Integrand size = 26, antiderivative size = 142 \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{5/4}} \, dx=-\frac {2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}}-\frac {2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}+\frac {16 (12 b c-11 a d) \left (a+b x^2\right )^{3/4}}{77 a^3 e^3 (e x)^{7/2}}-\frac {64 b (12 b c-11 a d) \left (a+b x^2\right )^{3/4}}{231 a^4 e^5 (e x)^{3/2}} \] Output:
-2/11*c/a/e/(e*x)^(11/2)/(b*x^2+a)^(1/4)-2/11*(-11*a*d+12*b*c)/a^2/e^3/(e* x)^(7/2)/(b*x^2+a)^(1/4)+16/77*(-11*a*d+12*b*c)*(b*x^2+a)^(3/4)/a^3/e^3/(e *x)^(7/2)-64/231*b*(-11*a*d+12*b*c)*(b*x^2+a)^(3/4)/a^4/e^5/(e*x)^(3/2)
Time = 0.86 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.64 \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{5/4}} \, dx=-\frac {2 x \left (21 a^3 c-36 a^2 b c x^2+33 a^3 d x^2+96 a b^2 c x^4-88 a^2 b d x^4+384 b^3 c x^6-352 a b^2 d x^6\right )}{231 a^4 (e x)^{13/2} \sqrt [4]{a+b x^2}} \] Input:
Integrate[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(5/4)),x]
Output:
(-2*x*(21*a^3*c - 36*a^2*b*c*x^2 + 33*a^3*d*x^2 + 96*a*b^2*c*x^4 - 88*a^2* b*d*x^4 + 384*b^3*c*x^6 - 352*a*b^2*d*x^6))/(231*a^4*(e*x)^(13/2)*(a + b*x ^2)^(1/4))
Time = 0.23 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {359, 246, 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.
\(\displaystyle \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{5/4}} \, dx\) |
\(\Big \downarrow \) 359 |
\(\displaystyle -\frac {(12 b c-11 a d) \int \frac {1}{(e x)^{9/2} \left (b x^2+a\right )^{5/4}}dx}{11 a e^2}-\frac {2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}}\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {(12 b c-11 a d) \left (\frac {8 \int \frac {1}{(e x)^{9/2} \sqrt [4]{b x^2+a}}dx}{a}+\frac {2}{a e (e x)^{7/2} \sqrt [4]{a+b x^2}}\right )}{11 a e^2}-\frac {2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}}\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {(12 b c-11 a d) \left (\frac {8 \left (-\frac {4 \int \frac {\left (b x^2+a\right )^{3/4}}{(e x)^{9/2}}dx}{3 a}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 a e (e x)^{7/2}}\right )}{a}+\frac {2}{a e (e x)^{7/2} \sqrt [4]{a+b x^2}}\right )}{11 a e^2}-\frac {2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {(12 b c-11 a d) \left (\frac {8 \left (\frac {8 \left (a+b x^2\right )^{7/4}}{21 a^2 e (e x)^{7/2}}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 a e (e x)^{7/2}}\right )}{a}+\frac {2}{a e (e x)^{7/2} \sqrt [4]{a+b x^2}}\right )}{11 a e^2}-\frac {2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}}\) |
Input:
Int[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(5/4)),x]
Output:
(-2*c)/(11*a*e*(e*x)^(11/2)*(a + b*x^2)^(1/4)) - ((12*b*c - 11*a*d)*(2/(a* e*(e*x)^(7/2)*(a + b*x^2)^(1/4)) + (8*((-2*(a + b*x^2)^(3/4))/(3*a*e*(e*x) ^(7/2)) + (8*(a + b*x^2)^(7/4))/(21*a^2*e*(e*x)^(7/2))))/a))/(11*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 = 86, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(-\frac {2 x \left (-352 a \,b^{2} d \,x^{6}+384 b^{3} c \,x^{6}-88 a^{2} b d \,x^{4}+96 a \,b^{2} c \,x^{4}+33 a^{3} d \,x^{2}-36 a^{2} b c \,x^{2}+21 a^{3} c \right )}{231 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{4} \left (e x \right )^{\frac {13}{2}}}\) | \(86\) |
orering | \(-\frac {2 x \left (-352 a \,b^{2} d \,x^{6}+384 b^{3} c \,x^{6}-88 a^{2} b d \,x^{4}+96 a \,b^{2} c \,x^{4}+33 a^{3} d \,x^{2}-36 a^{2} b c \,x^{2}+21 a^{3} c \right )}{231 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{4} \left (e x \right )^{\frac {13}{2}}}\) | \(86\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (-121 a b d \,x^{4}+153 b^{2} c \,x^{4}+33 a^{2} d \,x^{2}-57 a b c \,x^{2}+21 a^{2} c \right )}{231 a^{4} x^{5} e^{6} \sqrt {e x}}+\frac {2 b^{2} x \left (a d -b c \right )}{a^{4} e^{6} \sqrt {e x}\, \left (b \,x^{2}+a \right )^{\frac {1}{4}}}\) | \(102\) |
Input:
int((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(5/4),x,method=_RETURNVERBOSE)
Output:
-2/231*x*(-352*a*b^2*d*x^6+384*b^3*c*x^6-88*a^2*b*d*x^4+96*a*b^2*c*x^4+33* a^3*d*x^2-36*a^2*b*c*x^2+21*a^3*c)/(b*x^2+a)^(1/4)/a^4/(e*x)^(13/2)
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.74 \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{5/4}} \, dx=-\frac {2 \, {\left (32 \, {\left (12 \, b^{3} c - 11 \, a b^{2} d\right )} x^{6} + 8 \, {\left (12 \, a b^{2} c - 11 \, a^{2} b d\right )} x^{4} + 21 \, a^{3} c - 3 \, {\left (12 \, a^{2} b c - 11 \, a^{3} d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x}}{231 \, {\left (a^{4} b e^{7} x^{8} + a^{5} e^{7} x^{6}\right )}} \] Input:
integrate((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(5/4),x, algorithm="fricas")
Output:
-2/231*(32*(12*b^3*c - 11*a*b^2*d)*x^6 + 8*(12*a*b^2*c - 11*a^2*b*d)*x^4 + 21*a^3*c - 3*(12*a^2*b*c - 11*a^3*d)*x^2)*(b*x^2 + a)^(3/4)*sqrt(e*x)/(a^ 4*b*e^7*x^8 + a^5*e^7*x^6)
Timed out. \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{5/4}} \, dx=\text {Timed out} \] Input:
integrate((d*x**2+c)/(e*x)**(13/2)/(b*x**2+a)**(5/4),x)
Output:
Timed out
\[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{5/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \left (e x\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(5/4),x, algorithm="maxima")
Output:
integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(13/2)), x)
\[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{5/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \left (e x\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(5/4),x, algorithm="giac")
Output:
integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(13/2)), x)
Time = 0.91 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.88 \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{5/4}} \, dx=-\frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {2\,c}{11\,a\,b\,e^6}-\frac {16\,x^4\,\left (11\,a\,d-12\,b\,c\right )}{231\,a^3\,e^6}+\frac {x^2\,\left (66\,a^3\,d-72\,a^2\,b\,c\right )}{231\,a^4\,b\,e^6}+\frac {x^6\,\left (768\,b^3\,c-704\,a\,b^2\,d\right )}{231\,a^4\,b\,e^6}\right )}{x^7\,\sqrt {e\,x}+\frac {a\,x^5\,\sqrt {e\,x}}{b}} \] Input:
int((c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(5/4)),x)
Output:
-((a + b*x^2)^(3/4)*((2*c)/(11*a*b*e^6) - (16*x^4*(11*a*d - 12*b*c))/(231* a^3*e^6) + (x^2*(66*a^3*d - 72*a^2*b*c))/(231*a^4*b*e^6) + (x^6*(768*b^3*c - 704*a*b^2*d))/(231*a^4*b*e^6)))/(x^7*(e*x)^(1/2) + (a*x^5*(e*x)^(1/2))/ b)
Time = 0.35 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.32 \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{5/4}} \, dx=\frac {\sqrt {e}\, \left (77 \left (b \,x^{2}+a \right ) a^{4} d -70 \left (b \,x^{2}+a \right ) a^{3} b c -132 \left (b \,x^{2}+a \right ) a^{3} b d \,x^{2}+120 \left (b \,x^{2}+a \right ) a^{2} b^{2} c \,x^{2}+352 \left (b \,x^{2}+a \right ) a^{2} b^{2} d \,x^{4}-320 \left (b \,x^{2}+a \right ) a \,b^{3} c \,x^{4}+1408 \left (b \,x^{2}+a \right ) a \,b^{3} d \,x^{6}-1280 \left (b \,x^{2}+a \right ) b^{4} c \,x^{6}-77 a^{5} d -77 a^{4} b d \,x^{2}\right )}{385 \left (b \,x^{2}+a \right )^{\frac {3}{4}} \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a^{4} b \,e^{7} x^{5}} \] Input:
int((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(5/4),x)
Output:
(sqrt(e)*(a + b*x**2)**(1/4)*(77*(a + b*x**2)*a**4*d - 70*(a + b*x**2)*a** 3*b*c - 132*(a + b*x**2)*a**3*b*d*x**2 + 120*(a + b*x**2)*a**2*b**2*c*x**2 + 352*(a + b*x**2)*a**2*b**2*d*x**4 - 320*(a + b*x**2)*a*b**3*c*x**4 + 14 08*(a + b*x**2)*a*b**3*d*x**6 - 1280*(a + b*x**2)*b**4*c*x**6 - 77*a**5*d - 77*a**4*b*d*x**2))/(385*sqrt(x)*sqrt(a + b*x**2)*a**4*b*e**7*x**5*(a + b *x**2))