Integrand size = 26, antiderivative size = 374 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{2 (d e-c f) (e+f x)^2}+\frac {(3 b d e+b c f-4 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{6 (b e-a f) (d e-c f)^2 (e+f x)}+\frac {(b c-a d) (3 b d e-b c f-2 a d f) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{3 \sqrt {3} (b e-a f)^{5/3} (d e-c f)^{7/3}}-\frac {(b c-a d) (3 b d e-b c f-2 a d f) \log (e+f x)}{18 (b e-a f)^{5/3} (d e-c f)^{7/3}}+\frac {(b c-a d) (3 b d e-b c f-2 a d f) \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{6 (b e-a f)^{5/3} (d e-c f)^{7/3}} \] Output:
1/2*(b*x+a)^(1/3)*(d*x+c)^(2/3)/(-c*f+d*e)/(f*x+e)^2+1/6*(-4*a*d*f+b*c*f+3 *b*d*e)*(b*x+a)^(1/3)*(d*x+c)^(2/3)/(-a*f+b*e)/(-c*f+d*e)^2/(f*x+e)+1/9*(- a*d+b*c)*(-2*a*d*f-b*c*f+3*b*d*e)*arctan(1/3*3^(1/2)+2/3*(-a*f+b*e)^(1/3)* (d*x+c)^(1/3)*3^(1/2)/(-c*f+d*e)^(1/3)/(b*x+a)^(1/3))*3^(1/2)/(-a*f+b*e)^( 5/3)/(-c*f+d*e)^(7/3)-1/18*(-a*d+b*c)*(-2*a*d*f-b*c*f+3*b*d*e)*ln(f*x+e)/( -a*f+b*e)^(5/3)/(-c*f+d*e)^(7/3)+1/6*(-a*d+b*c)*(-2*a*d*f-b*c*f+3*b*d*e)*l n(-(b*x+a)^(1/3)+(-a*f+b*e)^(1/3)*(d*x+c)^(1/3)/(-c*f+d*e)^(1/3))/(-a*f+b* e)^(5/3)/(-c*f+d*e)^(7/3)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.14 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\frac {\sqrt [3]{a+b x} \left (-3 f (a+b x) (c+d x)+\frac {2 (3 b d e-b c f-2 a d f) (e+f x) \left ((b e-a f) (c+d x)-(b c-a d) (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{(b e-a f) (d e-c f)}\right )}{6 (b e-a f) (d e-c f) \sqrt [3]{c+d x} (e+f x)^2} \] Input:
Integrate[(a + b*x)^(1/3)/((c + d*x)^(1/3)*(e + f*x)^3),x]
Output:
((a + b*x)^(1/3)*(-3*f*(a + b*x)*(c + d*x) + (2*(3*b*d*e - b*c*f - 2*a*d*f )*(e + f*x)*((b*e - a*f)*(c + d*x) - (b*c - a*d)*(e + f*x)*Hypergeometric2 F1[1/3, 1, 4/3, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]))/((b*e - a*f)*(d*e - c*f))))/(6*(b*e - a*f)*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)^ 2)
Time = 0.33 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {107, 105, 102}
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.
| \(\displaystyle \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {(-2 a d f-b c f+3 b d e) \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2}dx}{3 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{4/3} (c+d x)^{2/3}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {(-2 a d f-b c f+3 b d e) \left (\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (d e-c f)}-\frac {(b c-a d) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}dx}{3 (d e-c f)}\right )}{3 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{4/3} (c+d x)^{2/3}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle \frac {(-2 a d f-b c f+3 b d e) \left (\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (d e-c f)}-\frac {(b c-a d) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{(b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac {\log (e+f x)}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}\right )}{3 (d e-c f)}\right )}{3 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{4/3} (c+d x)^{2/3}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\) |
Input:
Int[(a + b*x)^(1/3)/((c + d*x)^(1/3)*(e + f*x)^3),x]
Output:
-1/2*(f*(a + b*x)^(4/3)*(c + d*x)^(2/3))/((b*e - a*f)*(d*e - c*f)*(e + f*x )^2) + ((3*b*d*e - b*c*f - 2*a*d*f)*(((a + b*x)^(1/3)*(c + d*x)^(2/3))/((d *e - c*f)*(e + f*x)) - ((b*c - a*d)*(-((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*(d*e - c*f)^(1/3)*(a + b*x)^(1/3)) ])/((b*e - a*f)^(2/3)*(d*e - c*f)^(1/3))) + Log[e + f*x]/(2*(b*e - a*f)^(2 /3)*(d*e - c*f)^(1/3)) - (3*Log[-(a + b*x)^(1/3) + ((b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(d*e - c*f)^(1/3)])/(2*(b*e - a*f)^(2/3)*(d*e - c*f)^(1/3)))) /(3*(d*e - c*f))))/(3*(b*e - a*f)*(d*e - c*f))
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.) *(x_))), x_] :> With[{q = Rt[(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])* q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))]/(d*e - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q *(a + b*x)^(1/3) - (c + d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, c, d, e, f}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
\[\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (x d +c \right )^{\frac {1}{3}} \left (f x +e \right )^{3}}d x\]
Input:
int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^3,x)
Output:
int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^3,x)
Leaf count of result is larger than twice the leaf count of optimal. 2311 vs. \(2 (324) = 648\).
Time = 0.73 (sec) , antiderivative size = 4778, normalized size of antiderivative = 12.78 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\int \frac {\sqrt [3]{a + b x}}{\sqrt [3]{c + d x} \left (e + f x\right )^{3}}\, dx \] Input:
integrate((b*x+a)**(1/3)/(d*x+c)**(1/3)/(f*x+e)**3,x)
Output:
Integral((a + b*x)**(1/3)/((c + d*x)**(1/3)*(e + f*x)**3), x)
\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}^{3}} \,d x } \] Input:
integrate((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^3,x, algorithm="maxima")
Output:
integrate((b*x + a)^(1/3)/((d*x + c)^(1/3)*(f*x + e)^3), x)
\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}^{3}} \,d x } \] Input:
integrate((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^3,x, algorithm="giac")
Output:
integrate((b*x + a)^(1/3)/((d*x + c)^(1/3)*(f*x + e)^3), x)
Timed out. \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/3}}{{\left (e+f\,x\right )}^3\,{\left (c+d\,x\right )}^{1/3}} \,d x \] Input:
int((a + b*x)^(1/3)/((e + f*x)^3*(c + d*x)^(1/3)),x)
Output:
int((a + b*x)^(1/3)/((e + f*x)^3*(c + d*x)^(1/3)), x)
\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {1}{3}} \left (f x +e \right )^{3}}d x \] Input:
int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^3,x)
Output:
int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^3,x)