Integrand size = 19, antiderivative size = 56 \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{2/3}} \, dx=\frac {3 \sqrt [3]{a+b x} (c+d x)^{4/3} \operatorname {Hypergeometric2F1}\left (1,\frac {5}{3},\frac {4}{3},-\frac {d (a+b x)}{b c-a d}\right )}{b c-a d} \] Output:
3*(b*x+a)^(1/3)*(d*x+c)^(4/3)*hypergeom([1, 5/3],[4/3],-d*(b*x+a)/(-a*d+b* c))/(-a*d+b*c)
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{2/3}} \, dx=\frac {3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{3},\frac {4}{3},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \] Input:
Integrate[(c + d*x)^(1/3)/(a + b*x)^(2/3),x]
Output:
(3*(a + b*x)^(1/3)*(c + d*x)^(1/3)*Hypergeometric2F1[-1/3, 1/3, 4/3, (d*(a + b*x))/(-(b*c) + a*d)])/(b*((b*(c + d*x))/(b*c - a*d))^(1/3))
Time = 0.16 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.29, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {80, 79}
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.
\(\displaystyle \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{2/3}} \, dx\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {\sqrt [3]{c+d x} \int \frac {\sqrt [3]{\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}{(a+b x)^{2/3}}dx}{\sqrt [3]{\frac {b (c+d x)}{b c-a d}}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {d (a+b x)}{b c-a d}\right )}{b \sqrt [3]{\frac {b (c+d x)}{b c-a d}}}\) |
Input:
Int[(c + d*x)^(1/3)/(a + b*x)^(2/3),x]
Output:
(3*(a + b*x)^(1/3)*(c + d*x)^(1/3)*Hypergeometric2F1[-1/3, 1/3, 4/3, -((d* (a + b*x))/(b*c - a*d))])/(b*((b*(c + d*x))/(b*c - a*d))^(1/3))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
\[\int \frac {\left (x d +c \right )^{\frac {1}{3}}}{\left (b x +a \right )^{\frac {2}{3}}}d x\]
Input:
int((d*x+c)^(1/3)/(b*x+a)^(2/3),x)
Output:
int((d*x+c)^(1/3)/(b*x+a)^(2/3),x)
\[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{2/3}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{3}}}{{\left (b x + a\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate((d*x+c)^(1/3)/(b*x+a)^(2/3),x, algorithm="fricas")
Output:
integral((d*x + c)^(1/3)/(b*x + a)^(2/3), x)
\[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{2/3}} \, dx=\int \frac {\sqrt [3]{c + d x}}{\left (a + b x\right )^{\frac {2}{3}}}\, dx \] Input:
integrate((d*x+c)**(1/3)/(b*x+a)**(2/3),x)
Output:
Integral((c + d*x)**(1/3)/(a + b*x)**(2/3), x)
\[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{2/3}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{3}}}{{\left (b x + a\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate((d*x+c)^(1/3)/(b*x+a)^(2/3),x, algorithm="maxima")
Output:
integrate((d*x + c)^(1/3)/(b*x + a)^(2/3), x)
\[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{2/3}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {1}{3}}}{{\left (b x + a\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate((d*x+c)^(1/3)/(b*x+a)^(2/3),x, algorithm="giac")
Output:
integrate((d*x + c)^(1/3)/(b*x + a)^(2/3), x)
Timed out. \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{2/3}} \, dx=\int \frac {{\left (c+d\,x\right )}^{1/3}}{{\left (a+b\,x\right )}^{2/3}} \,d x \] Input:
int((c + d*x)^(1/3)/(a + b*x)^(2/3),x)
Output:
int((c + d*x)^(1/3)/(a + b*x)^(2/3), x)
\[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{2/3}} \, dx=\int \frac {\left (d x +c \right )^{\frac {1}{3}}}{\left (b x +a \right )^{\frac {2}{3}}}d x \] Input:
int((d*x+c)^(1/3)/(b*x+a)^(2/3),x)
Output:
int((c + d*x)**(1/3)/(a + b*x)**(2/3),x)