Integrand size = 17, antiderivative size = 82 \[ \int \frac {\left (a x+b x^2\right )^{3/4}}{x} \, dx=\frac {2}{3} \left (a x+b x^2\right )^{3/4}+\frac {a^2 \sqrt [4]{-\frac {b x}{a}-\frac {b^2 x^2}{a^2}} E\left (\left .\frac {1}{2} \arcsin \left (1+\frac {2 b x}{a}\right )\right |2\right )}{\sqrt {2} b \sqrt [4]{a x+b x^2}} \] Output:
2/3*(b*x^2+a*x)^(3/4)+1/2*a^2*(-b*x/a-b^2*x^2/a^2)^(1/4)*EllipticE(sin(1/2 *arcsin(1+2*b*x/a)),2^(1/2))*2^(1/2)/b/(b*x^2+a*x)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.54 \[ \int \frac {\left (a x+b x^2\right )^{3/4}}{x} \, dx=\frac {4 (x (a+b x))^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {3}{4},\frac {7}{4},-\frac {b x}{a}\right )}{3 \left (1+\frac {b x}{a}\right )^{3/4}} \] Input:
Integrate[(a*x + b*x^2)^(3/4)/x,x]
Output:
(4*(x*(a + b*x))^(3/4)*Hypergeometric2F1[-3/4, 3/4, 7/4, -((b*x)/a)])/(3*( 1 + (b*x)/a)^(3/4))
Time = 0.47 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.63, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {1137, 60, 73, 839, 813, 858, 807, 212}
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.
\(\displaystyle \int \frac {\left (a x+b x^2\right )^{3/4}}{x} \, dx\) |
\(\Big \downarrow \) 1137 |
\(\displaystyle \frac {\left (a x+b x^2\right )^{3/4} \int \frac {(a+b x)^{3/4}}{\sqrt [4]{x}}dx}{x^{3/4} (a+b x)^{3/4}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {\left (a x+b x^2\right )^{3/4} \left (\frac {1}{2} a \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a+b x}}dx+\frac {2}{3} x^{3/4} (a+b x)^{3/4}\right )}{x^{3/4} (a+b x)^{3/4}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\left (a x+b x^2\right )^{3/4} \left (2 a \int \frac {\sqrt {x}}{\sqrt [4]{a+b x}}d\sqrt [4]{x}+\frac {2}{3} x^{3/4} (a+b x)^{3/4}\right )}{x^{3/4} (a+b x)^{3/4}}\) |
\(\Big \downarrow \) 839 |
\(\displaystyle \frac {\left (a x+b x^2\right )^{3/4} \left (2 a \left (\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}-\frac {1}{2} a \int \frac {\sqrt {x}}{(a+b x)^{5/4}}d\sqrt [4]{x}\right )+\frac {2}{3} x^{3/4} (a+b x)^{3/4}\right )}{x^{3/4} (a+b x)^{3/4}}\) |
\(\Big \downarrow \) 813 |
\(\displaystyle \frac {\left (a x+b x^2\right )^{3/4} \left (2 a \left (\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}-\frac {a \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} \int \frac {1}{\left (\frac {a}{b x}+1\right )^{5/4} x^{3/4}}d\sqrt [4]{x}}{2 b \sqrt [4]{a+b x}}\right )+\frac {2}{3} x^{3/4} (a+b x)^{3/4}\right )}{x^{3/4} (a+b x)^{3/4}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {\left (a x+b x^2\right )^{3/4} \left (2 a \left (\frac {a \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} \int \frac {1}{\sqrt [4]{x} \left (\frac {a x}{b}+1\right )^{5/4}}d\frac {1}{\sqrt [4]{x}}}{2 b \sqrt [4]{a+b x}}+\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}\right )+\frac {2}{3} x^{3/4} (a+b x)^{3/4}\right )}{x^{3/4} (a+b x)^{3/4}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {\left (a x+b x^2\right )^{3/4} \left (2 a \left (\frac {a \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} \int \frac {1}{\left (\frac {\sqrt {x} a}{b}+1\right )^{5/4}}d\sqrt {x}}{4 b \sqrt [4]{a+b x}}+\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}\right )+\frac {2}{3} x^{3/4} (a+b x)^{3/4}\right )}{x^{3/4} (a+b x)^{3/4}}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {\left (a x+b x^2\right )^{3/4} \left (2 a \left (\frac {\sqrt {a} \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |2\right )}{2 \sqrt {b} \sqrt [4]{a+b x}}+\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}\right )+\frac {2}{3} x^{3/4} (a+b x)^{3/4}\right )}{x^{3/4} (a+b x)^{3/4}}\) |
Input:
Int[(a*x + b*x^2)^(3/4)/x,x]
Output:
((a*x + b*x^2)^(3/4)*((2*x^(3/4)*(a + b*x)^(3/4))/3 + 2*a*(x^(3/4)/(2*(a + b*x)^(1/4)) + (Sqrt[a]*(1 + a/(b*x))^(1/4)*x^(1/4)*EllipticE[ArcTan[(Sqrt [a]*Sqrt[x])/Sqrt[b]]/2, 2])/(2*Sqrt[b]*(a + b*x)^(1/4)))))/(x^(3/4)*(a + b*x)^(3/4))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) )*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(5/4), x_Symbol] :> Simp[x*((1 + a/(b*x^4) )^(1/4)/(b*(a + b*x^4)^(1/4))) Int[1/(x^3*(1 + a/(b*x^4))^(5/4)), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(1/4), x_Symbol] :> Simp[x^3/(2*(a + b*x^4 )^(1/4)), x] - Simp[a/2 Int[x^2/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a, b} , x] && PosQ[b/a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Int[((e_.)*(x_))^(m_)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[( e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) Int[x^(m + p)*(b + c*x)^ p, x], x] /; FreeQ[{b, c, e, m}, x]
\[\int \frac {\left (b \,x^{2}+a x \right )^{\frac {3}{4}}}{x}d x\]
Input:
int((b*x^2+a*x)^(3/4)/x,x)
Output:
int((b*x^2+a*x)^(3/4)/x,x)
\[ \int \frac {\left (a x+b x^2\right )^{3/4}}{x} \, dx=\int { \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{4}}}{x} \,d x } \] Input:
integrate((b*x^2+a*x)^(3/4)/x,x, algorithm="fricas")
Output:
integral((b*x^2 + a*x)^(3/4)/x, x)
\[ \int \frac {\left (a x+b x^2\right )^{3/4}}{x} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {3}{4}}}{x}\, dx \] Input:
integrate((b*x**2+a*x)**(3/4)/x,x)
Output:
Integral((x*(a + b*x))**(3/4)/x, x)
\[ \int \frac {\left (a x+b x^2\right )^{3/4}}{x} \, dx=\int { \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{4}}}{x} \,d x } \] Input:
integrate((b*x^2+a*x)^(3/4)/x,x, algorithm="maxima")
Output:
integrate((b*x^2 + a*x)^(3/4)/x, x)
\[ \int \frac {\left (a x+b x^2\right )^{3/4}}{x} \, dx=\int { \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{4}}}{x} \,d x } \] Input:
integrate((b*x^2+a*x)^(3/4)/x,x, algorithm="giac")
Output:
integrate((b*x^2 + a*x)^(3/4)/x, x)
Timed out. \[ \int \frac {\left (a x+b x^2\right )^{3/4}}{x} \, dx=\int \frac {{\left (b\,x^2+a\,x\right )}^{3/4}}{x} \,d x \] Input:
int((a*x + b*x^2)^(3/4)/x,x)
Output:
int((a*x + b*x^2)^(3/4)/x, x)
\[ \int \frac {\left (a x+b x^2\right )^{3/4}}{x} \, dx=\frac {2 x^{\frac {3}{4}} \left (b x +a \right )^{\frac {3}{4}}}{3}+\frac {\left (\int \frac {\left (b x +a \right )^{\frac {3}{4}}}{x^{\frac {1}{4}} a +x^{\frac {5}{4}} b}d x \right ) a}{2} \] Input:
int((b*x^2+a*x)^(3/4)/x,x)
Output:
(4*x**(3/4)*(a + b*x)**(3/4) + 3*int((a + b*x)**(3/4)/(x**(1/4)*a + x**(1/ 4)*b*x),x)*a)/6