3.2870 \(\int \frac{(c+d x)^4}{\left (a+b (c+d x)^3\right )^3} \, dx\)

Optimal. Leaf size=205 \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{4/3} b^{5/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{4/3} b^{5/3} d}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{5/3} d}+\frac{(c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac{(c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2} \]

[Out]

-(c + d*x)^2/(6*b*d*(a + b*(c + d*x)^3)^2) + (c + d*x)^2/(9*a*b*d*(a + b*(c + d*
x)^3)) - ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))]/(9*Sqrt[3]*a^
(4/3)*b^(5/3)*d) - Log[a^(1/3) + b^(1/3)*(c + d*x)]/(27*a^(4/3)*b^(5/3)*d) + Log
[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2]/(54*a^(4/3)*b^(5/3)*
d)

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Rubi [A]  time = 0.405762, antiderivative size = 205, 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 \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{4/3} b^{5/3} d}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{4/3} b^{5/3} d}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{5/3} d}+\frac{(c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac{(c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^4/(a + b*(c + d*x)^3)^3,x]

[Out]

-(c + d*x)^2/(6*b*d*(a + b*(c + d*x)^3)^2) + (c + d*x)^2/(9*a*b*d*(a + b*(c + d*
x)^3)) - ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))]/(9*Sqrt[3]*a^
(4/3)*b^(5/3)*d) - Log[a^(1/3) + b^(1/3)*(c + d*x)]/(27*a^(4/3)*b^(5/3)*d) + Log
[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2]/(54*a^(4/3)*b^(5/3)*
d)

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Rubi in Sympy [A]  time = 45.4543, size = 182, normalized size = 0.89 \[ - \frac{\left (c + d x\right )^{2}}{6 b d \left (a + b \left (c + d x\right )^{3}\right )^{2}} + \frac{\left (c + d x\right )^{2}}{9 a b d \left (a + b \left (c + d x\right )^{3}\right )} - \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{27 a^{\frac{4}{3}} b^{\frac{5}{3}} d} + \frac{\log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{54 a^{\frac{4}{3}} b^{\frac{5}{3}} d} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{4}{3}} b^{\frac{5}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**4/(a+b*(d*x+c)**3)**3,x)

[Out]

-(c + d*x)**2/(6*b*d*(a + b*(c + d*x)**3)**2) + (c + d*x)**2/(9*a*b*d*(a + b*(c
+ d*x)**3)) - log(a**(1/3) + b**(1/3)*(c + d*x))/(27*a**(4/3)*b**(5/3)*d) + log(
a**(2/3) + a**(1/3)*b**(1/3)*(-c - d*x) + b**(2/3)*(c + d*x)**2)/(54*a**(4/3)*b*
*(5/3)*d) - sqrt(3)*atan(sqrt(3)*(a**(1/3)/3 + b**(1/3)*(-2*c/3 - 2*d*x/3))/a**(
1/3))/(27*a**(4/3)*b**(5/3)*d)

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Mathematica [A]  time = 0.339446, size = 182, normalized size = 0.89 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{a^{4/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{a^{4/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{4/3}}+\frac{6 b^{2/3} (c+d x)^2}{a \left (a+b (c+d x)^3\right )}-\frac{9 b^{2/3} (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}}{54 b^{5/3} d} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^4/(a + b*(c + d*x)^3)^3,x]

[Out]

((-9*b^(2/3)*(c + d*x)^2)/(a + b*(c + d*x)^3)^2 + (6*b^(2/3)*(c + d*x)^2)/(a*(a
+ b*(c + d*x)^3)) + (2*Sqrt[3]*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*
a^(1/3))])/a^(4/3) - (2*Log[a^(1/3) + b^(1/3)*(c + d*x)])/a^(4/3) + Log[a^(2/3)
- a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2]/a^(4/3))/(54*b^(5/3)*d)

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Maple [C]  time = 0.028, size = 214, normalized size = 1. \[{\frac{1}{ \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}} \left ({\frac{{d}^{4}{x}^{5}}{9\,a}}+{\frac{5\,c{d}^{3}{x}^{4}}{9\,a}}+{\frac{10\,{c}^{2}{d}^{2}{x}^{3}}{9\,a}}-{\frac{d \left ( -20\,b{c}^{3}+a \right ){x}^{2}}{18\,ab}}-{\frac{c \left ( -5\,b{c}^{3}+a \right ) x}{9\,ab}}-{\frac{{c}^{2} \left ( -2\,b{c}^{3}+a \right ) }{18\,bda}} \right ) }+{\frac{1}{27\,a{b}^{2}d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^4/(a+b*(d*x+c)^3)^3,x)

[Out]

(1/9*d^4/a*x^5+5/9*c*d^3/a*x^4+10/9*c^2*d^2/a*x^3-1/18/b*d*(-20*b*c^3+a)/a*x^2-1
/9/b*c*(-5*b*c^3+a)/a*x-1/18/b*c^2/d*(-2*b*c^3+a)/a)/(b*d^3*x^3+3*b*c*d^2*x^2+3*
b*c^2*d*x+b*c^3+a)^2+1/27/a/b^2/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. \[ \frac{2 \, b d^{5} x^{5} + 10 \, b c d^{4} x^{4} + 20 \, b c^{2} d^{3} x^{3} + 2 \, b c^{5} +{\left (20 \, b c^{3} - a\right )} d^{2} x^{2} - a c^{2} + 2 \,{\left (5 \, b c^{4} - a c\right )} d x}{18 \,{\left (a b^{3} d^{7} x^{6} + 6 \, a b^{3} c d^{6} x^{5} + 15 \, a b^{3} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, a b^{3} c^{3} + a^{2} b^{2}\right )} d^{4} x^{3} + 3 \,{\left (5 \, a b^{3} c^{4} + 2 \, a^{2} b^{2} c\right )} d^{3} x^{2} + 6 \,{\left (a b^{3} c^{5} + a^{2} b^{2} c^{2}\right )} d^{2} x +{\left (a b^{3} c^{6} + 2 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d\right )}} + \frac{\int \frac{d x + c}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{9 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^4/((d*x + c)^3*b + a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.241558, size = 952, normalized size = 4.64 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^4/((d*x + c)^3*b + a)^3,x, algorithm="fricas")

[Out]

1/162*sqrt(3)*(2*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)) - sqr
t(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)) - 6*(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)*(2*b*d^5*x^5 + 10*b*c*d^4*x^4 + 20*
b*c^2*d^3*x^3 + 2*b*c^5 + (20*b*c^3 - a)*d^2*x^2 - a*c^2 + 2*(5*b*c^4 - a*c)*d*x
)*(-a*b^2)^(1/3))/((a*b^3*d^7*x^6 + 6*a*b^3*c*d^6*x^5 + 15*a*b^3*c^2*d^5*x^4 + 2
*(10*a*b^3*c^3 + a^2*b^2)*d^4*x^3 + 3*(5*a*b^3*c^4 + 2*a^2*b^2*c)*d^3*x^2 + 6*(a
*b^3*c^5 + a^2*b^2*c^2)*d^2*x + (a*b^3*c^6 + 2*a^2*b^2*c^3 + a^3*b)*d)*(-a*b^2)^
(1/3))

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Sympy [A]  time = 118.13, size = 287, normalized size = 1.4 \[ \frac{- a c^{2} + 2 b c^{5} + 20 b c^{2} d^{3} x^{3} + 10 b c d^{4} x^{4} + 2 b d^{5} x^{5} + x^{2} \left (- a d^{2} + 20 b c^{3} d^{2}\right ) + x \left (- 2 a c d + 10 b c^{4} d\right )}{18 a^{3} b d + 36 a^{2} b^{2} c^{3} d + 18 a b^{3} c^{6} d + 270 a b^{3} c^{2} d^{5} x^{4} + 108 a b^{3} c d^{6} x^{5} + 18 a b^{3} d^{7} x^{6} + x^{3} \left (36 a^{2} b^{2} d^{4} + 360 a b^{3} c^{3} d^{4}\right ) + x^{2} \left (108 a^{2} b^{2} c d^{3} + 270 a b^{3} c^{4} d^{3}\right ) + x \left (108 a^{2} b^{2} c^{2} d^{2} + 108 a b^{3} c^{5} d^{2}\right )} + \frac{\operatorname{RootSum}{\left (19683 t^{3} a^{4} b^{5} + 1, \left ( t \mapsto t \log{\left (x + \frac{729 t^{2} a^{3} b^{3} + c}{d} \right )} \right )\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**4/(a+b*(d*x+c)**3)**3,x)

[Out]

(-a*c**2 + 2*b*c**5 + 20*b*c**2*d**3*x**3 + 10*b*c*d**4*x**4 + 2*b*d**5*x**5 + x
**2*(-a*d**2 + 20*b*c**3*d**2) + x*(-2*a*c*d + 10*b*c**4*d))/(18*a**3*b*d + 36*a
**2*b**2*c**3*d + 18*a*b**3*c**6*d + 270*a*b**3*c**2*d**5*x**4 + 108*a*b**3*c*d*
*6*x**5 + 18*a*b**3*d**7*x**6 + x**3*(36*a**2*b**2*d**4 + 360*a*b**3*c**3*d**4)
+ x**2*(108*a**2*b**2*c*d**3 + 270*a*b**3*c**4*d**3) + x*(108*a**2*b**2*c**2*d**
2 + 108*a*b**3*c**5*d**2)) + RootSum(19683*_t**3*a**4*b**5 + 1, Lambda(_t, _t*lo
g(x + (729*_t**2*a**3*b**3 + c)/d)))/d

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{4}}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^4/((d*x + c)^3*b + a)^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^4/((d*x + c)^3*b + a)^3, x)