3.2873 \(\int \frac{c+d x}{\left (a+b (c+d x)^3\right )^3} \, dx\)

Optimal. Leaf size=202 \[ -\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3} d}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.379607, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3} d}+\frac{2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac{(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 45.0929, size = 187, normalized size = 0.93 \[ \frac{\left (c + d x\right )^{2}}{6 a d \left (a + b \left (c + d x\right )^{3}\right )^{2}} + \frac{2 \left (c + d x\right )^{2}}{9 a^{2} d \left (a + b \left (c + d x\right )^{3}\right )} - \frac{2 \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{27 a^{\frac{7}{3}} b^{\frac{2}{3}} d} + \frac{\log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{27 a^{\frac{7}{3}} b^{\frac{2}{3}} d} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{7}{3}} b^{\frac{2}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.214277, size = 180, normalized size = 0.89 \[ \frac{\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{b^{2/3}}+\frac{9 a^{4/3} (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}-\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{b^{2/3}}+\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{2/3}}+\frac{12 \sqrt [3]{a} (c+d x)^2}{a+b (c+d x)^3}}{54 a^{7/3} d} \]

Antiderivative was successfully verified.

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

[Out]

((9*a^(4/3)*(c + d*x)^2)/(a + b*(c + d*x)^3)^2 + (12*a^(1/3)*(c + d*x)^2)/(a + b
*(c + d*x)^3) + (4*Sqrt[3]*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1
/3))])/b^(2/3) - (4*Log[a^(1/3) + b^(1/3)*(c + d*x)])/b^(2/3) + (2*Log[a^(2/3) -
a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/b^(2/3))/(54*a^(7/3)*d)

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Maple [C]  time = 0.024, size = 214, normalized size = 1.1 \[{\frac{1}{ \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}} \left ({\frac{2\,b{d}^{4}{x}^{5}}{9\,{a}^{2}}}+{\frac{10\,bc{d}^{3}{x}^{4}}{9\,{a}^{2}}}+{\frac{20\,{c}^{2}{d}^{2}b{x}^{3}}{9\,{a}^{2}}}+{\frac{d \left ( 40\,b{c}^{3}+7\,a \right ){x}^{2}}{18\,{a}^{2}}}+{\frac{c \left ( 10\,b{c}^{3}+7\,a \right ) x}{9\,{a}^{2}}}+{\frac{{c}^{2} \left ( 4\,b{c}^{3}+7\,a \right ) }{18\,{a}^{2}d}} \right ) }+{\frac{2}{27\,{a}^{2}bd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2/9*b*d^4/a^2*x^5+10/9*b*c*d^3/a^2*x^4+20/9*c^2*d^2*b/a^2*x^3+1/18*d*(40*b*c^3+
7*a)/a^2*x^2+1/9*c*(10*b*c^3+7*a)/a^2*x+1/18*c^2/d*(4*b*c^3+7*a)/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+2/27/a^2/b/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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{4 \, b d^{5} x^{5} + 20 \, b c d^{4} x^{4} + 40 \, b c^{2} d^{3} x^{3} + 4 \, b c^{5} +{\left (40 \, b c^{3} + 7 \, a\right )} d^{2} x^{2} + 7 \, a c^{2} + 2 \,{\left (10 \, b c^{4} + 7 \, a c\right )} d x}{18 \,{\left (a^{2} b^{2} d^{7} x^{6} + 6 \, a^{2} b^{2} c d^{6} x^{5} + 15 \, a^{2} b^{2} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d^{4} x^{3} + 3 \,{\left (5 \, a^{2} b^{2} c^{4} + 2 \, a^{3} b c\right )} d^{3} x^{2} + 6 \,{\left (a^{2} b^{2} c^{5} + a^{3} b c^{2}\right )} d^{2} x +{\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{3} + a^{4}\right )} d\right )}} + \frac{2 \, \int \frac{d x + c}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{9 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*(4*b*d^5*x^5 + 20*b*c*d^4*x^4 + 40*b*c^2*d^3*x^3 + 4*b*c^5 + (40*b*c^3 + 7*
a)*d^2*x^2 + 7*a*c^2 + 2*(10*b*c^4 + 7*a*c)*d*x)/(a^2*b^2*d^7*x^6 + 6*a^2*b^2*c*
d^6*x^5 + 15*a^2*b^2*c^2*d^5*x^4 + 2*(10*a^2*b^2*c^3 + a^3*b)*d^4*x^3 + 3*(5*a^2
*b^2*c^4 + 2*a^3*b*c)*d^3*x^2 + 6*(a^2*b^2*c^5 + a^3*b*c^2)*d^2*x + (a^2*b^2*c^6
+ 2*a^3*b*c^3 + a^4)*d) + 2/9*integrate((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^2

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Fricas [A]  time = 0.241543, size = 957, normalized size = 4.74 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/162*sqrt(3)*(4*sqrt(3)*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + b
^2*c^6 + 2*(10*b^2*c^3 + a*b)*d^3*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*
x^2 + 6*(b^2*c^5 + a*b*c^2)*d*x + a^2)*log(a*b + (-a*b^2)^(2/3)*(d*x + c)) - 2*s
qrt(3)*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + b^2*c^6 + 2*(10*b^2
*c^3 + a*b)*d^3*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*x^2 + 6*(b^2*c^5 +
a*b*c^2)*d*x + a^2)*log(-a*b + (-a*b^2)^(2/3)*(d*x + c) + (b*d^2*x^2 + 2*b*c*d*
x + b*c^2)*(-a*b^2)^(1/3)) - 12*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*
x^4 + b^2*c^6 + 2*(10*b^2*c^3 + a*b)*d^3*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*
c)*d^2*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*x + a^2)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(
3)*(-a*b^2)^(2/3)*(d*x + c))/(a*b)) + 3*sqrt(3)*(4*b*d^5*x^5 + 20*b*c*d^4*x^4 +
40*b*c^2*d^3*x^3 + 4*b*c^5 + (40*b*c^3 + 7*a)*d^2*x^2 + 7*a*c^2 + 2*(10*b*c^4 +
7*a*c)*d*x)*(-a*b^2)^(1/3))/((a^2*b^2*d^7*x^6 + 6*a^2*b^2*c*d^6*x^5 + 15*a^2*b^2
*c^2*d^5*x^4 + 2*(10*a^2*b^2*c^3 + a^3*b)*d^4*x^3 + 3*(5*a^2*b^2*c^4 + 2*a^3*b*c
)*d^3*x^2 + 6*(a^2*b^2*c^5 + a^3*b*c^2)*d^2*x + (a^2*b^2*c^6 + 2*a^3*b*c^3 + a^4
)*d)*(-a*b^2)^(1/3))

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Sympy [A]  time = 70.1458, size = 296, normalized size = 1.47 \[ \frac{7 a c^{2} + 4 b c^{5} + 40 b c^{2} d^{3} x^{3} + 20 b c d^{4} x^{4} + 4 b d^{5} x^{5} + x^{2} \left (7 a d^{2} + 40 b c^{3} d^{2}\right ) + x \left (14 a c d + 20 b c^{4} d\right )}{18 a^{4} d + 36 a^{3} b c^{3} d + 18 a^{2} b^{2} c^{6} d + 270 a^{2} b^{2} c^{2} d^{5} x^{4} + 108 a^{2} b^{2} c d^{6} x^{5} + 18 a^{2} b^{2} d^{7} x^{6} + x^{3} \left (36 a^{3} b d^{4} + 360 a^{2} b^{2} c^{3} d^{4}\right ) + x^{2} \left (108 a^{3} b c d^{3} + 270 a^{2} b^{2} c^{4} d^{3}\right ) + x \left (108 a^{3} b c^{2} d^{2} + 108 a^{2} b^{2} c^{5} d^{2}\right )} + \frac{\operatorname{RootSum}{\left (19683 t^{3} a^{7} b^{2} + 8, \left ( t \mapsto t \log{\left (x + \frac{729 t^{2} a^{5} b + 4 c}{4 d} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(7*a*c**2 + 4*b*c**5 + 40*b*c**2*d**3*x**3 + 20*b*c*d**4*x**4 + 4*b*d**5*x**5 +
x**2*(7*a*d**2 + 40*b*c**3*d**2) + x*(14*a*c*d + 20*b*c**4*d))/(18*a**4*d + 36*a
**3*b*c**3*d + 18*a**2*b**2*c**6*d + 270*a**2*b**2*c**2*d**5*x**4 + 108*a**2*b**
2*c*d**6*x**5 + 18*a**2*b**2*d**7*x**6 + x**3*(36*a**3*b*d**4 + 360*a**2*b**2*c*
*3*d**4) + x**2*(108*a**3*b*c*d**3 + 270*a**2*b**2*c**4*d**3) + x*(108*a**3*b*c*
*2*d**2 + 108*a**2*b**2*c**5*d**2)) + RootSum(19683*_t**3*a**7*b**2 + 8, Lambda(
_t, _t*log(x + (729*_t**2*a**5*b + 4*c)/(4*d))))/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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