Integrand size = 26, antiderivative size = 104 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=-\frac {2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {2 (8 b c-7 a d)}{7 a^2 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}+\frac {8 (8 b c-7 a d) \left (a+b x^2\right )^{3/4}}{21 a^3 e^3 (e x)^{3/2}} \] Output:
-2/7*c/a/e/(e*x)^(7/2)/(b*x^2+a)^(1/4)-2/7*(-7*a*d+8*b*c)/a^2/e^3/(e*x)^(3 /2)/(b*x^2+a)^(1/4)+8/21*(-7*a*d+8*b*c)*(b*x^2+a)^(3/4)/a^3/e^3/(e*x)^(3/2 )
Time = 0.58 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=-\frac {2 x \left (3 a^2 c-8 a b c x^2+7 a^2 d x^2-32 b^2 c x^4+28 a b d x^4\right )}{21 a^3 (e x)^{9/2} \sqrt [4]{a+b x^2}} \] Input:
Integrate[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(5/4)),x]
Output:
(-2*x*(3*a^2*c - 8*a*b*c*x^2 + 7*a^2*d*x^2 - 32*b^2*c*x^4 + 28*a*b*d*x^4)) /(21*a^3*(e*x)^(9/2)*(a + b*x^2)^(1/4))
Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {359, 246, 242}
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.
\(\displaystyle \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx\) |
\(\Big \downarrow \) 359 |
\(\displaystyle -\frac {(8 b c-7 a d) \int \frac {1}{(e x)^{5/2} \left (b x^2+a\right )^{5/4}}dx}{7 a e^2}-\frac {2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}}\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {(8 b c-7 a d) \left (\frac {4 \int \frac {1}{(e x)^{5/2} \sqrt [4]{b x^2+a}}dx}{a}+\frac {2}{a e (e x)^{3/2} \sqrt [4]{a+b x^2}}\right )}{7 a e^2}-\frac {2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {(8 b c-7 a d) \left (\frac {2}{a e (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac {8 \left (a+b x^2\right )^{3/4}}{3 a^2 e (e x)^{3/2}}\right )}{7 a e^2}-\frac {2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}}\) |
Input:
Int[(c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(5/4)),x]
Output:
(-2*c)/(7*a*e*(e*x)^(7/2)*(a + b*x^2)^(1/4)) - ((8*b*c - 7*a*d)*(2/(a*e*(e *x)^(3/2)*(a + b*x^2)^(1/4)) - (8*(a + b*x^2)^(3/4))/(3*a^2*e*(e*x)^(3/2)) ))/(7*a*e^2)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(a*c*2*(p + 1))), x] + Simp[(m + 2*p + 3)/( a*2*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m , p}, x] && ILtQ[Simplify[(m + 1)/2 + p + 1], 0] && NeQ[p, -1]
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]
Time = 0.42 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(-\frac {2 x \left (28 a b d \,x^{4}-32 b^{2} c \,x^{4}+7 a^{2} d \,x^{2}-8 a b c \,x^{2}+3 a^{2} c \right )}{21 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3} \left (e x \right )^{\frac {9}{2}}}\) | \(62\) |
orering | \(-\frac {2 x \left (28 a b d \,x^{4}-32 b^{2} c \,x^{4}+7 a^{2} d \,x^{2}-8 a b c \,x^{2}+3 a^{2} c \right )}{21 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3} \left (e x \right )^{\frac {9}{2}}}\) | \(62\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (7 a d \,x^{2}-11 x^{2} b c +3 a c \right )}{21 a^{3} x^{3} e^{4} \sqrt {e x}}-\frac {2 b x \left (a d -b c \right )}{a^{3} e^{4} \sqrt {e x}\, \left (b \,x^{2}+a \right )^{\frac {1}{4}}}\) | \(78\) |
Input:
int((d*x^2+c)/(e*x)^(9/2)/(b*x^2+a)^(5/4),x,method=_RETURNVERBOSE)
Output:
-2/21*x*(28*a*b*d*x^4-32*b^2*c*x^4+7*a^2*d*x^2-8*a*b*c*x^2+3*a^2*c)/(b*x^2 +a)^(1/4)/a^3/(e*x)^(9/2)
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.77 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=\frac {2 \, {\left (4 \, {\left (8 \, b^{2} c - 7 \, a b d\right )} x^{4} - 3 \, a^{2} c + {\left (8 \, a b c - 7 \, a^{2} d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x}}{21 \, {\left (a^{3} b e^{5} x^{6} + a^{4} e^{5} x^{4}\right )}} \] Input:
integrate((d*x^2+c)/(e*x)^(9/2)/(b*x^2+a)^(5/4),x, algorithm="fricas")
Output:
2/21*(4*(8*b^2*c - 7*a*b*d)*x^4 - 3*a^2*c + (8*a*b*c - 7*a^2*d)*x^2)*(b*x^ 2 + a)^(3/4)*sqrt(e*x)/(a^3*b*e^5*x^6 + a^4*e^5*x^4)
Timed out. \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=\text {Timed out} \] Input:
integrate((d*x**2+c)/(e*x)**(9/2)/(b*x**2+a)**(5/4),x)
Output:
Timed out
\[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \left (e x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((d*x^2+c)/(e*x)^(9/2)/(b*x^2+a)^(5/4),x, algorithm="maxima")
Output:
integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(9/2)), x)
\[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \left (e x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((d*x^2+c)/(e*x)^(9/2)/(b*x^2+a)^(5/4),x, algorithm="giac")
Output:
integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(9/2)), x)
Time = 0.90 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=-\frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {2\,c}{7\,a\,b\,e^4}+\frac {x^2\,\left (14\,a^2\,d-16\,a\,b\,c\right )}{21\,a^3\,b\,e^4}-\frac {x^4\,\left (64\,b^2\,c-56\,a\,b\,d\right )}{21\,a^3\,b\,e^4}\right )}{x^5\,\sqrt {e\,x}+\frac {a\,x^3\,\sqrt {e\,x}}{b}} \] Input:
int((c + d*x^2)/((e*x)^(9/2)*(a + b*x^2)^(5/4)),x)
Output:
-((a + b*x^2)^(3/4)*((2*c)/(7*a*b*e^4) + (x^2*(14*a^2*d - 16*a*b*c))/(21*a ^3*b*e^4) - (x^4*(64*b^2*c - 56*a*b*d))/(21*a^3*b*e^4)))/(x^5*(e*x)^(1/2) + (a*x^3*(e*x)^(1/2))/b)
Time = 0.32 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.44 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=\frac {\sqrt {e}\, \left (21 \left (b \,x^{2}+a \right ) a^{3} d -18 \left (b \,x^{2}+a \right ) a^{2} b c -56 \left (b \,x^{2}+a \right ) a^{2} b d \,x^{2}+48 \left (b \,x^{2}+a \right ) a \,b^{2} c \,x^{2}-224 \left (b \,x^{2}+a \right ) a \,b^{2} d \,x^{4}+192 \left (b \,x^{2}+a \right ) b^{3} c \,x^{4}-21 a^{4} d -21 a^{3} b d \,x^{2}\right )}{63 \left (b \,x^{2}+a \right )^{\frac {3}{4}} \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a^{3} b \,e^{5} x^{3}} \] Input:
int((d*x^2+c)/(e*x)^(9/2)/(b*x^2+a)^(5/4),x)
Output:
(sqrt(e)*(a + b*x**2)**(1/4)*(21*(a + b*x**2)*a**3*d - 18*(a + b*x**2)*a** 2*b*c - 56*(a + b*x**2)*a**2*b*d*x**2 + 48*(a + b*x**2)*a*b**2*c*x**2 - 22 4*(a + b*x**2)*a*b**2*d*x**4 + 192*(a + b*x**2)*b**3*c*x**4 - 21*a**4*d - 21*a**3*b*d*x**2))/(63*sqrt(x)*sqrt(a + b*x**2)*a**3*b*e**5*x**3*(a + b*x* *2))