3.1.28 \(\int \frac {\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx\) [28]

3.1.28.1 Optimal result

Integrand size = 19, antiderivative size = 818 \[ \int \frac {\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx=-\frac {c^2 \sqrt [3]{a+b x^3}}{d \left (c^3+d^3 x^3\right )}-\frac {d x^2 \sqrt [3]{a+b x^3}}{c^3+d^3 x^3}+\frac {x \sqrt [3]{a+b x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},2,\frac {4}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{c^2 \sqrt [3]{1+\frac {b x^3}{a}}}-\frac {d^3 x^4 \sqrt [3]{a+b x^3} \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{2 c^5 \sqrt [3]{1+\frac {b x^3}{a}}}-\frac {\sqrt [3]{b} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {2 a d \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c \left (b c^3-a d^3\right )^{2/3}}+\frac {\left (3 b c^3-2 a d^3\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {b c^2 \arctan \left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {b c^2 \log \left (c^3+d^3 x^3\right )}{6 d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {a d \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{2/3}}-\frac {\left (3 b c^3-2 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 c d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 d^2}+\frac {a d \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{a+b x^3}\right )}{3 c \left (b c^3-a d^3\right )^{2/3}}+\frac {\left (3 b c^3-2 a d^3\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{a+b x^3}\right )}{6 c d^2 \left (b c^3-a d^3\right )^{2/3}}+\frac {b c^2 \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{2 d^2 \left (b c^3-a d^3\right )^{2/3}} \]

output

-c^2*(b*x^3+a)^(1/3)/d/(d^3*x^3+c^3)-d*x^2*(b*x^3+a)^(1/3)/(d^3*x^3+c^3)+x 
*(b*x^3+a)^(1/3)*AppellF1(1/3,-1/3,2,4/3,-b*x^3/a,-d^3*x^3/c^3)/c^2/(1+b*x 
^3/a)^(1/3)-1/2*d^3*x^4*(b*x^3+a)^(1/3)*AppellF1(4/3,-1/3,2,7/3,-b*x^3/a,- 
d^3*x^3/c^3)/c^5/(1+b*x^3/a)^(1/3)-1/6*b*c^2*ln(d^3*x^3+c^3)/d^2/(-a*d^3+b 
*c^3)^(2/3)-1/9*a*d*ln(d^3*x^3+c^3)/c/(-a*d^3+b*c^3)^(2/3)-1/18*(-2*a*d^3+ 
3*b*c^3)*ln(d^3*x^3+c^3)/c/d^2/(-a*d^3+b*c^3)^(2/3)-1/2*b^(1/3)*ln(b^(1/3) 
*x-(b*x^3+a)^(1/3))/d^2+1/3*a*d*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3 
))/c/(-a*d^3+b*c^3)^(2/3)+1/6*(-2*a*d^3+3*b*c^3)*ln((-a*d^3+b*c^3)^(1/3)*x 
/c-(b*x^3+a)^(1/3))/c/d^2/(-a*d^3+b*c^3)^(2/3)+1/2*b*c^2*ln((-a*d^3+b*c^3) 
^(1/3)+d*(b*x^3+a)^(1/3))/d^2/(-a*d^3+b*c^3)^(2/3)-1/3*b^(1/3)*arctan(1/3* 
(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/d^2*3^(1/2)+2/9*a*d*arctan(1/3*(1 
+2*(-a*d^3+b*c^3)^(1/3)*x/c/(b*x^3+a)^(1/3))*3^(1/2))/c/(-a*d^3+b*c^3)^(2/ 
3)*3^(1/2)+1/9*(-2*a*d^3+3*b*c^3)*arctan(1/3*(1+2*(-a*d^3+b*c^3)^(1/3)*x/c 
/(b*x^3+a)^(1/3))*3^(1/2))/c/d^2/(-a*d^3+b*c^3)^(2/3)*3^(1/2)-1/3*b*c^2*ar 
ctan(1/3*(1-2*d*(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)^(1/3))*3^(1/2))/d^2/(-a*d^3 
+b*c^3)^(2/3)*3^(1/2)
 

3.1.28.2 Mathematica [F]

\[ \int \frac {\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx=\int \frac {\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx \]

input

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

output

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

3.1.28.3 Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 818, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2581, 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.

\(\displaystyle \int \frac {\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx\)

\(\Big \downarrow \) 2581

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d^3 \sqrt [3]{b x^3+a} \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) x^4}{2 c^5 \sqrt [3]{\frac {b x^3}{a}+1}}-\frac {d \sqrt [3]{b x^3+a} x^2}{c^3+d^3 x^3}+\frac {\sqrt [3]{b x^3+a} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},2,\frac {4}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) x}{c^2 \sqrt [3]{\frac {b x^3}{a}+1}}-\frac {\sqrt [3]{b} \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {2 a d \arctan \left (\frac {\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \left (b c^3-a d^3\right )^{2/3}}+\frac {\left (3 b c^3-2 a d^3\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {b c^2 \arctan \left (\frac {1-\frac {2 d \sqrt [3]{b x^3+a}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {a d \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{2/3}}-\frac {\left (3 b c^3-2 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 c d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {b c^2 \log \left (c^3+d^3 x^3\right )}{6 d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b} x-\sqrt [3]{b x^3+a}\right )}{2 d^2}+\frac {a d \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{3 c \left (b c^3-a d^3\right )^{2/3}}+\frac {\left (3 b c^3-2 a d^3\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{6 c d^2 \left (b c^3-a d^3\right )^{2/3}}+\frac {b c^2 \log \left (\sqrt [3]{b x^3+a} d+\sqrt [3]{b c^3-a d^3}\right )}{2 d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {c^2 \sqrt [3]{b x^3+a}}{d \left (c^3+d^3 x^3\right )}\)

input

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

output

-((c^2*(a + b*x^3)^(1/3))/(d*(c^3 + d^3*x^3))) - (d*x^2*(a + b*x^3)^(1/3)) 
/(c^3 + d^3*x^3) + (x*(a + b*x^3)^(1/3)*AppellF1[1/3, -1/3, 2, 4/3, -((b*x 
^3)/a), -((d^3*x^3)/c^3)])/(c^2*(1 + (b*x^3)/a)^(1/3)) - (d^3*x^4*(a + b*x 
^3)^(1/3)*AppellF1[4/3, -1/3, 2, 7/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/(2* 
c^5*(1 + (b*x^3)/a)^(1/3)) - (b^(1/3)*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3 
)^(1/3))/Sqrt[3]])/(Sqrt[3]*d^2) + (2*a*d*ArcTan[(1 + (2*(b*c^3 - a*d^3)^( 
1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]*c*(b*c^3 - a*d^3)^(2/3 
)) + ((3*b*c^3 - 2*a*d^3)*ArcTan[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + 
b*x^3)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]*c*d^2*(b*c^3 - a*d^3)^(2/3)) - (b*c^2* 
ArcTan[(1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/Sqrt[3]])/(Sqrt 
[3]*d^2*(b*c^3 - a*d^3)^(2/3)) - (b*c^2*Log[c^3 + d^3*x^3])/(6*d^2*(b*c^3 
- a*d^3)^(2/3)) - (a*d*Log[c^3 + d^3*x^3])/(9*c*(b*c^3 - a*d^3)^(2/3)) - ( 
(3*b*c^3 - 2*a*d^3)*Log[c^3 + d^3*x^3])/(18*c*d^2*(b*c^3 - a*d^3)^(2/3)) - 
 (b^(1/3)*Log[b^(1/3)*x - (a + b*x^3)^(1/3)])/(2*d^2) + (a*d*Log[((b*c^3 - 
 a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)])/(3*c*(b*c^3 - a*d^3)^(2/3)) + ((3 
*b*c^3 - 2*a*d^3)*Log[((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)])/(6 
*c*d^2*(b*c^3 - a*d^3)^(2/3)) + (b*c^2*Log[(b*c^3 - a*d^3)^(1/3) + d*(a + 
b*x^3)^(1/3)])/(2*d^2*(b*c^3 - a*d^3)^(2/3))
 

3.1.28.3.1 Defintions of rubi rules used

rule 2009

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

rule 2581

Int[(Px_.)*((c_) + (d_.)*(x_))^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] 
:> Int[ExpandIntegrand[(c^3 + d^3*x^3)^q*(a + b*x^3)^p, Px/(c^2 - c*d*x + d 
^2*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p}, x] && PolyQ[Px, x] && ILtQ[q, 0 
] && RationalQ[p] && EqQ[Denominator[p], 3]
 

3.1.28.4 Maple [F]

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

input

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

output

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

3.1.28.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.1.28.6 Sympy [F]

\[ \int \frac {\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx=\int \frac {\sqrt [3]{a + b x^{3}}}{\left (c + d x\right )^{2}}\, dx \]

input

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

output

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

3.1.28.7 Maxima [F]

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

input

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

output

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

3.1.28.8 Giac [F]

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

input

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

output

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

3.1.28.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{1/3}}{{\left (c+d\,x\right )}^2} \,d x \]

input

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

output

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