Integrand size = 15, antiderivative size = 87 \[ \int \frac {x}{\left (a x+b x^2\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{a x+b x^2}}{b}+\frac {2 \sqrt {a} \left (\frac {b x}{a+b x}\right )^{3/4} (a+b x)^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x}}\right ),2\right )}{b^2 \left (a x+b x^2\right )^{3/4}} \] Output:
2*(b*x^2+a*x)^(1/4)/b+2*a^(1/2)*(b*x/(b*x+a))^(3/4)*(b*x+a)^(3/2)*InverseJ acobiAM(1/2*arcsin(a^(1/2)/(b*x+a)^(1/2)),2^(1/2))/b^2/(b*x^2+a*x)^(3/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.54 \[ \int \frac {x}{\left (a x+b x^2\right )^{3/4}} \, dx=\frac {4 x^2 \left (1+\frac {b x}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{4},\frac {9}{4},-\frac {b x}{a}\right )}{5 (x (a+b x))^{3/4}} \] Input:
Integrate[x/(a*x + b*x^2)^(3/4),x]
Output:
(4*x^2*(1 + (b*x)/a)^(3/4)*Hypergeometric2F1[3/4, 5/4, 9/4, -((b*x)/a)])/( 5*(x*(a + b*x))^(3/4))
Time = 0.38 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1160, 1093, 1090, 230}
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.
\(\displaystyle \int \frac {x}{\left (a x+b x^2\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {2 \sqrt [4]{a x+b x^2}}{b}-\frac {a \int \frac {1}{\left (b x^2+a x\right )^{3/4}}dx}{2 b}\) |
\(\Big \downarrow \) 1093 |
\(\displaystyle \frac {2 \sqrt [4]{a x+b x^2}}{b}-\frac {a \left (-\frac {b \left (a x+b x^2\right )}{a^2}\right )^{3/4} \int \frac {1}{\left (-\frac {b^2 x^2}{a^2}-\frac {b x}{a}\right )^{3/4}}dx}{2 b \left (a x+b x^2\right )^{3/4}}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {a^3 \left (-\frac {b \left (a x+b x^2\right )}{a^2}\right )^{3/4} \int \frac {1}{\left (1-\frac {a^2 \left (-\frac {2 x b^2}{a^2}-\frac {b}{a}\right )^2}{b^2}\right )^{3/4}}d\left (-\frac {2 x b^2}{a^2}-\frac {b}{a}\right )}{\sqrt {2} b^3 \left (a x+b x^2\right )^{3/4}}+\frac {2 \sqrt [4]{a x+b x^2}}{b}\) |
\(\Big \downarrow \) 230 |
\(\displaystyle \frac {\sqrt {2} a^2 \left (-\frac {b \left (a x+b x^2\right )}{a^2}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {a \left (-\frac {2 x b^2}{a^2}-\frac {b}{a}\right )}{b}\right ),2\right )}{b^2 \left (a x+b x^2\right )^{3/4}}+\frac {2 \sqrt [4]{a x+b x^2}}{b}\) |
Input:
Int[x/(a*x + b*x^2)^(3/4),x]
Output:
(2*(a*x + b*x^2)^(1/4))/b + (Sqrt[2]*a^2*(-((b*(a*x + b*x^2))/a^2))^(3/4)* EllipticF[ArcSin[(a*(-(b/a) - (2*b^2*x)/a^2))/b]/2, 2])/(b^2*(a*x + b*x^2) ^(3/4))
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2] ))*EllipticF[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ [a, 0] && NegQ[b/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[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
\[\int \frac {x}{\left (b \,x^{2}+a x \right )^{\frac {3}{4}}}d x\]
Input:
int(x/(b*x^2+a*x)^(3/4),x)
Output:
int(x/(b*x^2+a*x)^(3/4),x)
\[ \int \frac {x}{\left (a x+b x^2\right )^{3/4}} \, dx=\int { \frac {x}{{\left (b x^{2} + a x\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x/(b*x^2+a*x)^(3/4),x, algorithm="fricas")
Output:
integral((b*x^2 + a*x)^(1/4)/(b*x + a), x)
\[ \int \frac {x}{\left (a x+b x^2\right )^{3/4}} \, dx=\int \frac {x}{\left (x \left (a + b x\right )\right )^{\frac {3}{4}}}\, dx \] Input:
integrate(x/(b*x**2+a*x)**(3/4),x)
Output:
Integral(x/(x*(a + b*x))**(3/4), x)
\[ \int \frac {x}{\left (a x+b x^2\right )^{3/4}} \, dx=\int { \frac {x}{{\left (b x^{2} + a x\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x/(b*x^2+a*x)^(3/4),x, algorithm="maxima")
Output:
integrate(x/(b*x^2 + a*x)^(3/4), x)
\[ \int \frac {x}{\left (a x+b x^2\right )^{3/4}} \, dx=\int { \frac {x}{{\left (b x^{2} + a x\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x/(b*x^2+a*x)^(3/4),x, algorithm="giac")
Output:
integrate(x/(b*x^2 + a*x)^(3/4), x)
Timed out. \[ \int \frac {x}{\left (a x+b x^2\right )^{3/4}} \, dx=\int \frac {x}{{\left (b\,x^2+a\,x\right )}^{3/4}} \,d x \] Input:
int(x/(a*x + b*x^2)^(3/4),x)
Output:
int(x/(a*x + b*x^2)^(3/4), x)
\[ \int \frac {x}{\left (a x+b x^2\right )^{3/4}} \, dx=\int \frac {x^{\frac {1}{4}}}{\left (b x +a \right )^{\frac {3}{4}}}d x \] Input:
int(x/(b*x^2+a*x)^(3/4),x)
Output:
int(x/(x**(3/4)*(a + b*x)**(3/4)),x)