\(\int \frac {c+d x^2}{\sqrt {e x} (a+b x^2)^{5/4}} \, dx\) [451]

Optimal result

Integrand size = 26, antiderivative size = 122 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{5/4}} \, dx=\frac {2 (b c-a d) \sqrt {e x}}{a b e \sqrt [4]{a+b x^2}}+\frac {d \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{5/4} \sqrt {e}}+\frac {d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{5/4} \sqrt {e}} \] Output:

2*(-a*d+b*c)*(e*x)^(1/2)/a/b/e/(b*x^2+a)^(1/4)+d*arctan(b^(1/4)*(e*x)^(1/2 
)/e^(1/2)/(b*x^2+a)^(1/4))/b^(5/4)/e^(1/2)+d*arctanh(b^(1/4)*(e*x)^(1/2)/e 
^(1/2)/(b*x^2+a)^(1/4))/b^(5/4)/e^(1/2)
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.84 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{5/4}} \, dx=\frac {\sqrt {x} \left (\frac {2 \sqrt [4]{b} (b c-a d) \sqrt {x}}{a \sqrt [4]{a+b x^2}}+d \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )+d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )\right )}{b^{5/4} \sqrt {e x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {357, 266, 770, 756, 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.

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

\(\Big \downarrow \) 357

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 770

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(2*(b*c - a*d)*Sqrt[e*x])/(a*b*e*(a + b*x^2)^(1/4)) + (2*d*((Sqrt[e]*ArcTa 
n[(b^(1/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(2*b^(1/4)) + (Sqrt[e] 
*ArcTanh[(b^(1/4)*Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(2*b^(1/4))))/( 
b*e)
 

Defintions of rubi rules used

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 756

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

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p + 1 
/n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 389, normalized size of antiderivative = 3.19 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{5/4}} \, dx=\frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (b c - a d\right )} \sqrt {e x} + {\left (a b^{2} e x^{2} + a^{2} b e\right )} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} d + {\left (b^{2} e x^{2} + a b e\right )} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) - {\left (a b^{2} e x^{2} + a^{2} b e\right )} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} d - {\left (b^{2} e x^{2} + a b e\right )} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) + {\left (-i \, a b^{2} e x^{2} - i \, a^{2} b e\right )} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} d - {\left (i \, b^{2} e x^{2} + i \, a b e\right )} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) + {\left (i \, a b^{2} e x^{2} + i \, a^{2} b e\right )} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} d - {\left (-i \, b^{2} e x^{2} - i \, a b e\right )} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right )}{2 \, {\left (a b^{2} e x^{2} + a^{2} b e\right )}} \] Input:

integrate((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(5/4),x, algorithm="fricas")
 

Output:

1/2*(4*(b*x^2 + a)^(3/4)*(b*c - a*d)*sqrt(e*x) + (a*b^2*e*x^2 + a^2*b*e)*( 
d^4/(b^5*e^2))^(1/4)*log(((b*x^2 + a)^(3/4)*sqrt(e*x)*d + (b^2*e*x^2 + a*b 
*e)*(d^4/(b^5*e^2))^(1/4))/(b*x^2 + a)) - (a*b^2*e*x^2 + a^2*b*e)*(d^4/(b^ 
5*e^2))^(1/4)*log(((b*x^2 + a)^(3/4)*sqrt(e*x)*d - (b^2*e*x^2 + a*b*e)*(d^ 
4/(b^5*e^2))^(1/4))/(b*x^2 + a)) + (-I*a*b^2*e*x^2 - I*a^2*b*e)*(d^4/(b^5* 
e^2))^(1/4)*log(((b*x^2 + a)^(3/4)*sqrt(e*x)*d - (I*b^2*e*x^2 + I*a*b*e)*( 
d^4/(b^5*e^2))^(1/4))/(b*x^2 + a)) + (I*a*b^2*e*x^2 + I*a^2*b*e)*(d^4/(b^5 
*e^2))^(1/4)*log(((b*x^2 + a)^(3/4)*sqrt(e*x)*d - (-I*b^2*e*x^2 - I*a*b*e) 
*(d^4/(b^5*e^2))^(1/4))/(b*x^2 + a)))/(a*b^2*e*x^2 + a^2*b*e)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.97 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.68 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{5/4}} \, dx=\frac {c \Gamma \left (\frac {1}{4}\right )}{2 a \sqrt [4]{b} \sqrt {e} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {5}{4}\right )} + \frac {d x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{4}} \sqrt {e} \Gamma \left (\frac {9}{4}\right )} \] Input:

integrate((d*x**2+c)/(e*x)**(1/2)/(b*x**2+a)**(5/4),x)
 

Output:

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

Maxima [F]

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

integrate((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(5/4),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(5/4),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.56 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{5/4}} \, dx=\frac {\sqrt {x}\, \sqrt {e}\, \left (-\left (b \,x^{2}+a \right ) a d +2 \left (b \,x^{2}+a \right ) b c +a^{2} d +a b d \,x^{2}\right )}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} \sqrt {b \,x^{2}+a}\, a b e} \] Input:

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

Output:

(sqrt(x)*sqrt(e)*(a + b*x**2)**(1/4)*( - (a + b*x**2)*a*d + 2*(a + b*x**2) 
*b*c + a**2*d + a*b*d*x**2))/(sqrt(a + b*x**2)*a*b*e*(a + b*x**2))