Integrand size = 28, antiderivative size = 533 \[ \int \frac {x^3}{\sqrt [5]{c+d x^5} \left (a c+2 a d x^5\right )} \, dx=\frac {\sqrt {5-\sqrt {5}} \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\frac {2 \sqrt [10]{2} c^{2/5}}{\sqrt {5-\sqrt {5}} \sqrt [5]{d} x \sqrt [5]{c+d x^5}}\right )}{20 \sqrt [10]{2} a c^{2/5} d^{4/5}}-\frac {\sqrt {5+\sqrt {5}} \arctan \left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}+\frac {2 \sqrt [10]{2} c^{2/5}}{\sqrt {5+\sqrt {5}} \sqrt [5]{d} x \sqrt [5]{c+d x^5}}\right )}{20 \sqrt [10]{2} a c^{2/5} d^{4/5}}-\frac {\log \left (2^{2/5} \sqrt [5]{d}+\frac {c^{2/5}}{x \sqrt [5]{c+d x^5}}\right )}{10\ 2^{3/5} a c^{2/5} d^{4/5}}+\frac {\left (1+\sqrt {5}\right ) \log \left (\frac {2 c^{4/5}-2^{2/5} c^{2/5} \sqrt [5]{d} x \sqrt [5]{c+d x^5}-2^{2/5} \sqrt {5} c^{2/5} \sqrt [5]{d} x \sqrt [5]{c+d x^5}+2\ 2^{4/5} d^{2/5} x^2 \left (c+d x^5\right )^{2/5}}{x^2 \left (c+d x^5\right )^{2/5}}\right )}{40\ 2^{3/5} a c^{2/5} d^{4/5}}+\frac {\left (1-\sqrt {5}\right ) \log \left (\frac {2 c^{4/5}-2^{2/5} c^{2/5} \sqrt [5]{d} x \sqrt [5]{c+d x^5}+2^{2/5} \sqrt {5} c^{2/5} \sqrt [5]{d} x \sqrt [5]{c+d x^5}+2\ 2^{4/5} d^{2/5} x^2 \left (c+d x^5\right )^{2/5}}{x^2 \left (c+d x^5\right )^{2/5}}\right )}{40\ 2^{3/5} a c^{2/5} d^{4/5}} \] Output:
-1/40*(5-5^(1/2))^(1/2)*arctan(-1/5*(25+10*5^(1/2))^(1/2)+2*2^(1/10)*c^(2/ 5)/(5-5^(1/2))^(1/2)/d^(1/5)/x/(d*x^5+c)^(1/5))*2^(9/10)/a/c^(2/5)/d^(4/5) -1/40*(5+5^(1/2))^(1/2)*arctan(1/5*(25-10*5^(1/2))^(1/2)+2*2^(1/10)*c^(2/5 )/(5+5^(1/2))^(1/2)/d^(1/5)/x/(d*x^5+c)^(1/5))*2^(9/10)/a/c^(2/5)/d^(4/5)- 1/20*ln(2^(2/5)*d^(1/5)+c^(2/5)/x/(d*x^5+c)^(1/5))*2^(2/5)/a/c^(2/5)/d^(4/ 5)+1/80*(5^(1/2)+1)*ln((2*c^(4/5)-2^(2/5)*c^(2/5)*d^(1/5)*x*(d*x^5+c)^(1/5 )-2^(2/5)*5^(1/2)*c^(2/5)*d^(1/5)*x*(d*x^5+c)^(1/5)+2*2^(4/5)*d^(2/5)*x^2* (d*x^5+c)^(2/5))/x^2/(d*x^5+c)^(2/5))*2^(2/5)/a/c^(2/5)/d^(4/5)+1/80*(-5^( 1/2)+1)*ln((2*c^(4/5)-2^(2/5)*c^(2/5)*d^(1/5)*x*(d*x^5+c)^(1/5)+2^(2/5)*5^ (1/2)*c^(2/5)*d^(1/5)*x*(d*x^5+c)^(1/5)+2*2^(4/5)*d^(2/5)*x^2*(d*x^5+c)^(2 /5))/x^2/(d*x^5+c)^(2/5))*2^(2/5)/a/c^(2/5)/d^(4/5)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.13 \[ \int \frac {x^3}{\sqrt [5]{c+d x^5} \left (a c+2 a d x^5\right )} \, dx=\frac {x^4 \sqrt [5]{\frac {c+d x^5}{c}} \operatorname {AppellF1}\left (\frac {4}{5},\frac {1}{5},1,\frac {9}{5},-\frac {d x^5}{c},-\frac {2 d x^5}{c}\right )}{4 a c \sqrt [5]{c+d x^5}} \] Input:
Integrate[x^3/((c + d*x^5)^(1/5)*(a*c + 2*a*d*x^5)),x]
Output:
(x^4*((c + d*x^5)/c)^(1/5)*AppellF1[4/5, 1/5, 1, 9/5, -((d*x^5)/c), (-2*d* x^5)/c])/(4*a*c*(c + d*x^5)^(1/5))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.36 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1013, 27, 1012}
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.
\(\displaystyle \int \frac {x^3}{\sqrt [5]{c+d x^5} \left (a c+2 a d x^5\right )} \, dx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {\sqrt [5]{\frac {d x^5}{c}+1} \int \frac {x^3}{a \left (2 d x^5+c\right ) \sqrt [5]{\frac {d x^5}{c}+1}}dx}{\sqrt [5]{c+d x^5}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt [5]{\frac {d x^5}{c}+1} \int \frac {x^3}{\left (2 d x^5+c\right ) \sqrt [5]{\frac {d x^5}{c}+1}}dx}{a \sqrt [5]{c+d x^5}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {x^4 \sqrt [5]{\frac {d x^5}{c}+1} \operatorname {AppellF1}\left (\frac {4}{5},1,\frac {1}{5},\frac {9}{5},-\frac {2 d x^5}{c},-\frac {d x^5}{c}\right )}{4 a c \sqrt [5]{c+d x^5}}\) |
Input:
Int[x^3/((c + d*x^5)^(1/5)*(a*c + 2*a*d*x^5)),x]
Output:
(x^4*(1 + (d*x^5)/c)^(1/5)*AppellF1[4/5, 1, 1/5, 9/5, (-2*d*x^5)/c, -((d*x ^5)/c)])/(4*a*c*(c + d*x^5)^(1/5))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ n/a))^FracPart[p]) Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
\[\int \frac {x^{3}}{\left (d \,x^{5}+c \right )^{\frac {1}{5}} \left (2 x^{5} a d +a c \right )}d x\]
Input:
int(x^3/(d*x^5+c)^(1/5)/(2*a*d*x^5+a*c),x)
Output:
int(x^3/(d*x^5+c)^(1/5)/(2*a*d*x^5+a*c),x)
Exception generated. \[ \int \frac {x^3}{\sqrt [5]{c+d x^5} \left (a c+2 a d x^5\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3/(d*x^5+c)^(1/5)/(2*a*d*x^5+a*c),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (trace 0)
\[ \int \frac {x^3}{\sqrt [5]{c+d x^5} \left (a c+2 a d x^5\right )} \, dx=\frac {\int \frac {x^{3}}{c \sqrt [5]{c + d x^{5}} + 2 d x^{5} \sqrt [5]{c + d x^{5}}}\, dx}{a} \] Input:
integrate(x**3/(d*x**5+c)**(1/5)/(2*a*d*x**5+a*c),x)
Output:
Integral(x**3/(c*(c + d*x**5)**(1/5) + 2*d*x**5*(c + d*x**5)**(1/5)), x)/a
\[ \int \frac {x^3}{\sqrt [5]{c+d x^5} \left (a c+2 a d x^5\right )} \, dx=\int { \frac {x^{3}}{{\left (2 \, a d x^{5} + a c\right )} {\left (d x^{5} + c\right )}^{\frac {1}{5}}} \,d x } \] Input:
integrate(x^3/(d*x^5+c)^(1/5)/(2*a*d*x^5+a*c),x, algorithm="maxima")
Output:
integrate(x^3/((2*a*d*x^5 + a*c)*(d*x^5 + c)^(1/5)), x)
\[ \int \frac {x^3}{\sqrt [5]{c+d x^5} \left (a c+2 a d x^5\right )} \, dx=\int { \frac {x^{3}}{{\left (2 \, a d x^{5} + a c\right )} {\left (d x^{5} + c\right )}^{\frac {1}{5}}} \,d x } \] Input:
integrate(x^3/(d*x^5+c)^(1/5)/(2*a*d*x^5+a*c),x, algorithm="giac")
Output:
integrate(x^3/((2*a*d*x^5 + a*c)*(d*x^5 + c)^(1/5)), x)
Timed out. \[ \int \frac {x^3}{\sqrt [5]{c+d x^5} \left (a c+2 a d x^5\right )} \, dx=\int \frac {x^3}{{\left (d\,x^5+c\right )}^{1/5}\,\left (2\,a\,d\,x^5+a\,c\right )} \,d x \] Input:
int(x^3/((c + d*x^5)^(1/5)*(a*c + 2*a*d*x^5)),x)
Output:
int(x^3/((c + d*x^5)^(1/5)*(a*c + 2*a*d*x^5)), x)
\[ \int \frac {x^3}{\sqrt [5]{c+d x^5} \left (a c+2 a d x^5\right )} \, dx=\frac {\int \frac {x^{3}}{\left (d \,x^{5}+c \right )^{\frac {1}{5}} c +2 \left (d \,x^{5}+c \right )^{\frac {1}{5}} d \,x^{5}}d x}{a} \] Input:
int(x^3/(d*x^5+c)^(1/5)/(2*a*d*x^5+a*c),x)
Output:
int(x**3/((c + d*x**5)**(1/5)*c + 2*(c + d*x**5)**(1/5)*d*x**5),x)/a