\(\int \frac {c+d x^2}{(e x)^{13/2} (a+b x^2)^{9/4}} \, dx\) [476]

Optimal result

Integrand size = 26, antiderivative size = 179 \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{9/4}} \, dx=-\frac {2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac {24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}+\frac {192 (16 b c-11 a d) \left (a+b x^2\right )^{3/4}}{385 a^4 e^3 (e x)^{7/2}}-\frac {256 b (16 b c-11 a d) \left (a+b x^2\right )^{3/4}}{385 a^5 e^5 (e x)^{3/2}} \] Output:

-2/11*c/a/e/(e*x)^(11/2)/(b*x^2+a)^(5/4)-2/55*(-11*a*d+16*b*c)/a^2/e^3/(e* 
x)^(7/2)/(b*x^2+a)^(5/4)-24/55*(-11*a*d+16*b*c)/a^3/e^3/(e*x)^(7/2)/(b*x^2 
+a)^(1/4)+192/385*(-11*a*d+16*b*c)*(b*x^2+a)^(3/4)/a^4/e^3/(e*x)^(7/2)-256 
/385*b*(-11*a*d+16*b*c)*(b*x^2+a)^(3/4)/a^5/e^5/(e*x)^(3/2)
 

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.64 \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{9/4}} \, dx=-\frac {2 x \left (35 a^4 c-80 a^3 b c x^2+55 a^4 d x^2+320 a^2 b^2 c x^4-220 a^3 b d x^4+2560 a b^3 c x^6-1760 a^2 b^2 d x^6+2048 b^4 c x^8-1408 a b^3 d x^8\right )}{385 a^5 (e x)^{13/2} \left (a+b x^2\right )^{5/4}} \] Input:

Integrate[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(9/4)),x]
 

Output:

(-2*x*(35*a^4*c - 80*a^3*b*c*x^2 + 55*a^4*d*x^2 + 320*a^2*b^2*c*x^4 - 220* 
a^3*b*d*x^4 + 2560*a*b^3*c*x^6 - 1760*a^2*b^2*d*x^6 + 2048*b^4*c*x^8 - 140 
8*a*b^3*d*x^8))/(385*a^5*(e*x)^(13/2)*(a + b*x^2)^(5/4))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {359, 246, 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.

\(\displaystyle \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{9/4}} \, dx\)

\(\Big \downarrow \) 359

\(\displaystyle -\frac {(16 b c-11 a d) \int \frac {1}{(e x)^{9/2} \left (b x^2+a\right )^{9/4}}dx}{11 a e^2}-\frac {2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}\)

\(\Big \downarrow \) 246

\(\displaystyle -\frac {(16 b c-11 a d) \left (\frac {12 \int \frac {1}{(e x)^{9/2} \left (b x^2+a\right )^{5/4}}dx}{5 a}+\frac {2}{5 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}}\right )}{11 a e^2}-\frac {2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}\)

\(\Big \downarrow \) 246

\(\displaystyle -\frac {(16 b c-11 a d) \left (\frac {12 \left (\frac {8 \int \frac {1}{(e x)^{9/2} \sqrt [4]{b x^2+a}}dx}{a}+\frac {2}{a e (e x)^{7/2} \sqrt [4]{a+b x^2}}\right )}{5 a}+\frac {2}{5 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}}\right )}{11 a e^2}-\frac {2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}\)

\(\Big \downarrow \) 246

\(\displaystyle -\frac {(16 b c-11 a d) \left (\frac {12 \left (\frac {8 \left (-\frac {4 \int \frac {\left (b x^2+a\right )^{3/4}}{(e x)^{9/2}}dx}{3 a}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 a e (e x)^{7/2}}\right )}{a}+\frac {2}{a e (e x)^{7/2} \sqrt [4]{a+b x^2}}\right )}{5 a}+\frac {2}{5 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}}\right )}{11 a e^2}-\frac {2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}\)

\(\Big \downarrow \) 242

\(\displaystyle -\frac {(16 b c-11 a d) \left (\frac {12 \left (\frac {8 \left (\frac {8 \left (a+b x^2\right )^{7/4}}{21 a^2 e (e x)^{7/2}}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 a e (e x)^{7/2}}\right )}{a}+\frac {2}{a e (e x)^{7/2} \sqrt [4]{a+b x^2}}\right )}{5 a}+\frac {2}{5 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}}\right )}{11 a e^2}-\frac {2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}\)

Input:

Int[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(9/4)),x]
 

Output:

(-2*c)/(11*a*e*(e*x)^(11/2)*(a + b*x^2)^(5/4)) - ((16*b*c - 11*a*d)*(2/(5* 
a*e*(e*x)^(7/2)*(a + b*x^2)^(5/4)) + (12*(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))/(5*a)))/(11*a*e^2)
 

Defintions of rubi rules used

rule 242

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]
 

rule 246

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]
 

rule 359

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]
 

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.61

method	result	size
gosper	\(-\frac {2 x \left (-1408 a \,b^{3} d \,x^{8}+2048 b^{4} c \,x^{8}-1760 a^{2} b^{2} d \,x^{6}+2560 a \,b^{3} c \,x^{6}-220 a^{3} b d \,x^{4}+320 a^{2} b^{2} c \,x^{4}+55 a^{4} d \,x^{2}-80 a^{3} b c \,x^{2}+35 c \,a^{4}\right )}{385 \left (b \,x^{2}+a \right )^{\frac {5}{4}} a^{5} \left (e x \right )^{\frac {13}{2}}}\)	\(110\)
orering	\(-\frac {2 x \left (-1408 a \,b^{3} d \,x^{8}+2048 b^{4} c \,x^{8}-1760 a^{2} b^{2} d \,x^{6}+2560 a \,b^{3} c \,x^{6}-220 a^{3} b d \,x^{4}+320 a^{2} b^{2} c \,x^{4}+55 a^{4} d \,x^{2}-80 a^{3} b c \,x^{2}+35 c \,a^{4}\right )}{385 \left (b \,x^{2}+a \right )^{\frac {5}{4}} a^{5} \left (e x \right )^{\frac {13}{2}}}\)	\(110\)
risch	\(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (-66 a b d \,x^{4}+117 b^{2} c \,x^{4}+11 a^{2} d \,x^{2}-30 a b c \,x^{2}+7 a^{2} c \right )}{77 a^{5} x^{5} e^{6} \sqrt {e x}}+\frac {2 b^{2} \left (14 a b d \,x^{2}-19 b^{2} c \,x^{2}+15 d \,a^{2}-20 a b c \right ) x}{5 \left (b \,x^{2}+a \right )^{\frac {5}{4}} a^{5} e^{6} \sqrt {e x}}\)	\(123\)

Input:

int((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(9/4),x,method=_RETURNVERBOSE)
 

Output:

-2/385*x*(-1408*a*b^3*d*x^8+2048*b^4*c*x^8-1760*a^2*b^2*d*x^6+2560*a*b^3*c 
*x^6-220*a^3*b*d*x^4+320*a^2*b^2*c*x^4+55*a^4*d*x^2-80*a^3*b*c*x^2+35*a^4* 
c)/(b*x^2+a)^(5/4)/a^5/(e*x)^(13/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.80 \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{9/4}} \, dx=-\frac {2 \, {\left (128 \, {\left (16 \, b^{4} c - 11 \, a b^{3} d\right )} x^{8} + 160 \, {\left (16 \, a b^{3} c - 11 \, a^{2} b^{2} d\right )} x^{6} + 35 \, a^{4} c + 20 \, {\left (16 \, a^{2} b^{2} c - 11 \, a^{3} b d\right )} x^{4} - 5 \, {\left (16 \, a^{3} b c - 11 \, a^{4} d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x}}{385 \, {\left (a^{5} b^{2} e^{7} x^{10} + 2 \, a^{6} b e^{7} x^{8} + a^{7} e^{7} x^{6}\right )}} \] Input:

integrate((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(9/4),x, algorithm="fricas")
 

Output:

-2/385*(128*(16*b^4*c - 11*a*b^3*d)*x^8 + 160*(16*a*b^3*c - 11*a^2*b^2*d)* 
x^6 + 35*a^4*c + 20*(16*a^2*b^2*c - 11*a^3*b*d)*x^4 - 5*(16*a^3*b*c - 11*a 
^4*d)*x^2)*(b*x^2 + a)^(3/4)*sqrt(e*x)/(a^5*b^2*e^7*x^10 + 2*a^6*b*e^7*x^8 
 + a^7*e^7*x^6)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{9/4}} \, dx=\text {Timed out} \] Input:

integrate((d*x**2+c)/(e*x)**(13/2)/(b*x**2+a)**(9/4),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} \left (e x\right )^{\frac {13}{2}}} \,d x } \] Input:

integrate((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(9/4),x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(13/2)), x)
 

Giac [F]

\[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} \left (e x\right )^{\frac {13}{2}}} \,d x } \] Input:

integrate((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(9/4),x, algorithm="giac")
 

Output:

integrate((d*x^2 + c)/((b*x^2 + a)^(9/4)*(e*x)^(13/2)), x)
 

Mupad [B] (verification not implemented)

Time = 0.95 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.87 \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{9/4}} \, dx=\frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {64\,x^6\,\left (11\,a\,d-16\,b\,c\right )}{77\,a^4\,e^6}-\frac {2\,c}{11\,a\,b^2\,e^6}+\frac {8\,x^4\,\left (11\,a\,d-16\,b\,c\right )}{77\,a^3\,b\,e^6}-\frac {x^2\,\left (110\,a^4\,d-160\,a^3\,b\,c\right )}{385\,a^5\,b^2\,e^6}+\frac {256\,b\,x^8\,\left (11\,a\,d-16\,b\,c\right )}{385\,a^5\,e^6}\right )}{x^9\,\sqrt {e\,x}+\frac {a^2\,x^5\,\sqrt {e\,x}}{b^2}+\frac {2\,a\,x^7\,\sqrt {e\,x}}{b}} \] Input:

int((c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(9/4)),x)
 

Output:

((a + b*x^2)^(3/4)*((64*x^6*(11*a*d - 16*b*c))/(77*a^4*e^6) - (2*c)/(11*a* 
b^2*e^6) + (8*x^4*(11*a*d - 16*b*c))/(77*a^3*b*e^6) - (x^2*(110*a^4*d - 16 
0*a^3*b*c))/(385*a^5*b^2*e^6) + (256*b*x^8*(11*a*d - 16*b*c))/(385*a^5*e^6 
)))/(x^9*(e*x)^(1/2) + (a^2*x^5*(e*x)^(1/2))/b^2 + (2*a*x^7*(e*x)^(1/2))/b 
)
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.37 \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{9/4}} \, dx=\frac {\sqrt {e}\, \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (385 \left (b \,x^{2}+a \right ) a^{5} d -490 \left (b \,x^{2}+a \right ) a^{4} b c -880 \left (b \,x^{2}+a \right ) a^{4} b d \,x^{2}+1120 \left (b \,x^{2}+a \right ) a^{3} b^{2} c \,x^{2}+3520 \left (b \,x^{2}+a \right ) a^{3} b^{2} d \,x^{4}-4480 \left (b \,x^{2}+a \right ) a^{2} b^{3} c \,x^{4}+28160 \left (b \,x^{2}+a \right ) a^{2} b^{3} d \,x^{6}-35840 \left (b \,x^{2}+a \right ) a \,b^{4} c \,x^{6}+22528 \left (b \,x^{2}+a \right ) a \,b^{4} d \,x^{8}-28672 \left (b \,x^{2}+a \right ) b^{5} c \,x^{8}-385 a^{6} d -385 a^{5} b d \,x^{2}\right )}{2695 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a^{5} b \,e^{7} x^{5} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:

int((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(9/4),x)
 

Output:

(sqrt(e)*(a + b*x**2)**(1/4)*(385*(a + b*x**2)*a**5*d - 490*(a + b*x**2)*a 
**4*b*c - 880*(a + b*x**2)*a**4*b*d*x**2 + 1120*(a + b*x**2)*a**3*b**2*c*x 
**2 + 3520*(a + b*x**2)*a**3*b**2*d*x**4 - 4480*(a + b*x**2)*a**2*b**3*c*x 
**4 + 28160*(a + b*x**2)*a**2*b**3*d*x**6 - 35840*(a + b*x**2)*a*b**4*c*x* 
*6 + 22528*(a + b*x**2)*a*b**4*d*x**8 - 28672*(a + b*x**2)*b**5*c*x**8 - 3 
85*a**6*d - 385*a**5*b*d*x**2))/(2695*sqrt(x)*sqrt(a + b*x**2)*a**5*b*e**7 
*x**5*(a**2 + 2*a*b*x**2 + b**2*x**4))