\(\int \frac {(a+b x^2)^2}{(e x)^{13/2} (c+d x)^{3/2}} \, dx\) [843]

Optimal result

Integrand size = 26, antiderivative size = 301 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} (c+d x)^{3/2}} \, dx=-\frac {2 a^2}{11 c e (e x)^{11/2} \sqrt {c+d x}}+\frac {4 a d \left (13 b c^2+10 a d^2\right )}{35 c^4 e^4 (e x)^{5/2} \sqrt {c+d x}}+\frac {2 \left (385 b^2 c^4+1056 a b c^2 d^2+640 a^2 d^4\right )}{385 c^5 e^5 (e x)^{3/2} \sqrt {c+d x}}+\frac {8 a^2 d \sqrt {c+d x}}{33 c^3 e^2 (e x)^{9/2}}-\frac {4 a \left (33 b c^2+34 a d^2\right ) \sqrt {c+d x}}{231 c^4 e^3 (e x)^{7/2}}-\frac {8 \left (385 b^2 c^4+1056 a b c^2 d^2+640 a^2 d^4\right ) \sqrt {c+d x}}{1155 c^6 e^5 (e x)^{3/2}}+\frac {16 d \left (385 b^2 c^4+1056 a b c^2 d^2+640 a^2 d^4\right ) \sqrt {c+d x}}{1155 c^7 e^6 \sqrt {e x}} \] Output:

-2/11*a^2/c/e/(e*x)^(11/2)/(d*x+c)^(1/2)+4/35*a*d*(10*a*d^2+13*b*c^2)/c^4/ 
e^4/(e*x)^(5/2)/(d*x+c)^(1/2)+2/385*(640*a^2*d^4+1056*a*b*c^2*d^2+385*b^2* 
c^4)/c^5/e^5/(e*x)^(3/2)/(d*x+c)^(1/2)+8/33*a^2*d*(d*x+c)^(1/2)/c^3/e^2/(e 
*x)^(9/2)-4/231*a*(34*a*d^2+33*b*c^2)*(d*x+c)^(1/2)/c^4/e^3/(e*x)^(7/2)-8/ 
1155*(640*a^2*d^4+1056*a*b*c^2*d^2+385*b^2*c^4)*(d*x+c)^(1/2)/c^6/e^5/(e*x 
)^(3/2)+16/1155*d*(640*a^2*d^4+1056*a*b*c^2*d^2+385*b^2*c^4)*(d*x+c)^(1/2) 
/c^7/e^6/(e*x)^(1/2)
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.59 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} (c+d x)^{3/2}} \, dx=\frac {2 \sqrt {e x} \left (-385 b^2 c^4 x^4 \left (c^2-4 c d x-8 d^2 x^2\right )+66 a b c^2 x^2 \left (-5 c^4+8 c^3 d x-16 c^2 d^2 x^2+64 c d^3 x^3+128 d^4 x^4\right )-5 a^2 \left (21 c^6-28 c^5 d x+40 c^4 d^2 x^2-64 c^3 d^3 x^3+128 c^2 d^4 x^4-512 c d^5 x^5-1024 d^6 x^6\right )\right )}{1155 c^7 e^7 x^6 \sqrt {c+d x}} \] Input:

Integrate[(a + b*x^2)^2/((e*x)^(13/2)*(c + d*x)^(3/2)),x]
 

Output:

(2*Sqrt[e*x]*(-385*b^2*c^4*x^4*(c^2 - 4*c*d*x - 8*d^2*x^2) + 66*a*b*c^2*x^ 
2*(-5*c^4 + 8*c^3*d*x - 16*c^2*d^2*x^2 + 64*c*d^3*x^3 + 128*d^4*x^4) - 5*a 
^2*(21*c^6 - 28*c^5*d*x + 40*c^4*d^2*x^2 - 64*c^3*d^3*x^3 + 128*c^2*d^4*x^ 
4 - 512*c*d^5*x^5 - 1024*d^6*x^6)))/(1155*c^7*e^7*x^6*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.20, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {519, 27, 2124, 27, 1193, 27, 87, 55, 55, 48}

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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} (c+d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 519

\(\displaystyle \frac {2 \left (a d^2+b c^2\right )^2}{c d^4 e (e x)^{11/2} \sqrt {c+d x}}-\frac {2 \int -\frac {\frac {11 b^2 c^4}{d^4}-\frac {b^2 x^2 c^2}{d^2}+\frac {22 a b c^2}{d^2}+\frac {b^2 x^3 c}{d}+\frac {b \left (b c^2+2 a d^2\right ) x c}{d^3}+12 a^2}{2 (e x)^{13/2} \sqrt {c+d x}}dx}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\frac {11 b^2 c^4}{d^4}-\frac {b^2 x^2 c^2}{d^2}+\frac {22 a b c^2}{d^2}+\frac {b^2 x^3 c}{d}+\frac {b \left (b c^2+2 a d^2\right ) x c}{d^3}+12 a^2}{(e x)^{13/2} \sqrt {c+d x}}dx}{c}+\frac {2 \left (a d^2+b c^2\right )^2}{c d^4 e (e x)^{11/2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 2124

\(\displaystyle \frac {-\frac {2 \int \frac {\frac {11 b^2 x c^3}{d^2}-\frac {11 b^2 x^2 c^2}{d}+3 \left (\frac {33 b^2 c^4}{d^3}+\frac {66 a b c^2}{d}+40 a^2 d\right )}{2 (e x)^{11/2} \sqrt {c+d x}}dx}{11 c e}-\frac {2 \sqrt {c+d x} \left (12 a^2+\frac {22 a b c^2}{d^2}+\frac {11 b^2 c^4}{d^4}\right )}{11 c e (e x)^{11/2}}}{c}+\frac {2 \left (a d^2+b c^2\right )^2}{c d^4 e (e x)^{11/2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {\frac {11 b^2 x c^3}{d^2}-\frac {11 b^2 x^2 c^2}{d}+3 \left (\frac {33 b^2 c^4}{d^3}+\frac {66 a b c^2}{d}+40 a^2 d\right )}{(e x)^{11/2} \sqrt {c+d x}}dx}{11 c e}-\frac {2 \sqrt {c+d x} \left (12 a^2+\frac {22 a b c^2}{d^2}+\frac {11 b^2 c^4}{d^4}\right )}{11 c e (e x)^{11/2}}}{c}+\frac {2 \left (a d^2+b c^2\right )^2}{c d^4 e (e x)^{11/2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 1193

\(\displaystyle \frac {-\frac {-\frac {2 \int \frac {3 \left (33 b^2 x c^3+d \left (\frac {231 b^2 c^4}{d^2}+528 a b c^2+320 a^2 d^2\right )\right )}{2 d (e x)^{9/2} \sqrt {c+d x}}dx}{9 c e}-\frac {2 \sqrt {c+d x} \left (40 a^2 d+\frac {66 a b c^2}{d}+\frac {33 b^2 c^4}{d^3}\right )}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 \sqrt {c+d x} \left (12 a^2+\frac {22 a b c^2}{d^2}+\frac {11 b^2 c^4}{d^4}\right )}{11 c e (e x)^{11/2}}}{c}+\frac {2 \left (a d^2+b c^2\right )^2}{c d^4 e (e x)^{11/2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\int \frac {\frac {231 b^2 c^4}{d}+33 b^2 x c^3+528 a b d c^2+320 a^2 d^3}{(e x)^{9/2} \sqrt {c+d x}}dx}{3 c d e}-\frac {2 \sqrt {c+d x} \left (40 a^2 d+\frac {66 a b c^2}{d}+\frac {33 b^2 c^4}{d^3}\right )}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 \sqrt {c+d x} \left (12 a^2+\frac {22 a b c^2}{d^2}+\frac {11 b^2 c^4}{d^4}\right )}{11 c e (e x)^{11/2}}}{c}+\frac {2 \left (a d^2+b c^2\right )^2}{c d^4 e (e x)^{11/2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {-\frac {-\frac {-\frac {3 \left (640 a^2 d^4+1056 a b c^2 d^2+385 b^2 c^4\right ) \int \frac {1}{(e x)^{7/2} \sqrt {c+d x}}dx}{7 c e}-\frac {2 \sqrt {c+d x} \left (320 a^2 d^3+528 a b c^2 d+\frac {231 b^2 c^4}{d}\right )}{7 c e (e x)^{7/2}}}{3 c d e}-\frac {2 \sqrt {c+d x} \left (40 a^2 d+\frac {66 a b c^2}{d}+\frac {33 b^2 c^4}{d^3}\right )}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 \sqrt {c+d x} \left (12 a^2+\frac {22 a b c^2}{d^2}+\frac {11 b^2 c^4}{d^4}\right )}{11 c e (e x)^{11/2}}}{c}+\frac {2 \left (a d^2+b c^2\right )^2}{c d^4 e (e x)^{11/2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {-\frac {-\frac {-\frac {3 \left (640 a^2 d^4+1056 a b c^2 d^2+385 b^2 c^4\right ) \left (-\frac {4 d \int \frac {1}{(e x)^{5/2} \sqrt {c+d x}}dx}{5 c e}-\frac {2 \sqrt {c+d x}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {2 \sqrt {c+d x} \left (320 a^2 d^3+528 a b c^2 d+\frac {231 b^2 c^4}{d}\right )}{7 c e (e x)^{7/2}}}{3 c d e}-\frac {2 \sqrt {c+d x} \left (40 a^2 d+\frac {66 a b c^2}{d}+\frac {33 b^2 c^4}{d^3}\right )}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 \sqrt {c+d x} \left (12 a^2+\frac {22 a b c^2}{d^2}+\frac {11 b^2 c^4}{d^4}\right )}{11 c e (e x)^{11/2}}}{c}+\frac {2 \left (a d^2+b c^2\right )^2}{c d^4 e (e x)^{11/2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {-\frac {-\frac {-\frac {3 \left (640 a^2 d^4+1056 a b c^2 d^2+385 b^2 c^4\right ) \left (-\frac {4 d \left (-\frac {2 d \int \frac {1}{(e x)^{3/2} \sqrt {c+d x}}dx}{3 c e}-\frac {2 \sqrt {c+d x}}{3 c e (e x)^{3/2}}\right )}{5 c e}-\frac {2 \sqrt {c+d x}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {2 \sqrt {c+d x} \left (320 a^2 d^3+528 a b c^2 d+\frac {231 b^2 c^4}{d}\right )}{7 c e (e x)^{7/2}}}{3 c d e}-\frac {2 \sqrt {c+d x} \left (40 a^2 d+\frac {66 a b c^2}{d}+\frac {33 b^2 c^4}{d^3}\right )}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 \sqrt {c+d x} \left (12 a^2+\frac {22 a b c^2}{d^2}+\frac {11 b^2 c^4}{d^4}\right )}{11 c e (e x)^{11/2}}}{c}+\frac {2 \left (a d^2+b c^2\right )^2}{c d^4 e (e x)^{11/2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {-\frac {2 \sqrt {c+d x} \left (12 a^2+\frac {22 a b c^2}{d^2}+\frac {11 b^2 c^4}{d^4}\right )}{11 c e (e x)^{11/2}}-\frac {-\frac {2 \sqrt {c+d x} \left (40 a^2 d+\frac {66 a b c^2}{d}+\frac {33 b^2 c^4}{d^3}\right )}{3 c e (e x)^{9/2}}-\frac {-\frac {2 \sqrt {c+d x} \left (320 a^2 d^3+528 a b c^2 d+\frac {231 b^2 c^4}{d}\right )}{7 c e (e x)^{7/2}}-\frac {3 \left (640 a^2 d^4+1056 a b c^2 d^2+385 b^2 c^4\right ) \left (-\frac {4 d \left (\frac {4 d \sqrt {c+d x}}{3 c^2 e^2 \sqrt {e x}}-\frac {2 \sqrt {c+d x}}{3 c e (e x)^{3/2}}\right )}{5 c e}-\frac {2 \sqrt {c+d x}}{5 c e (e x)^{5/2}}\right )}{7 c e}}{3 c d e}}{11 c e}}{c}+\frac {2 \left (a d^2+b c^2\right )^2}{c d^4 e (e x)^{11/2} \sqrt {c+d x}}\)

Input:

Int[(a + b*x^2)^2/((e*x)^(13/2)*(c + d*x)^(3/2)),x]
 

Output:

(2*(b*c^2 + a*d^2)^2)/(c*d^4*e*(e*x)^(11/2)*Sqrt[c + d*x]) + ((-2*(12*a^2 
+ (11*b^2*c^4)/d^4 + (22*a*b*c^2)/d^2)*Sqrt[c + d*x])/(11*c*e*(e*x)^(11/2) 
) - ((-2*((33*b^2*c^4)/d^3 + (66*a*b*c^2)/d + 40*a^2*d)*Sqrt[c + d*x])/(3* 
c*e*(e*x)^(9/2)) - ((-2*((231*b^2*c^4)/d + 528*a*b*c^2*d + 320*a^2*d^3)*Sq 
rt[c + d*x])/(7*c*e*(e*x)^(7/2)) - (3*(385*b^2*c^4 + 1056*a*b*c^2*d^2 + 64 
0*a^2*d^4)*((-2*Sqrt[c + d*x])/(5*c*e*(e*x)^(5/2)) - (4*d*((-2*Sqrt[c + d* 
x])/(3*c*e*(e*x)^(3/2)) + (4*d*Sqrt[c + d*x])/(3*c^2*e^2*Sqrt[e*x])))/(5*c 
*e)))/(7*c*e))/(3*c*d*e))/(11*c*e))/c
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 519

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2)^p, c + d*x, x], R = 
 PolynomialRemainder[(a + b*x^2)^p, c + d*x, x]}, Simp[(-R)*(e*x)^(m + 1)*( 
(c + d*x)^(n + 1)/(c*e*(n + 1))), x] + Simp[1/(c*(n + 1))   Int[(e*x)^m*(c 
+ d*x)^(n + 1)*ExpandToSum[c*(n + 1)*Qx + R*(m + n + 2), x], x], x]] /; Fre 
eQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0] && LtQ[n, -1] &&  !IntegerQ[m]
 

rule 1193

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x 
 + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p, d + 
e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g)) 
), x] + Simp[1/((m + 1)*(e*f - d*g))   Int[(d + e*x)^(m + 1)*(f + g*x)^n*Ex 
pandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a 
, b, c, d, e, f, g, n}, x] && IGtQ[p, 0] && ILtQ[2*m, -2] &&  !IntegerQ[n] 
&&  !(EqQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0])
 

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.66

method	result	size
gosper	\(-\frac {2 x \left (-5120 a^{2} d^{6} x^{6}-8448 a b \,c^{2} d^{4} x^{6}-3080 b^{2} c^{4} d^{2} x^{6}-2560 a^{2} c \,d^{5} x^{5}-4224 a b \,c^{3} d^{3} x^{5}-1540 b^{2} c^{5} d \,x^{5}+640 a^{2} c^{2} d^{4} x^{4}+1056 a b \,c^{4} d^{2} x^{4}+385 b^{2} c^{6} x^{4}-320 a^{2} c^{3} d^{3} x^{3}-528 a b \,c^{5} d \,x^{3}+200 a^{2} c^{4} d^{2} x^{2}+330 a b \,c^{6} x^{2}-140 a^{2} c^{5} d x +105 a^{2} c^{6}\right )}{1155 \sqrt {d x +c}\, c^{7} \left (e x \right )^{\frac {13}{2}}}\)	\(200\)
orering	\(-\frac {2 x \left (-5120 a^{2} d^{6} x^{6}-8448 a b \,c^{2} d^{4} x^{6}-3080 b^{2} c^{4} d^{2} x^{6}-2560 a^{2} c \,d^{5} x^{5}-4224 a b \,c^{3} d^{3} x^{5}-1540 b^{2} c^{5} d \,x^{5}+640 a^{2} c^{2} d^{4} x^{4}+1056 a b \,c^{4} d^{2} x^{4}+385 b^{2} c^{6} x^{4}-320 a^{2} c^{3} d^{3} x^{3}-528 a b \,c^{5} d \,x^{3}+200 a^{2} c^{4} d^{2} x^{2}+330 a b \,c^{6} x^{2}-140 a^{2} c^{5} d x +105 a^{2} c^{6}\right )}{1155 \sqrt {d x +c}\, c^{7} \left (e x \right )^{\frac {13}{2}}}\)	\(200\)
default	\(-\frac {2 \left (-5120 a^{2} d^{6} x^{6}-8448 a b \,c^{2} d^{4} x^{6}-3080 b^{2} c^{4} d^{2} x^{6}-2560 a^{2} c \,d^{5} x^{5}-4224 a b \,c^{3} d^{3} x^{5}-1540 b^{2} c^{5} d \,x^{5}+640 a^{2} c^{2} d^{4} x^{4}+1056 a b \,c^{4} d^{2} x^{4}+385 b^{2} c^{6} x^{4}-320 a^{2} c^{3} d^{3} x^{3}-528 a b \,c^{5} d \,x^{3}+200 a^{2} c^{4} d^{2} x^{2}+330 a b \,c^{6} x^{2}-140 a^{2} c^{5} d x +105 a^{2} c^{6}\right )}{1155 \sqrt {e x}\, e^{6} \sqrt {d x +c}\, c^{7} x^{5}}\)	\(205\)
risch	\(-\frac {2 \sqrt {d x +c}\, \left (-3965 a^{2} d^{5} x^{5}-6138 a b \,c^{2} d^{3} x^{5}-1925 b^{2} d \,x^{5} c^{4}+1405 a^{2} c \,d^{4} x^{4}+1914 a b \,c^{3} d^{2} x^{4}+385 b^{2} c^{5} x^{4}-765 a^{2} c^{2} d^{3} x^{3}-858 a b \,c^{4} d \,x^{3}+445 a^{2} c^{3} d^{2} x^{2}+330 a b \,c^{5} x^{2}-245 a^{2} c^{4} d x +105 a^{2} c^{5}\right )}{1155 c^{7} x^{5} e^{6} \sqrt {e x}}+\frac {2 d^{2} \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) x}{c^{7} e^{6} \sqrt {e x}\, \sqrt {d x +c}}\)	\(214\)

Input:

int((b*x^2+a)^2/(e*x)^(13/2)/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-2/1155*x*(-5120*a^2*d^6*x^6-8448*a*b*c^2*d^4*x^6-3080*b^2*c^4*d^2*x^6-256 
0*a^2*c*d^5*x^5-4224*a*b*c^3*d^3*x^5-1540*b^2*c^5*d*x^5+640*a^2*c^2*d^4*x^ 
4+1056*a*b*c^4*d^2*x^4+385*b^2*c^6*x^4-320*a^2*c^3*d^3*x^3-528*a*b*c^5*d*x 
^3+200*a^2*c^4*d^2*x^2+330*a*b*c^6*x^2-140*a^2*c^5*d*x+105*a^2*c^6)/(d*x+c 
)^(1/2)/c^7/(e*x)^(13/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} (c+d x)^{3/2}} \, dx=\frac {2 \, {\left (140 \, a^{2} c^{5} d x - 105 \, a^{2} c^{6} + 8 \, {\left (385 \, b^{2} c^{4} d^{2} + 1056 \, a b c^{2} d^{4} + 640 \, a^{2} d^{6}\right )} x^{6} + 4 \, {\left (385 \, b^{2} c^{5} d + 1056 \, a b c^{3} d^{3} + 640 \, a^{2} c d^{5}\right )} x^{5} - {\left (385 \, b^{2} c^{6} + 1056 \, a b c^{4} d^{2} + 640 \, a^{2} c^{2} d^{4}\right )} x^{4} + 16 \, {\left (33 \, a b c^{5} d + 20 \, a^{2} c^{3} d^{3}\right )} x^{3} - 10 \, {\left (33 \, a b c^{6} + 20 \, a^{2} c^{4} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{1155 \, {\left (c^{7} d e^{7} x^{7} + c^{8} e^{7} x^{6}\right )}} \] Input:

integrate((b*x^2+a)^2/(e*x)^(13/2)/(d*x+c)^(3/2),x, algorithm="fricas")
 

Output:

2/1155*(140*a^2*c^5*d*x - 105*a^2*c^6 + 8*(385*b^2*c^4*d^2 + 1056*a*b*c^2* 
d^4 + 640*a^2*d^6)*x^6 + 4*(385*b^2*c^5*d + 1056*a*b*c^3*d^3 + 640*a^2*c*d 
^5)*x^5 - (385*b^2*c^6 + 1056*a*b*c^4*d^2 + 640*a^2*c^2*d^4)*x^4 + 16*(33* 
a*b*c^5*d + 20*a^2*c^3*d^3)*x^3 - 10*(33*a*b*c^6 + 20*a^2*c^4*d^2)*x^2)*sq 
rt(d*x + c)*sqrt(e*x)/(c^7*d*e^7*x^7 + c^8*e^7*x^6)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} (c+d x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**2/(e*x)**(13/2)/(d*x+c)**(3/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x^2+a)^2/(e*x)^(13/2)/(d*x+c)^(3/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (259) = 518\).

Time = 0.29 (sec) , antiderivative size = 673, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} (c+d x)^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^2/(e*x)^(13/2)/(d*x+c)^(3/2),x, algorithm="giac")
 

Output:

2/1155*((((d*x + c)*((d*x + c)*((1925*b^2*c^24*d^12*e^5*abs(d) + 6138*a*b* 
c^22*d^14*e^5*abs(d) + 3965*a^2*c^20*d^16*e^5*abs(d))*(d*x + c)/(c^27*d^6* 
e^6) - 22*(455*b^2*c^25*d^12*e^5*abs(d) + 1482*a*b*c^23*d^14*e^5*abs(d) + 
965*a^2*c^21*d^16*e^5*abs(d))/(c^27*d^6*e^6)) + 99*(210*b^2*c^26*d^12*e^5* 
abs(d) + 706*a*b*c^24*d^14*e^5*abs(d) + 465*a^2*c^22*d^16*e^5*abs(d))/(c^2 
7*d^6*e^6)) - 308*(70*b^2*c^27*d^12*e^5*abs(d) + 246*a*b*c^25*d^14*e^5*abs 
(d) + 165*a^2*c^23*d^16*e^5*abs(d))/(c^27*d^6*e^6))*(d*x + c) + 385*(29*b^ 
2*c^28*d^12*e^5*abs(d) + 108*a*b*c^26*d^14*e^5*abs(d) + 75*a^2*c^24*d^16*e 
^5*abs(d))/(c^27*d^6*e^6))*(d*x + c) - 2310*(b^2*c^29*d^12*e^5*abs(d) + 4* 
a*b*c^27*d^14*e^5*abs(d) + 3*a^2*c^25*d^16*e^5*abs(d))/(c^27*d^6*e^6))*sqr 
t(d*x + c)/((d*x + c)*d*e - c*d*e)^(11/2) + 4*(b^4*c^8*d^7 + 4*a*b^3*c^6*d 
^9 + 6*a^2*b^2*c^4*d^11 + 4*a^3*b*c^2*d^13 + a^4*d^15)/((sqrt(d*e)*b^2*c^5 
*d^4*e + 2*sqrt(d*e)*a*b*c^3*d^6*e + sqrt(d*e)*a^2*c*d^8*e + sqrt(d*e)*(sq 
rt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2*b^2*c^4*d^3 + 2*sqr 
t(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2*a*b*c^2*d 
^5 + sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2*a 
^2*d^7)*c^6*e^5*abs(d))
 

Mupad [B] (verification not implemented)

Time = 8.72 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} (c+d x)^{3/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {8\,a^2\,x}{33\,c^2\,e^6}-\frac {2\,a^2}{11\,c\,d\,e^6}-\frac {x^4\,\left (1280\,a^2\,c^2\,d^4+2112\,a\,b\,c^4\,d^2+770\,b^2\,c^6\right )}{1155\,c^7\,d\,e^6}+\frac {x^6\,\left (10240\,a^2\,d^6+16896\,a\,b\,c^2\,d^4+6160\,b^2\,c^4\,d^2\right )}{1155\,c^7\,d\,e^6}+\frac {x^5\,\left (5120\,a^2\,c\,d^5+8448\,a\,b\,c^3\,d^3+3080\,b^2\,c^5\,d\right )}{1155\,c^7\,d\,e^6}+\frac {32\,a\,x^3\,\left (33\,b\,c^2+20\,a\,d^2\right )}{1155\,c^4\,e^6}-\frac {4\,a\,x^2\,\left (33\,b\,c^2+20\,a\,d^2\right )}{231\,c^3\,d\,e^6}\right )}{x^6\,\sqrt {e\,x}+\frac {c\,x^5\,\sqrt {e\,x}}{d}} \] Input:

int((a + b*x^2)^2/((e*x)^(13/2)*(c + d*x)^(3/2)),x)
 

Output:

((c + d*x)^(1/2)*((8*a^2*x)/(33*c^2*e^6) - (2*a^2)/(11*c*d*e^6) - (x^4*(77 
0*b^2*c^6 + 1280*a^2*c^2*d^4 + 2112*a*b*c^4*d^2))/(1155*c^7*d*e^6) + (x^6* 
(10240*a^2*d^6 + 6160*b^2*c^4*d^2 + 16896*a*b*c^2*d^4))/(1155*c^7*d*e^6) + 
 (x^5*(5120*a^2*c*d^5 + 3080*b^2*c^5*d + 8448*a*b*c^3*d^3))/(1155*c^7*d*e^ 
6) + (32*a*x^3*(20*a*d^2 + 33*b*c^2))/(1155*c^4*e^6) - (4*a*x^2*(20*a*d^2 
+ 33*b*c^2))/(231*c^3*d*e^6)))/(x^6*(e*x)^(1/2) + (c*x^5*(e*x)^(1/2))/d)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} (c+d x)^{3/2}} \, dx=\frac {2 \sqrt {e}\, \left (-5120 \sqrt {d}\, \sqrt {d x +c}\, a^{2} d^{5} x^{6}-8448 \sqrt {d}\, \sqrt {d x +c}\, a b \,c^{2} d^{3} x^{6}-3080 \sqrt {d}\, \sqrt {d x +c}\, b^{2} c^{4} d \,x^{6}-105 \sqrt {x}\, a^{2} c^{6}+140 \sqrt {x}\, a^{2} c^{5} d x -200 \sqrt {x}\, a^{2} c^{4} d^{2} x^{2}+320 \sqrt {x}\, a^{2} c^{3} d^{3} x^{3}-640 \sqrt {x}\, a^{2} c^{2} d^{4} x^{4}+2560 \sqrt {x}\, a^{2} c \,d^{5} x^{5}+5120 \sqrt {x}\, a^{2} d^{6} x^{6}-330 \sqrt {x}\, a b \,c^{6} x^{2}+528 \sqrt {x}\, a b \,c^{5} d \,x^{3}-1056 \sqrt {x}\, a b \,c^{4} d^{2} x^{4}+4224 \sqrt {x}\, a b \,c^{3} d^{3} x^{5}+8448 \sqrt {x}\, a b \,c^{2} d^{4} x^{6}-385 \sqrt {x}\, b^{2} c^{6} x^{4}+1540 \sqrt {x}\, b^{2} c^{5} d \,x^{5}+3080 \sqrt {x}\, b^{2} c^{4} d^{2} x^{6}\right )}{1155 \sqrt {d x +c}\, c^{7} e^{7} x^{6}} \] Input:

int((b*x^2+a)^2/(e*x)^(13/2)/(d*x+c)^(3/2),x)
 

Output:

(2*sqrt(e)*( - 5120*sqrt(d)*sqrt(c + d*x)*a**2*d**5*x**6 - 8448*sqrt(d)*sq 
rt(c + d*x)*a*b*c**2*d**3*x**6 - 3080*sqrt(d)*sqrt(c + d*x)*b**2*c**4*d*x* 
*6 - 105*sqrt(x)*a**2*c**6 + 140*sqrt(x)*a**2*c**5*d*x - 200*sqrt(x)*a**2* 
c**4*d**2*x**2 + 320*sqrt(x)*a**2*c**3*d**3*x**3 - 640*sqrt(x)*a**2*c**2*d 
**4*x**4 + 2560*sqrt(x)*a**2*c*d**5*x**5 + 5120*sqrt(x)*a**2*d**6*x**6 - 3 
30*sqrt(x)*a*b*c**6*x**2 + 528*sqrt(x)*a*b*c**5*d*x**3 - 1056*sqrt(x)*a*b* 
c**4*d**2*x**4 + 4224*sqrt(x)*a*b*c**3*d**3*x**5 + 8448*sqrt(x)*a*b*c**2*d 
**4*x**6 - 385*sqrt(x)*b**2*c**6*x**4 + 1540*sqrt(x)*b**2*c**5*d*x**5 + 30 
80*sqrt(x)*b**2*c**4*d**2*x**6))/(1155*sqrt(c + d*x)*c**7*e**7*x**6)