\(\int \frac {(a+b x^2)^{3/4} (c+d x^2)}{(e x)^{9/2}} \, dx\) [428]

Optimal result

Integrand size = 26, antiderivative size = 140 \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{9/2}} \, dx=-\frac {2 d \left (a+b x^2\right )^{3/4}}{3 e^3 (e x)^{3/2}}-\frac {2 c \left (a+b x^2\right )^{7/4}}{7 a e (e x)^{7/2}}+\frac {b^{3/4} d \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{e^{9/2}}+\frac {b^{3/4} d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{e^{9/2}} \] Output:

-2/3*d*(b*x^2+a)^(3/4)/e^3/(e*x)^(3/2)-2/7*c*(b*x^2+a)^(7/4)/a/e/(e*x)^(7/ 
2)+b^(3/4)*d*arctan(b^(1/4)*(e*x)^(1/2)/e^(1/2)/(b*x^2+a)^(1/4))/e^(9/2)+b 
^(3/4)*d*arctanh(b^(1/4)*(e*x)^(1/2)/e^(1/2)/(b*x^2+a)^(1/4))/e^(9/2)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{9/2}} \, dx=\frac {\sqrt {e x} \left (-2 \left (a+b x^2\right )^{3/4} \left (3 a c+3 b c x^2+7 a d x^2\right )+21 a b^{3/4} d x^{7/2} \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )+21 a b^{3/4} d x^{7/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )\right )}{21 a e^5 x^4} \] Input:

Integrate[((a + b*x^2)^(3/4)*(c + d*x^2))/(e*x)^(9/2),x]
 

Output:

(Sqrt[e*x]*(-2*(a + b*x^2)^(3/4)*(3*a*c + 3*b*c*x^2 + 7*a*d*x^2) + 21*a*b^ 
(3/4)*d*x^(7/2)*ArcTan[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)] + 21*a*b^(3/4) 
*d*x^(7/2)*ArcTanh[(b^(1/4)*Sqrt[x])/(a + b*x^2)^(1/4)]))/(21*a*e^5*x^4)
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {358, 247, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{9/2}} \, dx\)

\(\Big \downarrow \) 358

\(\displaystyle \frac {d \int \frac {\left (b x^2+a\right )^{3/4}}{(e x)^{5/2}}dx}{e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{7 a e (e x)^{7/2}}\)

\(\Big \downarrow \) 247

\(\displaystyle \frac {d \left (\frac {b \int \frac {1}{\sqrt {e x} \sqrt [4]{b x^2+a}}dx}{e^2}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 e (e x)^{3/2}}\right )}{e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{7 a e (e x)^{7/2}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {d \left (\frac {2 b \int \frac {1}{\sqrt [4]{b x^2+a}}d\sqrt {e x}}{e^3}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 e (e x)^{3/2}}\right )}{e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{7 a e (e x)^{7/2}}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {d \left (\frac {2 b \int \frac {1}{1-b x^2}d\frac {\sqrt {e x}}{\sqrt [4]{b x^2+a}}}{e^3}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 e (e x)^{3/2}}\right )}{e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{7 a e (e x)^{7/2}}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {d \left (\frac {2 b \left (\frac {1}{2} e \int \frac {1}{e-\sqrt {b} e x}d\frac {\sqrt {e x}}{\sqrt [4]{b x^2+a}}+\frac {1}{2} e \int \frac {1}{\sqrt {b} x e+e}d\frac {\sqrt {e x}}{\sqrt [4]{b x^2+a}}\right )}{e^3}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 e (e x)^{3/2}}\right )}{e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{7 a e (e x)^{7/2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {d \left (\frac {2 b \left (\frac {1}{2} e \int \frac {1}{e-\sqrt {b} e x}d\frac {\sqrt {e x}}{\sqrt [4]{b x^2+a}}+\frac {\sqrt {e} \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{2 \sqrt [4]{b}}\right )}{e^3}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 e (e x)^{3/2}}\right )}{e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{7 a e (e x)^{7/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {2 b \left (\frac {\sqrt {e} \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{2 \sqrt [4]{b}}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{2 \sqrt [4]{b}}\right )}{e^3}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 e (e x)^{3/2}}\right )}{e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{7 a e (e x)^{7/2}}\)

Input:

Int[((a + b*x^2)^(3/4)*(c + d*x^2))/(e*x)^(9/2),x]
 

Output:

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

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 247

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1)))   Int[ 
(c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 
0] && LtQ[m, -1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, 
m, p, x]
 

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 358

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] + S 
imp[d/e^2   Int[(e*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e 
, m, p}, x] && NeQ[b*c - a*d, 0] && EqQ[Simplify[m + 2*p + 3], 0] && NeQ[m, 
 -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]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (x^{2} d +c \right )}{\left (e x \right )^{\frac {9}{2}}}d x\]

Input:

int((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(9/2),x)
 

Output:

int((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(9/2),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.58 \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{9/2}} \, dx=\frac {21 \, a e^{5} x^{4} \left (\frac {b^{3} d^{4}}{e^{18}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} b d + {\left (b e^{5} x^{2} + a e^{5}\right )} \left (\frac {b^{3} d^{4}}{e^{18}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) - 21 \, a e^{5} x^{4} \left (\frac {b^{3} d^{4}}{e^{18}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} b d - {\left (b e^{5} x^{2} + a e^{5}\right )} \left (\frac {b^{3} d^{4}}{e^{18}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) - 21 i \, a e^{5} x^{4} \left (\frac {b^{3} d^{4}}{e^{18}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} b d - {\left (i \, b e^{5} x^{2} + i \, a e^{5}\right )} \left (\frac {b^{3} d^{4}}{e^{18}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) + 21 i \, a e^{5} x^{4} \left (\frac {b^{3} d^{4}}{e^{18}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} b d - {\left (-i \, b e^{5} x^{2} - i \, a e^{5}\right )} \left (\frac {b^{3} d^{4}}{e^{18}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) - 4 \, {\left ({\left (3 \, b c + 7 \, a d\right )} x^{2} + 3 \, a c\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x}}{42 \, a e^{5} x^{4}} \] Input:

integrate((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(9/2),x, algorithm="fricas")
 

Output:

1/42*(21*a*e^5*x^4*(b^3*d^4/e^18)^(1/4)*log(((b*x^2 + a)^(3/4)*sqrt(e*x)*b 
*d + (b*e^5*x^2 + a*e^5)*(b^3*d^4/e^18)^(1/4))/(b*x^2 + a)) - 21*a*e^5*x^4 
*(b^3*d^4/e^18)^(1/4)*log(((b*x^2 + a)^(3/4)*sqrt(e*x)*b*d - (b*e^5*x^2 + 
a*e^5)*(b^3*d^4/e^18)^(1/4))/(b*x^2 + a)) - 21*I*a*e^5*x^4*(b^3*d^4/e^18)^ 
(1/4)*log(((b*x^2 + a)^(3/4)*sqrt(e*x)*b*d - (I*b*e^5*x^2 + I*a*e^5)*(b^3* 
d^4/e^18)^(1/4))/(b*x^2 + a)) + 21*I*a*e^5*x^4*(b^3*d^4/e^18)^(1/4)*log((( 
b*x^2 + a)^(3/4)*sqrt(e*x)*b*d - (-I*b*e^5*x^2 - I*a*e^5)*(b^3*d^4/e^18)^( 
1/4))/(b*x^2 + a)) - 4*((3*b*c + 7*a*d)*x^2 + 3*a*c)*(b*x^2 + a)^(3/4)*sqr 
t(e*x))/(a*e^5*x^4)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 33.97 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{9/2}} \, dx=\frac {a^{\frac {3}{4}} d \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {9}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {b^{\frac {3}{4}} c \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{2 e^{\frac {9}{2}} x^{2} \Gamma \left (- \frac {3}{4}\right )} + \frac {b^{\frac {7}{4}} c \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{2 a e^{\frac {9}{2}} \Gamma \left (- \frac {3}{4}\right )} \] Input:

integrate((b*x**2+a)**(3/4)*(d*x**2+c)/(e*x)**(9/2),x)
 

Output:

a**(3/4)*d*gamma(-3/4)*hyper((-3/4, -3/4), (1/4,), b*x**2*exp_polar(I*pi)/ 
a)/(2*e**(9/2)*x**(3/2)*gamma(1/4)) + b**(3/4)*c*(a/(b*x**2) + 1)**(3/4)*g 
amma(-7/4)/(2*e**(9/2)*x**2*gamma(-3/4)) + b**(7/4)*c*(a/(b*x**2) + 1)**(3 
/4)*gamma(-7/4)/(2*a*e**(9/2)*gamma(-3/4))
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}}{\left (e x\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(9/2),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(3/4)*(d*x^2 + c)/(e*x)^(9/2), x)
                                                                                    
                                                                                    
 

Giac [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(9/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{9/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (d\,x^2+c\right )}{{\left (e\,x\right )}^{9/2}} \,d x \] Input:

int(((a + b*x^2)^(3/4)*(c + d*x^2))/(e*x)^(9/2),x)
 

Output:

int(((a + b*x^2)^(3/4)*(c + d*x^2))/(e*x)^(9/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{9/2}} \, dx=\frac {\sqrt {e}\, \left (-2 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a c -2 \left (b \,x^{2}+a \right )^{\frac {3}{4}} b c \,x^{2}+7 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{\frac {3}{4}}}{x^{3}}d x \right ) a d \,x^{3}\right )}{7 \sqrt {x}\, a \,e^{5} x^{3}} \] Input:

int((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(9/2),x)
 

Output:

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