3.1.35 \(\int \frac {1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx\) [35]

3.1.35.1 Optimal result

Integrand size = 19, antiderivative size = 1513 \[ \int \frac {1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx =\text {Too large to display} \]

output

3/2*c^4*d^2*(b*x^3+a)^(2/3)/(-a*d^3+b*c^3)/(d^3*x^3+c^3)^2-3/2*c^3*d^3*x*( 
b*x^3+a)^(2/3)/(-a*d^3+b*c^3)/(d^3*x^3+c^3)^2+4/3*b*c^4*d^2*(b*x^3+a)^(2/3 
)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)-1/3*c*d^2*(-3*a*d^3+b*c^3)*(b*x^3+a)^(2/3 
)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)+1/18*d^3*(-7*a*d^3+3*b*c^3)*x*(b*x^3+a)^( 
2/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)-1/18*d^3*(-5*a*d^3+9*b*c^3)*x*(b*x^3+a 
)^(2/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)-7/18*d^3*(a*d^3+3*b*c^3)*x*(b*x^3+a 
)^(2/3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)-3/2*d*x^2*(1+b*x^3/a)^(1/3)*AppellF 
1(2/3,1/3,3,5/3,-b*x^3/a,-d^3*x^3/c^3)/c^4/(b*x^3+a)^(1/3)+6/5*d^4*x^5*(1+ 
b*x^3/a)^(1/3)*AppellF1(5/3,1/3,3,8/3,-b*x^3/a,-d^3*x^3/c^3)/c^7/(b*x^3+a) 
^(1/3)+2/9*b^2*c^4*ln(d^3*x^3+c^3)/(-a*d^3+b*c^3)^(7/3)+1/27*a^2*d^6*ln(d^ 
3*x^3+c^3)/c^2/(-a*d^3+b*c^3)^(7/3)-1/18*b*c*(-3*a*d^3+b*c^3)*ln(d^3*x^3+c 
^3)/(-a*d^3+b*c^3)^(7/3)+7/54*a*d^3*(-a*d^3+3*b*c^3)*ln(d^3*x^3+c^3)/c^2/( 
-a*d^3+b*c^3)^(7/3)+1/54*(5*a^2*d^6-12*a*b*c^3*d^3+9*b^2*c^6)*ln(d^3*x^3+c 
^3)/c^2/(-a*d^3+b*c^3)^(7/3)-1/9*a^2*d^6*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^ 
3+a)^(1/3))/c^2/(-a*d^3+b*c^3)^(7/3)-7/18*a*d^3*(-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^2/(-a*d^3+b*c^3)^(7/3)-1/18*(5*a^2*d 
^6-12*a*b*c^3*d^3+9*b^2*c^6)*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/ 
c^2/(-a*d^3+b*c^3)^(7/3)-2/3*b^2*c^4*ln((-a*d^3+b*c^3)^(1/3)+d*(b*x^3+a)^( 
1/3))/(-a*d^3+b*c^3)^(7/3)+1/6*b*c*(-3*a*d^3+b*c^3)*ln((-a*d^3+b*c^3)^(1/3 
)+d*(b*x^3+a)^(1/3))/(-a*d^3+b*c^3)^(7/3)+2/27*a^2*d^6*arctan(1/3*(1+2*...
 

3.1.35.2 Mathematica [F]

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

input

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

output

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

3.1.35.3 Rubi [A] (verified)

Time = 2.24 (sec) , antiderivative size = 1513, 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.

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

\(\Big \downarrow \) 2581

\(\displaystyle \int \left (-\frac {7 c^3 d^3 x^3}{\sqrt [3]{a+b x^3} \left (c^3+d^3 x^3\right )^3}+\frac {d^6 x^6}{\sqrt [3]{a+b x^3} \left (c^3+d^3 x^3\right )^3}-\frac {3 c d^5 x^5}{\sqrt [3]{a+b x^3} \left (c^3+d^3 x^3\right )^3}+\frac {c^6}{\sqrt [3]{a+b x^3} \left (c^3+d^3 x^3\right )^3}-\frac {3 c^5 d x}{\sqrt [3]{a+b x^3} \left (c^3+d^3 x^3\right )^3}+\frac {6 c^4 d^2 x^2}{\sqrt [3]{a+b x^3} \left (c^3+d^3 x^3\right )^3}+\frac {6 c^2 d^4 x^4}{\sqrt [3]{a+b x^3} \left (c^3+d^3 x^3\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 a^2 \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 ) d^6}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {a^2 \log \left (c^3+d^3 x^3\right ) d^6}{27 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {a^2 \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right ) d^6}{9 c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {6 x^5 \sqrt [3]{\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},3,\frac {8}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) d^4}{5 c^7 \sqrt [3]{b x^3+a}}+\frac {7 a \left (3 b c^3-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 ) d^3}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {7 a \left (3 b c^3-a d^3\right ) \log \left (c^3+d^3 x^3\right ) d^3}{54 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {7 a \left (3 b c^3-a d^3\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right ) d^3}{18 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {7 \left (3 b c^3+a d^3\right ) x \left (b x^3+a\right )^{2/3} d^3}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {\left (3 b c^3-7 a d^3\right ) x \left (b x^3+a\right )^{2/3} d^3}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {\left (9 b c^3-5 a d^3\right ) x \left (b x^3+a\right )^{2/3} d^3}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {3 c^3 x \left (b x^3+a\right )^{2/3} d^3}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}+\frac {4 b c^4 \left (b x^3+a\right )^{2/3} d^2}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {c \left (b c^3-3 a d^3\right ) \left (b x^3+a\right )^{2/3} d^2}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {3 c^4 \left (b x^3+a\right )^{2/3} d^2}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}-\frac {3 x^2 \sqrt [3]{\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},3,\frac {5}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) d}{2 c^4 \sqrt [3]{b x^3+a}}+\frac {\left (9 b^2 c^6-12 a b d^3 c^3+5 a^2 d^6\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 )}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {4 b^2 c^4 \arctan \left (\frac {1-\frac {2 d \sqrt [3]{b x^3+a}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{7/3}}+\frac {b c \left (b c^3-3 a d^3\right ) \arctan \left (\frac {1-\frac {2 d \sqrt [3]{b x^3+a}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{7/3}}+\frac {\left (9 b^2 c^6-12 a b d^3 c^3+5 a^2 d^6\right ) \log \left (c^3+d^3 x^3\right )}{54 c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {2 b^2 c^4 \log \left (c^3+d^3 x^3\right )}{9 \left (b c^3-a d^3\right )^{7/3}}-\frac {b c \left (b c^3-3 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 \left (b c^3-a d^3\right )^{7/3}}-\frac {\left (9 b^2 c^6-12 a b d^3 c^3+5 a^2 d^6\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{18 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {2 b^2 c^4 \log \left (\sqrt [3]{b x^3+a} d+\sqrt [3]{b c^3-a d^3}\right )}{3 \left (b c^3-a d^3\right )^{7/3}}+\frac {b c \left (b c^3-3 a d^3\right ) \log \left (\sqrt [3]{b x^3+a} d+\sqrt [3]{b c^3-a d^3}\right )}{6 \left (b c^3-a d^3\right )^{7/3}}\)

input

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

output

(3*c^4*d^2*(a + b*x^3)^(2/3))/(2*(b*c^3 - a*d^3)*(c^3 + d^3*x^3)^2) - (3*c 
^3*d^3*x*(a + b*x^3)^(2/3))/(2*(b*c^3 - a*d^3)*(c^3 + d^3*x^3)^2) + (4*b*c 
^4*d^2*(a + b*x^3)^(2/3))/(3*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) - (c*d^2*( 
b*c^3 - 3*a*d^3)*(a + b*x^3)^(2/3))/(3*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) 
+ (d^3*(3*b*c^3 - 7*a*d^3)*x*(a + b*x^3)^(2/3))/(18*(b*c^3 - a*d^3)^2*(c^3 
 + d^3*x^3)) - (d^3*(9*b*c^3 - 5*a*d^3)*x*(a + b*x^3)^(2/3))/(18*(b*c^3 - 
a*d^3)^2*(c^3 + d^3*x^3)) - (7*d^3*(3*b*c^3 + a*d^3)*x*(a + b*x^3)^(2/3))/ 
(18*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) - (3*d*x^2*(1 + (b*x^3)/a)^(1/3)*Ap 
pellF1[2/3, 1/3, 3, 5/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/(2*c^4*(a + b*x^ 
3)^(1/3)) + (6*d^4*x^5*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 3, 8/3, -( 
(b*x^3)/a), -((d^3*x^3)/c^3)])/(5*c^7*(a + b*x^3)^(1/3)) + (2*a^2*d^6*ArcT 
an[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(9*Sq 
rt[3]*c^2*(b*c^3 - a*d^3)^(7/3)) + (7*a*d^3*(3*b*c^3 - a*d^3)*ArcTan[(1 + 
(2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(9*Sqrt[3]*c^ 
2*(b*c^3 - a*d^3)^(7/3)) + ((9*b^2*c^6 - 12*a*b*c^3*d^3 + 5*a^2*d^6)*ArcTa 
n[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(9*Sqr 
t[3]*c^2*(b*c^3 - a*d^3)^(7/3)) - (4*b^2*c^4*ArcTan[(1 - (2*d*(a + b*x^3)^ 
(1/3))/(b*c^3 - a*d^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]*(b*c^3 - a*d^3)^(7/3)) 
+ (b*c*(b*c^3 - 3*a*d^3)*ArcTan[(1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^ 
3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]*(b*c^3 - a*d^3)^(7/3)) + (2*b^2*c^4*Log[...
 

3.1.35.3.1 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]
 

3.1.35.4 Maple [F]

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

input

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

output

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

3.1.35.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.1.35.6 Sympy [F]

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

input

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

output

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

3.1.35.7 Maxima [F]

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

input

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

output

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

3.1.35.8 Giac [F]

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

input

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

output

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

3.1.35.9 Mupad [F(-1)]

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

input

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

output

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