Integrand size = 26, antiderivative size = 173 \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{3/2}} \, dx=\frac {(6 b c+a d) (e x)^{3/2}}{2 e^3 \sqrt [4]{a+b x^2}}+\frac {(6 b c+a d) (e x)^{3/2} \left (a+b x^2\right )^{3/4}}{3 a e^3}-\frac {2 c \left (a+b x^2\right )^{7/4}}{a e \sqrt {e x}}+\frac {\sqrt {a} (6 b c+a d) \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {b} e^2 \sqrt [4]{a+b x^2}} \] Output:
1/2*(a*d+6*b*c)*(e*x)^(3/2)/e^3/(b*x^2+a)^(1/4)+1/3*(a*d+6*b*c)*(e*x)^(3/2 )*(b*x^2+a)^(3/4)/a/e^3-2*c*(b*x^2+a)^(7/4)/a/e/(e*x)^(1/2)+1/2*a^(1/2)*(a *d+6*b*c)*(1+a/b/x^2)^(1/4)*(e*x)^(1/2)*EllipticE(sin(1/2*arccot(b^(1/2)*x /a^(1/2))),2^(1/2))/b^(1/2)/e^2/(b*x^2+a)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.57 \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{3/2}} \, dx=-\frac {2 c x \left (a+b x^2\right )^{7/4}}{a (e x)^{3/2}}-\frac {4 \left (-3 b c-\frac {a d}{2}\right ) x^3 \left (a+b x^2\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )}{3 a (e x)^{3/2} \left (1+\frac {b x^2}{a}\right )^{3/4}} \] Input:
Integrate[((a + b*x^2)^(3/4)*(c + d*x^2))/(e*x)^(3/2),x]
Output:
(-2*c*x*(a + b*x^2)^(7/4))/(a*(e*x)^(3/2)) - (4*(-3*b*c - (a*d)/2)*x^3*(a + b*x^2)^(3/4)*Hypergeometric2F1[-3/4, 3/4, 7/4, -((b*x^2)/a)])/(3*a*(e*x) ^(3/2)*(1 + (b*x^2)/a)^(3/4))
Time = 0.28 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {359, 248, 255, 249, 858, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {(a d+6 b c) \int \sqrt {e x} \left (b x^2+a\right )^{3/4}dx}{a e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {(a d+6 b c) \left (\frac {1}{2} a \int \frac {\sqrt {e x}}{\sqrt [4]{b x^2+a}}dx+\frac {(e x)^{3/2} \left (a+b x^2\right )^{3/4}}{3 e}\right )}{a e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 255 |
| \(\displaystyle \frac {(a d+6 b c) \left (\frac {1}{2} a \left (\frac {x \sqrt {e x}}{\sqrt [4]{a+b x^2}}-\frac {1}{2} a \int \frac {\sqrt {e x}}{\left (b x^2+a\right )^{5/4}}dx\right )+\frac {(e x)^{3/2} \left (a+b x^2\right )^{3/4}}{3 e}\right )}{a e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 249 |
| \(\displaystyle \frac {(a d+6 b c) \left (\frac {1}{2} a \left (\frac {x \sqrt {e x}}{\sqrt [4]{a+b x^2}}-\frac {a \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{5/4} x^2}dx}{2 b \sqrt [4]{a+b x^2}}\right )+\frac {(e x)^{3/2} \left (a+b x^2\right )^{3/4}}{3 e}\right )}{a e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 858 |
| \(\displaystyle \frac {(a d+6 b c) \left (\frac {1}{2} a \left (\frac {a \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{5/4}}d\frac {1}{x}}{2 b \sqrt [4]{a+b x^2}}+\frac {x \sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )+\frac {(e x)^{3/2} \left (a+b x^2\right )^{3/4}}{3 e}\right )}{a e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 212 |
| \(\displaystyle \frac {(a d+6 b c) \left (\frac {1}{2} a \left (\frac {\sqrt {a} \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x}\right )\right |2\right )}{\sqrt {b} \sqrt [4]{a+b x^2}}+\frac {x \sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )+\frac {(e x)^{3/2} \left (a+b x^2\right )^{3/4}}{3 e}\right )}{a e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{a e \sqrt {e x}}\) |
Input:
Int[((a + b*x^2)^(3/4)*(c + d*x^2))/(e*x)^(3/2),x]
Output:
(-2*c*(a + b*x^2)^(7/4))/(a*e*Sqrt[e*x]) + ((6*b*c + a*d)*(((e*x)^(3/2)*(a + b*x^2)^(3/4))/(3*e) + (a*((x*Sqrt[e*x])/(a + b*x^2)^(1/4) + (Sqrt[a]*(1 + a/(b*x^2))^(1/4)*Sqrt[e*x]*EllipticE[ArcTan[Sqrt[a]/(Sqrt[b]*x)]/2, 2]) /(Sqrt[b]*(a + b*x^2)^(1/4))))/2))/(a*e^2)
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[Sqrt[(c_.)*(x_)]/((a_) + (b_.)*(x_)^2)^(5/4), x_Symbol] :> Simp[Sqrt[c* x]*((1 + a/(b*x^2))^(1/4)/(b*(a + b*x^2)^(1/4))) Int[1/(x^2*(1 + a/(b*x^2 ))^(5/4)), x], x] /; FreeQ[{a, b, c}, x] && PosQ[b/a]
Int[Sqrt[(c_)*(x_)]/((a_) + (b_.)*(x_)^2)^(1/4), x_Symbol] :> Simp[x*(Sqrt[ c*x]/(a + b*x^2)^(1/4)), x] - Simp[a/2 Int[Sqrt[c*x]/(a + b*x^2)^(5/4), x ], x] /; FreeQ[{a, b, c}, x] && PosQ[b/a]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
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]
\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (x^{2} d +c \right )}{\left (e x \right )^{\frac {3}{2}}}d x\]
Input:
int((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(3/2),x)
Output:
int((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(3/2),x)
\[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(3/2),x, algorithm="fricas")
Output:
integral((b*x^2 + a)^(3/4)*(d*x^2 + c)*sqrt(e*x)/(e^2*x^2), x)
Result contains complex when optimal does not.
Time = 7.90 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.48 \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{3/2}} \, dx=\frac {a^{\frac {3}{4}} d x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {b^{\frac {3}{4}} c x {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{e^{\frac {3}{2}}} \] Input:
integrate((b*x**2+a)**(3/4)*(d*x**2+c)/(e*x)**(3/2),x)
Output:
a**(3/4)*d*x**(3/2)*gamma(3/4)*hyper((-3/4, 3/4), (7/4,), b*x**2*exp_polar (I*pi)/a)/(2*e**(3/2)*gamma(7/4)) + b**(3/4)*c*x*hyper((-3/4, -1/2), (1/2, ), a*exp_polar(I*pi)/(b*x**2))/e**(3/2)
\[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(3/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/4)*(d*x^2 + c)/(e*x)^(3/2), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(3/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (d\,x^2+c\right )}{{\left (e\,x\right )}^{3/2}} \,d x \] Input:
int(((a + b*x^2)^(3/4)*(c + d*x^2))/(e*x)^(3/2),x)
Output:
int(((a + b*x^2)^(3/4)*(c + d*x^2))/(e*x)^(3/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{3/2}} \, dx=\frac {\sqrt {e}\, \left (6 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a d +12 \left (b \,x^{2}+a \right )^{\frac {3}{4}} b c +4 \left (b \,x^{2}+a \right )^{\frac {3}{4}} b d \,x^{2}+3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{\frac {3}{4}}}{b \,x^{4}+a \,x^{2}}d x \right ) a^{2} d +18 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{\frac {3}{4}}}{b \,x^{4}+a \,x^{2}}d x \right ) a b c \right )}{12 \sqrt {x}\, b \,e^{2}} \] Input:
int((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(3/2),x)
Output:
(sqrt(e)*(6*(a + b*x**2)**(3/4)*a*d + 12*(a + b*x**2)**(3/4)*b*c + 4*(a + b*x**2)**(3/4)*b*d*x**2 + 3*sqrt(x)*int((sqrt(x)*(a + b*x**2)**(3/4))/(a*x **2 + b*x**4),x)*a**2*d + 18*sqrt(x)*int((sqrt(x)*(a + b*x**2)**(3/4))/(a* x**2 + b*x**4),x)*a*b*c))/(12*sqrt(x)*b*e**2)