\(\int \frac {c+d x^2}{\sqrt {e x} (a+b x^2)^{3/4}} \, dx\) [446]

Optimal result

Integrand size = 26, antiderivative size = 102 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{3/4}} \, dx=\frac {d \sqrt {e x} \sqrt [4]{a+b x^2}}{b e}-\frac {(2 b c-a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{\sqrt {a} \sqrt {b} e^2 \left (a+b x^2\right )^{3/4}} \] Output:

d*(e*x)^(1/2)*(b*x^2+a)^(1/4)/b/e-(-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^(1/2)/b^(1/2)/ 
e^2/(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 = 10.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.75 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{3/4}} \, dx=\frac {d x \left (a+b x^2\right )+(2 b c-a d) x \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},-\frac {b x^2}{a}\right )}{b \sqrt {e x} \left (a+b x^2\right )^{3/4}} \] Input:

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

Output:

(d*x*(a + b*x^2) + (2*b*c - a*d)*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1 
[1/4, 3/4, 5/4, -((b*x^2)/a)])/(b*Sqrt[e*x]*(a + b*x^2)^(3/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.29 (sec) , antiderivative size = 105, 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 = {363, 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.

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

\(\Big \downarrow \) 363

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 768

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

\(\Big \downarrow \) 858

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

\(\Big \downarrow \) 807

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

\(\Big \downarrow \) 229

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

Input:

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

Output:

(d*Sqrt[e*x]*(a + b*x^2)^(1/4))/(b*e) - ((2*b*c - a*d)*(1 + a/(b*x^2))^(3/ 
4)*(e*x)^(3/2)*EllipticF[ArcTan[(Sqrt[a]*e^2*x)/Sqrt[b]]/2, 2])/(Sqrt[a]*S 
qrt[b]*e^2*(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 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((d*x^2+c)/(e*x)^(1/2)/(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*x^3 + a*e*x), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.24 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.76 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{3/4}} \, dx=- \frac {c {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{b^{\frac {3}{4}} \sqrt {e} x} + \frac {d x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} \sqrt {e} \Gamma \left (\frac {9}{4}\right )} \] Input:

integrate((d*x**2+c)/(e*x)**(1/2)/(b*x**2+a)**(3/4),x)
 

Output:

-c*hyper((1/2, 3/4), (3/2,), a*exp_polar(I*pi)/(b*x**2))/(b**(3/4)*sqrt(e) 
*x) + d*x**(5/2)*gamma(5/4)*hyper((3/4, 5/4), (9/4,), b*x**2*exp_polar(I*p 
i)/a)/(2*a**(3/4)*sqrt(e)*gamma(9/4))
 

Maxima [F]

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

integrate((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(3/4),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(3/4),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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