Integrand size = 13, antiderivative size = 33 \[ \int \frac {1}{\left (a x+b x^2\right )^{2/3}} \, dx=\frac {3 \sqrt [3]{a x+b x^2} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {4}{3},-\frac {b x}{a}\right )}{a} \] Output:
3*(b*x^2+a*x)^(1/3)*hypergeom([2/3, 1],[4/3],-b*x/a)/a
Time = 9.91 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\left (a x+b x^2\right )^{2/3}} \, dx=\frac {3 x \left (1+\frac {b x}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x}{a}\right )}{(x (a+b x))^{2/3}} \] Input:
Integrate[(a*x + b*x^2)^(-2/3),x]
Output:
(3*x*(1 + (b*x)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x)/a)])/(x* (a + b*x))^(2/3)
Leaf count is larger than twice the leaf count of optimal. \(278\) vs. \(2(33)=66\).
Time = 0.49 (sec) , antiderivative size = 278, normalized size of antiderivative = 8.42, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1093, 1090, 234, 760}
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.
\(\displaystyle \int \frac {1}{\left (a x+b x^2\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 1093 |
\(\displaystyle \frac {\left (-\frac {b \left (a x+b x^2\right )}{a^2}\right )^{2/3} \int \frac {1}{\left (-\frac {b^2 x^2}{a^2}-\frac {b x}{a}\right )^{2/3}}dx}{\left (a x+b x^2\right )^{2/3}}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle -\frac {\sqrt [3]{2} a^2 \left (-\frac {b \left (a x+b x^2\right )}{a^2}\right )^{2/3} \int \frac {1}{\left (1-\frac {a^2 \left (-\frac {2 x b^2}{a^2}-\frac {b}{a}\right )^2}{b^2}\right )^{2/3}}d\left (-\frac {2 x b^2}{a^2}-\frac {b}{a}\right )}{b^2 \left (a x+b x^2\right )^{2/3}}\) |
\(\Big \downarrow \) 234 |
\(\displaystyle \frac {3 \sqrt {-\frac {a^2 \left (-\frac {2 b^2 x}{a^2}-\frac {b}{a}\right )^2}{b^2}} \left (-\frac {b \left (a x+b x^2\right )}{a^2}\right )^{2/3} \int \frac {1}{\sqrt {-\frac {a^2 \left (-\frac {2 x b^2}{a^2}-\frac {b}{a}\right )^2}{b^2}}}d\sqrt [3]{1-\frac {a^2 \left (-\frac {2 x b^2}{a^2}-\frac {b}{a}\right )^2}{b^2}}}{2^{2/3} \left (-\frac {2 b^2 x}{a^2}-\frac {b}{a}\right ) \left (a x+b x^2\right )^{2/3}}\) |
\(\Big \downarrow \) 760 |
\(\displaystyle -\frac {\sqrt [3]{2} 3^{3/4} \sqrt {2-\sqrt {3}} \left (\frac {2 b^2 x}{a^2}+\frac {b}{a}+1\right ) \left (-\frac {b \left (a x+b x^2\right )}{a^2}\right )^{2/3} \sqrt {\frac {\left (-\frac {2 b^2 x}{a^2}-\frac {b}{a}\right )^2+\sqrt [3]{1-\frac {a^2 \left (-\frac {2 b^2 x}{a^2}-\frac {b}{a}\right )^2}{b^2}}+1}{\left (\frac {2 b^2 x}{a^2}+\frac {b}{a}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {2 x b^2}{a^2}+\frac {b}{a}+\sqrt {3}+1}{\frac {2 x b^2}{a^2}+\frac {b}{a}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\left (-\frac {2 b^2 x}{a^2}-\frac {b}{a}\right ) \sqrt {-\frac {\frac {2 b^2 x}{a^2}+\frac {b}{a}+1}{\left (\frac {2 b^2 x}{a^2}+\frac {b}{a}-\sqrt {3}+1\right )^2}} \left (a x+b x^2\right )^{2/3}}\) |
Input:
Int[(a*x + b*x^2)^(-2/3),x]
Output:
-((2^(1/3)*3^(3/4)*Sqrt[2 - Sqrt[3]]*(1 + b/a + (2*b^2*x)/a^2)*(-((b*(a*x + b*x^2))/a^2))^(2/3)*Sqrt[(1 + (-(b/a) - (2*b^2*x)/a^2)^2 + (1 - (a^2*(-( b/a) - (2*b^2*x)/a^2)^2)/b^2)^(1/3))/(1 - Sqrt[3] + b/a + (2*b^2*x)/a^2)^2 ]*EllipticF[ArcSin[(1 + Sqrt[3] + b/a + (2*b^2*x)/a^2)/(1 - Sqrt[3] + b/a + (2*b^2*x)/a^2)], -7 + 4*Sqrt[3]])/((-(b/a) - (2*b^2*x)/a^2)*Sqrt[-((1 + b/a + (2*b^2*x)/a^2)/(1 - Sqrt[3] + b/a + (2*b^2*x)/a^2)^2)]*(a*x + b*x^2) ^(2/3)))
Int[((a_) + (b_.)*(x_)^2)^(-2/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) Subst[Int[1/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b }, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) *s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x ] && NegQ[a]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[((b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b*x + c*x^2)^p/((- c)*((b*x + c*x^2)/b^2))^p Int[((-c)*(x/b) - c^2*(x^2/b^2))^p, x], x] /; F reeQ[{b, c}, x] && (IntegerQ[4*p] || IntegerQ[3*p])
\[\int \frac {1}{\left (b \,x^{2}+a x \right )^{\frac {2}{3}}}d x\]
Input:
int(1/(b*x^2+a*x)^(2/3),x)
Output:
int(1/(b*x^2+a*x)^(2/3),x)
\[ \int \frac {1}{\left (a x+b x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b x^{2} + a x\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(1/(b*x^2+a*x)^(2/3),x, algorithm="fricas")
Output:
integral((b*x^2 + a*x)^(-2/3), x)
\[ \int \frac {1}{\left (a x+b x^2\right )^{2/3}} \, dx=\int \frac {1}{\left (a x + b x^{2}\right )^{\frac {2}{3}}}\, dx \] Input:
integrate(1/(b*x**2+a*x)**(2/3),x)
Output:
Integral((a*x + b*x**2)**(-2/3), x)
\[ \int \frac {1}{\left (a x+b x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b x^{2} + a x\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(1/(b*x^2+a*x)^(2/3),x, algorithm="maxima")
Output:
integrate((b*x^2 + a*x)^(-2/3), x)
\[ \int \frac {1}{\left (a x+b x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b x^{2} + a x\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(1/(b*x^2+a*x)^(2/3),x, algorithm="giac")
Output:
integrate((b*x^2 + a*x)^(-2/3), x)
Time = 9.57 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (a x+b x^2\right )^{2/3}} \, dx=\frac {3\,x\,{\left (\frac {b\,x}{a}+1\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {2}{3};\ \frac {4}{3};\ -\frac {b\,x}{a}\right )}{{\left (b\,x^2+a\,x\right )}^{2/3}} \] Input:
int(1/(a*x + b*x^2)^(2/3),x)
Output:
(3*x*((b*x)/a + 1)^(2/3)*hypergeom([1/3, 2/3], 4/3, -(b*x)/a))/(a*x + b*x^ 2)^(2/3)
\[ \int \frac {1}{\left (a x+b x^2\right )^{2/3}} \, dx=\int \frac {1}{x^{\frac {2}{3}} \left (b x +a \right )^{\frac {2}{3}}}d x \] Input:
int(1/(b*x^2+a*x)^(2/3),x)
Output:
int(1/(x**(2/3)*(a + b*x)**(2/3)),x)