\(\int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{2/3}} \, dx\) [164]

Optimal result

Integrand size = 32, antiderivative size = 330 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{2/3}} \, dx=\frac {3 \left (a^2 d^2 D-a b d (C d-c D)-b^2 \left (c C d-B d^2-c^2 D\right )\right ) \sqrt [3]{c+d x}}{b^3 d^3}+\frac {3 (b C d-2 b c D-a d D) (c+d x)^{4/3}}{4 b^2 d^3}+\frac {3 D (c+d x)^{7/3}}{7 b d^3}-\frac {\sqrt {3} \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{b^{10/3} (b c-a d)^{2/3}}-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \log (a+b x)}{2 b^{10/3} (b c-a d)^{2/3}}+\frac {3 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{10/3} (b c-a d)^{2/3}} \] Output:

3*(a^2*d^2*D-a*b*d*(C*d-D*c)-b^2*(-B*d^2+C*c*d-D*c^2))*(d*x+c)^(1/3)/b^3/d 
^3+3/4*(C*b*d-D*a*d-2*D*b*c)*(d*x+c)^(4/3)/b^2/d^3+3/7*D*(d*x+c)^(7/3)/b/d 
^3-3^(1/2)*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*arctan(1/3*(1+2*b^(1/3)*(d*x+c)^( 
1/3)/(-a*d+b*c)^(1/3))*3^(1/2))/b^(10/3)/(-a*d+b*c)^(2/3)-1/2*(A*b^3-a*(B* 
b^2-C*a*b+D*a^2))*ln(b*x+a)/b^(10/3)/(-a*d+b*c)^(2/3)+3/2*(A*b^3-a*(B*b^2- 
C*a*b+D*a^2))*ln((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3))/b^(10/3)/(-a*d+b* 
c)^(2/3)
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{2/3}} \, dx=\frac {3 \sqrt [3]{b} (-b c+a d)^{2/3} \sqrt [3]{c+d x} \left (28 a^2 d^2 D-7 a b d (4 C d-3 c D+d D x)+b^2 \left (18 c^2 D-3 c d (7 C+2 D x)+d^2 \left (28 B+7 C x+4 D x^2\right )\right )\right )-28 \sqrt {3} d^3 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{-b c+a d}}}{\sqrt {3}}\right )+28 d^3 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \log \left (\sqrt [3]{-b c+a d}+\sqrt [3]{b} \sqrt [3]{c+d x}\right )-14 d^3 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \log \left ((-b c+a d)^{2/3}-\sqrt [3]{b} \sqrt [3]{-b c+a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{28 b^{10/3} d^3 (-b c+a d)^{2/3}} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*(c + d*x)^(2/3)),x]
 

Output:

(3*b^(1/3)*(-(b*c) + a*d)^(2/3)*(c + d*x)^(1/3)*(28*a^2*d^2*D - 7*a*b*d*(4 
*C*d - 3*c*D + d*D*x) + b^2*(18*c^2*D - 3*c*d*(7*C + 2*D*x) + d^2*(28*B + 
7*C*x + 4*D*x^2))) - 28*Sqrt[3]*d^3*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Ar 
cTan[(1 - (2*b^(1/3)*(c + d*x)^(1/3))/(-(b*c) + a*d)^(1/3))/Sqrt[3]] + 28* 
d^3*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Log[(-(b*c) + a*d)^(1/3) + b^(1/3) 
*(c + d*x)^(1/3)] - 14*d^3*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Log[(-(b*c) 
 + a*d)^(2/3) - b^(1/3)*(-(b*c) + a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c 
+ d*x)^(2/3)])/(28*b^(10/3)*d^3*(-(b*c) + a*d)^(2/3))
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2123, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{2/3}} \, dx\)

\(\Big \downarrow \) 2123

\(\displaystyle \int \left (\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{b^3 (a+b x) (c+d x)^{2/3}}+\frac {a^2 d^2 D-a b d (C d-c D)-\left (b^2 \left (-B d^2+c^2 (-D)+c C d\right )\right )}{b^3 d^2 (c+d x)^{2/3}}+\frac {\sqrt [3]{c+d x} (-a d D-2 b c D+b C d)}{b^2 d^2}+\frac {D (c+d x)^{4/3}}{b d^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {3} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}+1}{\sqrt {3}}\right )}{b^{10/3} (b c-a d)^{2/3}}-\frac {\log (a+b x) \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{2 b^{10/3} (b c-a d)^{2/3}}+\frac {3 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{10/3} (b c-a d)^{2/3}}+\frac {3 \sqrt [3]{c+d x} \left (a^2 d^2 D-a b d (C d-c D)-\left (b^2 \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{b^3 d^3}+\frac {3 (c+d x)^{4/3} (-a d D-2 b c D+b C d)}{4 b^2 d^3}+\frac {3 D (c+d x)^{7/3}}{7 b d^3}\)

Input:

Int[(A + B*x + C*x^2 + D*x^3)/((a + b*x)*(c + d*x)^(2/3)),x]
 

Output:

(3*(a^2*d^2*D - a*b*d*(C*d - c*D) - b^2*(c*C*d - B*d^2 - c^2*D))*(c + d*x) 
^(1/3))/(b^3*d^3) + (3*(b*C*d - 2*b*c*D - a*d*D)*(c + d*x)^(4/3))/(4*b^2*d 
^3) + (3*D*(c + d*x)^(7/3))/(7*b*d^3) - (Sqrt[3]*(A*b^3 - a*(b^2*B - a*b*C 
 + a^2*D))*ArcTan[(1 + (2*b^(1/3)*(c + d*x)^(1/3))/(b*c - a*d)^(1/3))/Sqrt 
[3]])/(b^(10/3)*(b*c - a*d)^(2/3)) - ((A*b^3 - a*(b^2*B - a*b*C + a^2*D))* 
Log[a + b*x])/(2*b^(10/3)*(b*c - a*d)^(2/3)) + (3*(A*b^3 - a*(b^2*B - a*b* 
C + a^2*D))*Log[(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)])/(2*b^(10/3)* 
(b*c - a*d)^(2/3))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2123

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c 
, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
 

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.80

method	result	size
pseudoelliptic	\(\frac {3 \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}} \left (\left (\left (\frac {1}{7} D x^{2}+\frac {1}{4} C x +B \right ) b^{2}-\left (\frac {D x}{4}+C \right ) a b +D a^{2}\right ) d^{2}-\frac {3 \left (\left (\frac {2 D x}{7}+C \right ) b -D a \right ) c b d}{4}+\frac {9 D b^{2} c^{2}}{14}\right ) \left (x d +c \right )^{\frac {1}{3}} b -\frac {\left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right ) \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (2 \left (x d +c \right )^{\frac {1}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (x d +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )-2 \ln \left (\left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )\right ) d^{3}}{2}}{d^{3} b^{4} \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}\)	\(264\)
derivativedivides	\(\frac {\frac {3 \left (\frac {D \left (x d +c \right )^{\frac {7}{3}} b^{2}}{7}+\frac {C \,b^{2} d \left (x d +c \right )^{\frac {4}{3}}}{4}-\frac {D a b d \left (x d +c \right )^{\frac {4}{3}}}{4}-\frac {D b^{2} c \left (x d +c \right )^{\frac {4}{3}}}{2}+B \,d^{2} b^{2} \left (x d +c \right )^{\frac {1}{3}}-C a \,d^{2} b \left (x d +c \right )^{\frac {1}{3}}-C \,b^{2} c d \left (x d +c \right )^{\frac {1}{3}}+D a^{2} d^{2} \left (x d +c \right )^{\frac {1}{3}}+D a b c d \left (x d +c \right )^{\frac {1}{3}}+D b^{2} c^{2} \left (x d +c \right )^{\frac {1}{3}}\right )}{b^{3}}+\frac {3 \left (\frac {\ln \left (\left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (x d +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (x d +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}\right ) d^{3} \left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right )}{b^{3}}}{d^{3}}\)	\(347\)
default	\(\frac {\frac {3 \left (\frac {D \left (x d +c \right )^{\frac {7}{3}} b^{2}}{7}+\frac {C \,b^{2} d \left (x d +c \right )^{\frac {4}{3}}}{4}-\frac {D a b d \left (x d +c \right )^{\frac {4}{3}}}{4}-\frac {D b^{2} c \left (x d +c \right )^{\frac {4}{3}}}{2}+B \,d^{2} b^{2} \left (x d +c \right )^{\frac {1}{3}}-C a \,d^{2} b \left (x d +c \right )^{\frac {1}{3}}-C \,b^{2} c d \left (x d +c \right )^{\frac {1}{3}}+D a^{2} d^{2} \left (x d +c \right )^{\frac {1}{3}}+D a b c d \left (x d +c \right )^{\frac {1}{3}}+D b^{2} c^{2} \left (x d +c \right )^{\frac {1}{3}}\right )}{b^{3}}+\frac {3 \left (\frac {\ln \left (\left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (x d +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (x d +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}\right ) d^{3} \left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right )}{b^{3}}}{d^{3}}\)	\(347\)

Input:

int((D*x^3+C*x^2+B*x+A)/(b*x+a)/(d*x+c)^(2/3),x,method=_RETURNVERBOSE)
 

Output:

3*(((a*d-b*c)/b)^(2/3)*(((1/7*D*x^2+1/4*C*x+B)*b^2-(1/4*D*x+C)*a*b+D*a^2)* 
d^2-3/4*((2/7*D*x+C)*b-D*a)*c*b*d+9/14*D*b^2*c^2)*(d*x+c)^(1/3)*b-1/6*(A*b 
^3-B*a*b^2+C*a^2*b-D*a^3)*(-2*arctan(1/3*3^(1/2)*(2*(d*x+c)^(1/3)-((a*d-b* 
c)/b)^(1/3))/((a*d-b*c)/b)^(1/3))*3^(1/2)+ln((d*x+c)^(2/3)-((a*d-b*c)/b)^( 
1/3)*(d*x+c)^(1/3)+((a*d-b*c)/b)^(2/3))-2*ln((d*x+c)^(1/3)+((a*d-b*c)/b)^( 
1/3)))*d^3)/((a*d-b*c)/b)^(2/3)/d^3/b^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 782 vs. \(2 (286) = 572\).

Time = 0.12 (sec) , antiderivative size = 1656, normalized size of antiderivative = 5.02 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{2/3}} \, dx=\text {Too large to display} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)/(d*x+c)^(2/3),x, algorithm="fricas")
 

Output:

[1/28*(14*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*(D*a^3 - C*a^2*b + B*a 
*b^2 - A*b^3)*d^3*log(-(b^2*c - a*b*d)*(d*x + c)^(2/3) - (b^3*c^2 - 2*a*b^ 
2*c*d + a^2*b*d^2)^(1/3)*(b*c - a*d) - (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2) 
^(2/3)*(d*x + c)^(1/3)) - 28*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*(D* 
a^3 - C*a^2*b + B*a*b^2 - A*b^3)*d^3*log(-(b^2*c - a*b*d)*(d*x + c)^(1/3) 
+ (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)) - 14*sqrt(3)*((D*a^3*b^2 - C* 
a^2*b^3 + B*a*b^4 - A*b^5)*c*d^3 - (D*a^4*b - C*a^3*b^2 + B*a^2*b^3 - A*a* 
b^4)*d^4)*sqrt(-(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)/b)*log(-(3*b^2*c 
^2 - 4*a*b*c*d + a^2*d^2 + 2*(b^2*c*d - a*b*d^2)*x - sqrt(3)*(2*(b^2*c - a 
*b*d)*(d*x + c)^(2/3) - (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)*(b*c - a 
*d) - (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*(d*x + c)^(1/3))*sqrt(-(b^ 
3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)/b) - 3*(b^3*c^2 - 2*a*b^2*c*d + a^2 
*b*d^2)^(1/3)*(b*c - a*d)*(d*x + c)^(1/3))/(b*x + a)) + 3*(18*D*b^5*c^4 - 
3*(5*D*a*b^4 + 7*C*b^5)*c^3*d + 2*(2*D*a^2*b^3 + 7*C*a*b^4 + 14*B*b^5)*c^2 
*d^2 - 7*(5*D*a^3*b^2 - 5*C*a^2*b^3 + 8*B*a*b^4)*c*d^3 + 28*(D*a^4*b - C*a 
^3*b^2 + B*a^2*b^3)*d^4 + 4*(D*b^5*c^2*d^2 - 2*D*a*b^4*c*d^3 + D*a^2*b^3*d 
^4)*x^2 - (6*D*b^5*c^3*d - (5*D*a*b^4 + 7*C*b^5)*c^2*d^2 - 2*(4*D*a^2*b^3 
- 7*C*a*b^4)*c*d^3 + 7*(D*a^3*b^2 - C*a^2*b^3)*d^4)*x)*(d*x + c)^(1/3))/(b 
^6*c^2*d^3 - 2*a*b^5*c*d^4 + a^2*b^4*d^5), 1/28*(14*(b^3*c^2 - 2*a*b^2*c*d 
 + a^2*b*d^2)^(2/3)*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*d^3*log(-(b^2*c...
 

Sympy [F]

\[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{2/3}} \, dx=\int \frac {A + B x + C x^{2} + D x^{3}}{\left (a + b x\right ) \left (c + d x\right )^{\frac {2}{3}}}\, dx \] Input:

integrate((D*x**3+C*x**2+B*x+A)/(b*x+a)/(d*x+c)**(2/3),x)
                                                                                    
                                                                                    
 

Output:

Integral((A + B*x + C*x**2 + D*x**3)/((a + b*x)*(c + d*x)**(2/3)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{2/3}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)/(d*x+c)^(2/3),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.44 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{2/3}} \, dx=-\frac {{\left (D a^{3} b^{4} d^{24} - C a^{2} b^{5} d^{24} + B a b^{6} d^{24} - A b^{7} d^{24}\right )} \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} \log \left ({\left | {\left (d x + c\right )}^{\frac {1}{3}} - \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} \right |}\right )}{b^{8} c d^{24} - a b^{7} d^{25}} + \frac {{\left (\sqrt {3} D a^{3} - \sqrt {3} C a^{2} b + \sqrt {3} B a b^{2} - \sqrt {3} A b^{3}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} + \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}}}\right )}{{\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} b^{2}} + \frac {{\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} \log \left ({\left (d x + c\right )}^{\frac {2}{3}} + {\left (d x + c\right )}^{\frac {1}{3}} \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} + \left (\frac {b c - a d}{b}\right )^{\frac {2}{3}}\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} b^{2}} + \frac {3 \, {\left (4 \, {\left (d x + c\right )}^{\frac {7}{3}} D b^{6} d^{18} - 14 \, {\left (d x + c\right )}^{\frac {4}{3}} D b^{6} c d^{18} + 28 \, {\left (d x + c\right )}^{\frac {1}{3}} D b^{6} c^{2} d^{18} - 7 \, {\left (d x + c\right )}^{\frac {4}{3}} D a b^{5} d^{19} + 7 \, {\left (d x + c\right )}^{\frac {4}{3}} C b^{6} d^{19} + 28 \, {\left (d x + c\right )}^{\frac {1}{3}} D a b^{5} c d^{19} - 28 \, {\left (d x + c\right )}^{\frac {1}{3}} C b^{6} c d^{19} + 28 \, {\left (d x + c\right )}^{\frac {1}{3}} D a^{2} b^{4} d^{20} - 28 \, {\left (d x + c\right )}^{\frac {1}{3}} C a b^{5} d^{20} + 28 \, {\left (d x + c\right )}^{\frac {1}{3}} B b^{6} d^{20}\right )}}{28 \, b^{7} d^{21}} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)/(d*x+c)^(2/3),x, algorithm="giac")
 

Output:

-(D*a^3*b^4*d^24 - C*a^2*b^5*d^24 + B*a*b^6*d^24 - A*b^7*d^24)*((b*c - a*d 
)/b)^(1/3)*log(abs((d*x + c)^(1/3) - ((b*c - a*d)/b)^(1/3)))/(b^8*c*d^24 - 
 a*b^7*d^25) + (sqrt(3)*D*a^3 - sqrt(3)*C*a^2*b + sqrt(3)*B*a*b^2 - sqrt(3 
)*A*b^3)*arctan(1/3*sqrt(3)*(2*(d*x + c)^(1/3) + ((b*c - a*d)/b)^(1/3))/(( 
b*c - a*d)/b)^(1/3))/((b^3*c - a*b^2*d)^(2/3)*b^2) + 1/2*(D*a^3 - C*a^2*b 
+ B*a*b^2 - A*b^3)*log((d*x + c)^(2/3) + (d*x + c)^(1/3)*((b*c - a*d)/b)^( 
1/3) + ((b*c - a*d)/b)^(2/3))/((b^3*c - a*b^2*d)^(2/3)*b^2) + 3/28*(4*(d*x 
 + c)^(7/3)*D*b^6*d^18 - 14*(d*x + c)^(4/3)*D*b^6*c*d^18 + 28*(d*x + c)^(1 
/3)*D*b^6*c^2*d^18 - 7*(d*x + c)^(4/3)*D*a*b^5*d^19 + 7*(d*x + c)^(4/3)*C* 
b^6*d^19 + 28*(d*x + c)^(1/3)*D*a*b^5*c*d^19 - 28*(d*x + c)^(1/3)*C*b^6*c* 
d^19 + 28*(d*x + c)^(1/3)*D*a^2*b^4*d^20 - 28*(d*x + c)^(1/3)*C*a*b^5*d^20 
 + 28*(d*x + c)^(1/3)*B*b^6*d^20)/(b^7*d^21)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{2/3}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{\left (a+b\,x\right )\,{\left (c+d\,x\right )}^{2/3}} \,d x \] Input:

int((A + B*x + C*x^2 + x^3*D)/((a + b*x)*(c + d*x)^(2/3)),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/((a + b*x)*(c + d*x)^(2/3)), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.14 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x) (c+d x)^{2/3}} \, dx =\text {Too large to display} \] Input:

int((D*x^3+C*x^2+B*x+A)/(b*x+a)/(d*x+c)^(2/3),x)
 

Output:

(28*sqrt(3)*atan((b**(1/6)*(a*d - b*c)**(1/6)*sqrt(3) - 2*b**(1/3)*(c + d* 
x)**(1/6))/(b**(1/6)*(a*d - b*c)**(1/6)))*a**3*d**3 - 28*sqrt(3)*atan((b** 
(1/6)*(a*d - b*c)**(1/6)*sqrt(3) - 2*b**(1/3)*(c + d*x)**(1/6))/(b**(1/6)* 
(a*d - b*c)**(1/6)))*a**2*b*c*d**2 + 28*sqrt(3)*atan((b**(1/6)*(a*d - b*c) 
**(1/6)*sqrt(3) + 2*b**(1/3)*(c + d*x)**(1/6))/(b**(1/6)*(a*d - b*c)**(1/6 
)))*a**3*d**3 - 28*sqrt(3)*atan((b**(1/6)*(a*d - b*c)**(1/6)*sqrt(3) + 2*b 
**(1/3)*(c + d*x)**(1/6))/(b**(1/6)*(a*d - b*c)**(1/6)))*a**2*b*c*d**2 + 8 
4*b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(2/3)*a**2*d**2 - 21*b**(1/3)*(c 
+ d*x)**(1/3)*(a*d - b*c)**(2/3)*a*b*c*d - 21*b**(1/3)*(c + d*x)**(1/3)*(a 
*d - b*c)**(2/3)*a*b*d**2*x + 84*b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(2 
/3)*b**3*d - 9*b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(2/3)*b**2*c**2 + 3* 
b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(2/3)*b**2*c*d*x + 12*b**(1/3)*(c + 
 d*x)**(1/3)*(a*d - b*c)**(2/3)*b**2*d**2*x**2 - 28*log((a*d - b*c)**(1/3) 
 + b**(1/3)*(c + d*x)**(1/3))*a**3*d**3 + 28*log((a*d - b*c)**(1/3) + b**( 
1/3)*(c + d*x)**(1/3))*a**2*b*c*d**2 + 14*log( - b**(1/6)*(c + d*x)**(1/6) 
*(a*d - b*c)**(1/6)*sqrt(3) + (a*d - b*c)**(1/3) + b**(1/3)*(c + d*x)**(1/ 
3))*a**3*d**3 - 14*log( - b**(1/6)*(c + d*x)**(1/6)*(a*d - b*c)**(1/6)*sqr 
t(3) + (a*d - b*c)**(1/3) + b**(1/3)*(c + d*x)**(1/3))*a**2*b*c*d**2 + 14* 
log(b**(1/6)*(c + d*x)**(1/6)*(a*d - b*c)**(1/6)*sqrt(3) + (a*d - b*c)**(1 
/3) + b**(1/3)*(c + d*x)**(1/3))*a**3*d**3 - 14*log(b**(1/6)*(c + d*x)*...