\(\int \frac {\sqrt {e x} (c+d x^2)}{(a+b x^2)^{7/4}} \, dx\) [462]

Optimal result

Integrand size = 26, antiderivative size = 125 \[ \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\frac {2 (b c-a d) (e x)^{3/2}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {d \sqrt {e} \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac {d \sqrt {e} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}} \] Output:

2/3*(-a*d+b*c)*(e*x)^(3/2)/a/b/e/(b*x^2+a)^(3/4)-d*e^(1/2)*arctan(b^(1/4)* 
(e*x)^(1/2)/e^(1/2)/(b*x^2+a)^(1/4))/b^(7/4)+d*e^(1/2)*arctanh(b^(1/4)*(e* 
x)^(1/2)/e^(1/2)/(b*x^2+a)^(1/4))/b^(7/4)
 

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\frac {\sqrt {e x} \left (\frac {2 b^{3/4} (b c-a d) x^{3/2}}{a \left (a+b x^2\right )^{3/4}}-3 d \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )+3 d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )\right )}{3 b^{7/4} \sqrt {x}} \] Input:

Integrate[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(7/4),x]
 

Output:

(Sqrt[e*x]*((2*b^(3/4)*(b*c - a*d)*x^(3/2))/(a*(a + b*x^2)^(3/4)) - 3*d*Ar 
cTan[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)] + 3*d*ArcTanh[(b^(1/4)*Sqrt[x])/ 
(a + b*x^2)^(1/4)]))/(3*b^(7/4)*Sqrt[x])
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {357, 266, 854, 27, 827, 218, 221}

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.

\(\displaystyle \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx\)

\(\Big \downarrow \) 357

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 854

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

Int[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(7/4),x]
 

Output:

(2*(b*c - a*d)*(e*x)^(3/2))/(3*a*b*e*(a + b*x^2)^(3/4)) + (2*d*e*(-1/2*Arc 
Tan[(b^(1/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))]/(b^(3/4)*Sqrt[e]) + A 
rcTanh[(b^(1/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))]/(2*b^(3/4)*Sqrt[e] 
)))/b
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

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

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

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 357

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

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n 
)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 
2^(-1)] && IntegersQ[m, p + (m + 1)/n]
 

Maple [F]

\[\int \frac {\sqrt {e x}\, \left (x^{2} d +c \right )}{\left (b \,x^{2}+a \right )^{\frac {7}{4}}}d x\]

Input:

int((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x)
 

Output:

int((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x)
 

Fricas [F(-1)]

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

integrate((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 11.92 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx=\frac {c \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )}{2 a^{\frac {7}{4}} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} + \frac {d \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {7}{4}} \Gamma \left (\frac {11}{4}\right )} \] Input:

integrate((e*x)**(1/2)*(d*x**2+c)/(b*x**2+a)**(7/4),x)
 

Output:

c*sqrt(e)*x**(3/2)*gamma(3/4)/(2*a**(7/4)*(1 + b*x**2/a)**(3/4)*gamma(7/4) 
) + d*sqrt(e)*x**(7/2)*gamma(7/4)*hyper((7/4, 7/4), (11/4,), b*x**2*exp_po 
lar(I*pi)/a)/(2*a**(7/4)*gamma(11/4))
 

Maxima [F]

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

integrate((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(7/4), x)
 

Giac [F]

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

integrate((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x, algorithm="giac")
 

Output:

integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(7/4), x)
 

Mupad [F(-1)]

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

int(((e*x)^(1/2)*(c + d*x^2))/(a + b*x^2)^(7/4),x)
 

Output:

int(((e*x)^(1/2)*(c + d*x^2))/(a + b*x^2)^(7/4), x)
 

Reduce [F]

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

int((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x)
 

Output:

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