\(\int \frac {c+d x^2}{(e x)^{5/2} (a+b x^2)^{3/4}} \, dx\) [447]

Optimal result

Integrand size = 26, antiderivative size = 107 \[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}+\frac {2 \sqrt {b} (2 b c-3 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 a^{3/2} e^4 \left (a+b x^2\right )^{3/4}} \] Output:

-2/3*c*(b*x^2+a)^(1/4)/a/e/(e*x)^(3/2)+2/3*b^(1/2)*(-3*a*d+2*b*c)*(1+a/b/x 
^2)^(3/4)*(e*x)^(3/2)*InverseJacobiAM(1/2*arccot(b^(1/2)*x/a^(1/2)),2^(1/2 
))/a^(3/2)/e^4/(b*x^2+a)^(3/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.79 \[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=\frac {x \left (-2 c \left (a+b x^2\right )+2 (-2 b c+3 a d) x^2 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{3 a (e x)^{5/2} \left (a+b x^2\right )^{3/4}} \] Input:

Integrate[(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/4)),x]
 

Output:

(x*(-2*c*(a + b*x^2) + 2*(-2*b*c + 3*a*d)*x^2*(1 + (b*x^2)/a)^(3/4)*Hyperg 
eometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(3*a*(e*x)^(5/2)*(a + b*x^2)^(3 
/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {359, 266, 768, 858, 807, 229}

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.

\(\displaystyle \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx\)

\(\Big \downarrow \) 359

\(\displaystyle -\frac {(2 b c-3 a d) \int \frac {1}{\sqrt {e x} \left (b x^2+a\right )^{3/4}}dx}{3 a e^2}-\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 266

\(\displaystyle -\frac {2 (2 b c-3 a d) \int \frac {1}{\left (b x^2+a\right )^{3/4}}d\sqrt {e x}}{3 a e^3}-\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 768

\(\displaystyle -\frac {2 (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (2 b c-3 a d) \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{3/4} (e x)^{3/2}}d\sqrt {e x}}{3 a e^3 \left (a+b x^2\right )^{3/4}}-\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 858

\(\displaystyle \frac {2 (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (2 b c-3 a d) \int \frac {1}{\sqrt {e x} \left (\frac {a x^2 e^4}{b}+1\right )^{3/4}}d\frac {1}{\sqrt {e x}}}{3 a e^3 \left (a+b x^2\right )^{3/4}}-\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {(e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (2 b c-3 a d) \int \frac {1}{\left (\frac {a x e^3}{b}+1\right )^{3/4}}d(e x)}{3 a e^3 \left (a+b x^2\right )^{3/4}}-\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 229

\(\displaystyle \frac {2 \sqrt {b} (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (2 b c-3 a d) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a} e^2 x}{\sqrt {b}}\right ),2\right )}{3 a^{3/2} e^4 \left (a+b x^2\right )^{3/4}}-\frac {2 c \sqrt [4]{a+b x^2}}{3 a e (e x)^{3/2}}\)

Input:

Int[(c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/4)),x]
 

Output:

(-2*c*(a + b*x^2)^(1/4))/(3*a*e*(e*x)^(3/2)) + (2*Sqrt[b]*(2*b*c - 3*a*d)* 
(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*EllipticF[ArcTan[(Sqrt[a]*e^2*x)/Sqrt[b] 
]/2, 2])/(3*a^(3/2)*e^4*(a + b*x^2)^(3/4))
 

Defintions of rubi rules used

rule 229

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) 
)*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 359

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]
 

rule 768

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Simp[x^3*((1 + a/(b*x^4))^(3 
/4)/(a + b*x^4)^(3/4))   Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ 
[{a, b}, x]
 

rule 807

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]
 

rule 858

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]
 

Maple [F]

\[\int \frac {x^{2} d +c}{\left (e x \right )^{\frac {5}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}d x\]

Input:

int((d*x^2+c)/(e*x)^(5/2)/(b*x^2+a)^(3/4),x)
 

Output:

int((d*x^2+c)/(e*x)^(5/2)/(b*x^2+a)^(3/4),x)
 

Fricas [F]

\[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((d*x^2+c)/(e*x)^(5/2)/(b*x^2+a)^(3/4),x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^(1/4)*(d*x^2 + c)*sqrt(e*x)/(b*e^3*x^5 + a*e^3*x^3), 
x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.61 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.77 \[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=- \frac {d {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{b^{\frac {3}{4}} e^{\frac {5}{2}} x} + \frac {c \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \] Input:

integrate((d*x**2+c)/(e*x)**(5/2)/(b*x**2+a)**(3/4),x)
 

Output:

-d*hyper((1/2, 3/4), (3/2,), a*exp_polar(I*pi)/(b*x**2))/(b**(3/4)*e**(5/2 
)*x) + c*gamma(-3/4)*hyper((-3/4, 3/4), (1/4,), b*x**2*exp_polar(I*pi)/a)/ 
(2*a**(3/4)*e**(5/2)*x**(3/2)*gamma(1/4))
 

Maxima [F]

\[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((d*x^2+c)/(e*x)^(5/2)/(b*x^2+a)^(3/4),x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(5/2)), x)
 

Giac [F]

\[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((d*x^2+c)/(e*x)^(5/2)/(b*x^2+a)^(3/4),x, algorithm="giac")
 

Output:

integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=\int \frac {d\,x^2+c}{{\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^{3/4}} \,d x \] Input:

int((c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/4)),x)
 

Output:

int((c + d*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/4)), x)
 

Reduce [F]

\[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx=\frac {\sqrt {e}\, \left (\left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{\frac {5}{4}}}{b^{2} x^{7}+2 a b \,x^{5}+a^{2} x^{3}}d x \right ) c +\left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{\frac {5}{4}}}{b^{2} x^{5}+2 a b \,x^{3}+a^{2} x}d x \right ) d \right )}{e^{3}} \] Input:

int((d*x^2+c)/(e*x)^(5/2)/(b*x^2+a)^(3/4),x)
 

Output:

(sqrt(e)*(int((sqrt(x)*(a + b*x**2)**(5/4))/(a**2*x**3 + 2*a*b*x**5 + b**2 
*x**7),x)*c + int((sqrt(x)*(a + b*x**2)**(5/4))/(a**2*x + 2*a*b*x**3 + b** 
2*x**5),x)*d))/e**3