{{a}^{3}d \left ( dx+c \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/9*b^2/a^3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*d^4*x^5-25/9*b^2/a^
3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c*d^3*x^4-50/9*b^2/a^3/(b*d^3*
x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c^2*d^2*x^3-50/9*b^2/a^3/(b*d^3*x^3+3*b
*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*x^2*c^3*d-13/18*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)^2*d*x^2-25/9*b^2/a^3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+
b*c^3+a)^2*x*c^4-13/9*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)^2*x*c-
5/9*b^2/a^3/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2*c^5/d-13/18*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)^2*c^2/d-14/27/a^3/d*sum((_R*d+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))-1/a^3/d/(d*x+c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{28 \, b^{2} d^{6} x^{6} + 168 \, b^{2} c d^{5} x^{5} + 420 \, b^{2} c^{2} d^{4} x^{4} + 28 \, b^{2} c^{6} + 7 \,{\left (80 \, b^{2} c^{3} + 7 \, a b\right )} d^{3} x^{3} + 49 \, a b c^{3} + 21 \,{\left (20 \, b^{2} c^{4} + 7 \, a b c\right )} d^{2} x^{2} + 21 \,{\left (8 \, b^{2} c^{5} + 7 \, a b c^{2}\right )} d x + 18 \, a^{2}}{18 \,{\left (a^{3} b^{2} d^{8} x^{7} + 7 \, a^{3} b^{2} c d^{7} x^{6} + 21 \, a^{3} b^{2} c^{2} d^{6} x^{5} +{\left (35 \, a^{3} b^{2} c^{3} + 2 \, a^{4} b\right )} d^{5} x^{4} +{\left (35 \, a^{3} b^{2} c^{4} + 8 \, a^{4} b c\right )} d^{4} x^{3} + 3 \,{\left (7 \, a^{3} b^{2} c^{5} + 4 \, a^{4} b c^{2}\right )} d^{3} x^{2} +{\left (7 \, a^{3} b^{2} c^{6} + 8 \, a^{4} b c^{3} + a^{5}\right )} d^{2} x +{\left (a^{3} b^{2} c^{7} + 2 \, a^{4} b c^{4} + a^{5} c\right )} d\right )}} - \frac{14 \, b \int \frac{d x + c}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{9 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*(28*b^2*d^6*x^6 + 168*b^2*c*d^5*x^5 + 420*b^2*c^2*d^4*x^4 + 28*b^2*c^6 + 7
*(80*b^2*c^3 + 7*a*b)*d^3*x^3 + 49*a*b*c^3 + 21*(20*b^2*c^4 + 7*a*b*c)*d^2*x^2 +
21*(8*b^2*c^5 + 7*a*b*c^2)*d*x + 18*a^2)/(a^3*b^2*d^8*x^7 + 7*a^3*b^2*c*d^7*x^6
+ 21*a^3*b^2*c^2*d^6*x^5 + (35*a^3*b^2*c^3 + 2*a^4*b)*d^5*x^4 + (35*a^3*b^2*c^4
+ 8*a^4*b*c)*d^4*x^3 + 3*(7*a^3*b^2*c^5 + 4*a^4*b*c^2)*d^3*x^2 + (7*a^3*b^2*c^6
+ 8*a^4*b*c^3 + a^5)*d^2*x + (a^3*b^2*c^7 + 2*a^4*b*c^4 + a^5*c)*d) - 14/9*b*in
tegrate((d*x + c)/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a), x)/a^3

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Fricas [A]  time = 0.305301, size = 1191, normalized size = 5.44 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(14*sqrt(3)*(b^2*d^7*x^7 + 7*b^2*c*d^6*x^6 + 21*b^2*c^2*d^5*x^5 +
b^2*c^7 + (35*b^2*c^3 + 2*a*b)*d^4*x^4 + (35*b^2*c^4 + 8*a*b*c)*d^3*x^3 + 2*a*b
*c^4 + 3*(7*b^2*c^5 + 4*a*b*c^2)*d^2*x^2 + a^2*c + (7*b^2*c^6 + 8*a*b*c^3 + a^2)
*d*x)*(b/a)^(1/3)*log(b*d^2*x^2 + 2*b*c*d*x + b*c^2 - (a*d*x + a*c)*(b/a)^(2/3)
+ a*(b/a)^(1/3)) - 28*sqrt(3)*(b^2*d^7*x^7 + 7*b^2*c*d^6*x^6 + 21*b^2*c^2*d^5*x^
5 + b^2*c^7 + (35*b^2*c^3 + 2*a*b)*d^4*x^4 + (35*b^2*c^4 + 8*a*b*c)*d^3*x^3 + 2*
a*b*c^4 + 3*(7*b^2*c^5 + 4*a*b*c^2)*d^2*x^2 + a^2*c + (7*b^2*c^6 + 8*a*b*c^3 + a
^2)*d*x)*(b/a)^(1/3)*log(b*d*x + b*c + a*(b/a)^(2/3)) - 84*(b^2*d^7*x^7 + 7*b^2*
c*d^6*x^6 + 21*b^2*c^2*d^5*x^5 + b^2*c^7 + (35*b^2*c^3 + 2*a*b)*d^4*x^4 + (35*b^
2*c^4 + 8*a*b*c)*d^3*x^3 + 2*a*b*c^4 + 3*(7*b^2*c^5 + 4*a*b*c^2)*d^2*x^2 + a^2*c
+ (7*b^2*c^6 + 8*a*b*c^3 + a^2)*d*x)*(b/a)^(1/3)*arctan(1/3*(sqrt(3)*a*(b/a)^(2
/3) - 2*sqrt(3)*(b*d*x + b*c))/(a*(b/a)^(2/3))) + 3*sqrt(3)*(28*b^2*d^6*x^6 + 16
8*b^2*c*d^5*x^5 + 420*b^2*c^2*d^4*x^4 + 28*b^2*c^6 + 7*(80*b^2*c^3 + 7*a*b)*d^3*
x^3 + 49*a*b*c^3 + 21*(20*b^2*c^4 + 7*a*b*c)*d^2*x^2 + 21*(8*b^2*c^5 + 7*a*b*c^2
)*d*x + 18*a^2))/(a^3*b^2*d^8*x^7 + 7*a^3*b^2*c*d^7*x^6 + 21*a^3*b^2*c^2*d^6*x^5
+ (35*a^3*b^2*c^3 + 2*a^4*b)*d^5*x^4 + (35*a^3*b^2*c^4 + 8*a^4*b*c)*d^4*x^3 + 3
*(7*a^3*b^2*c^5 + 4*a^4*b*c^2)*d^3*x^2 + (7*a^3*b^2*c^6 + 8*a^4*b*c^3 + a^5)*d^2
*x + (a^3*b^2*c^7 + 2*a^4*b*c^4 + a^5*c)*d)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229755, size = 301, normalized size = 1.37 \[ \frac{14 \, \left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | -\left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}} - \frac{1}{{\left (d x + c\right )} d} \right |}\right )}{27 \, a^{3}} - \frac{14 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (\left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}} - \frac{2}{{\left (d x + c\right )} d}\right )}}{3 \, \left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} d} - \frac{7 \, \left (a^{2} b\right )^{\frac{1}{3}}{\rm ln}\left (\left (\frac{b}{a d^{3}}\right )^{\frac{2}{3}} - \frac{\left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}}}{{\left (d x + c\right )} d} + \frac{1}{{\left (d x + c\right )}^{2} d^{2}}\right )}{27 \, a^{4} d} - \frac{\frac{10 \, b^{2}}{{\left (d x + c\right )} d} + \frac{13 \, a b}{{\left (d x + c\right )}^{4} d}}{18 \, a^{3}{\left (b + \frac{a}{{\left (d x + c\right )}^{3}}\right )}^{2}} - \frac{1}{{\left (d x + c\right )} a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14/27*(b/(a*d^3))^(1/3)*ln(abs(-(b/(a*d^3))^(1/3) - 1/((d*x + c)*d)))/a^3 - 14/2
7*sqrt(3)*(a^2*b)^(1/3)*arctan(1/3*sqrt(3)*((b/(a*d^3))^(1/3) - 2/((d*x + c)*d))
/(b/(a*d^3))^(1/3))/(a^4*d) - 7/27*(a^2*b)^(1/3)*ln((b/(a*d^3))^(2/3) - (b/(a*d^
3))^(1/3)/((d*x + c)*d) + 1/((d*x + c)^2*d^2))/(a^4*d) - 1/18*(10*b^2/((d*x + c)
*d) + 13*a*b/((d*x + c)^4*d))/(a^3*(b + a/(d*x + c)^3)^2) - 1/((d*x + c)*a^3*d)