\(\int \frac {(a+b x^2)^2}{(e x)^{3/2} (c+d x)^{5/2}} \, dx\) [848]

Optimal result

Integrand size = 26, antiderivative size = 209 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} (c+d x)^{5/2}} \, dx=-\frac {2 a^2}{c e \sqrt {e x} (c+d x)^{3/2}}-\frac {2 \left (b^2 c^4+2 a b c^2 d^2+4 a^2 d^4\right ) \sqrt {e x}}{3 c^2 d^3 e^2 (c+d x)^{3/2}}+\frac {2 \left (7 b^2 c^4+2 a b c^2 d^2-8 a^2 d^4\right ) \sqrt {e x}}{3 c^3 d^3 e^2 \sqrt {c+d x}}+\frac {b^2 \sqrt {e x} \sqrt {c+d x}}{d^3 e^2}-\frac {5 b^2 c \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{7/2} e^{3/2}} \] Output:

-2*a^2/c/e/(e*x)^(1/2)/(d*x+c)^(3/2)-2/3*(4*a^2*d^4+2*a*b*c^2*d^2+b^2*c^4) 
*(e*x)^(1/2)/c^2/d^3/e^2/(d*x+c)^(3/2)+2/3*(-8*a^2*d^4+2*a*b*c^2*d^2+7*b^2 
*c^4)*(e*x)^(1/2)/c^3/d^3/e^2/(d*x+c)^(1/2)+b^2*(e*x)^(1/2)*(d*x+c)^(1/2)/ 
d^3/e^2-5*b^2*c*arctanh(d^(1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/d^(7/2) 
/e^(3/2)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {x \left (\frac {\sqrt {d} \left (4 a b c^2 d^3 x^2+b^2 c^3 x \left (15 c^2+20 c d x+3 d^2 x^2\right )-2 a^2 d^3 \left (3 c^2+12 c d x+8 d^2 x^2\right )\right )}{(c+d x)^{3/2}}+30 b^2 c^4 \sqrt {x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}-\sqrt {c+d x}}\right )\right )}{3 c^3 d^{7/2} (e x)^{3/2}} \] Input:

Integrate[(a + b*x^2)^2/((e*x)^(3/2)*(c + d*x)^(5/2)),x]
 

Output:

(x*((Sqrt[d]*(4*a*b*c^2*d^3*x^2 + b^2*c^3*x*(15*c^2 + 20*c*d*x + 3*d^2*x^2 
) - 2*a^2*d^3*(3*c^2 + 12*c*d*x + 8*d^2*x^2)))/(c + d*x)^(3/2) + 30*b^2*c^ 
4*Sqrt[x]*ArcTanh[(Sqrt[d]*Sqrt[x])/(Sqrt[c] - Sqrt[c + d*x])]))/(3*c^3*d^ 
(7/2)*(e*x)^(3/2))
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {519, 27, 2124, 27, 1193, 27, 90, 65, 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.

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

\(\Big \downarrow \) 519

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2124

\(\displaystyle \frac {-\frac {2 \int -\frac {\frac {b^2 c^4}{d^3}-\frac {3 b^2 x c^3}{d^2}+\frac {3 b^2 x^2 c^2}{d}+\frac {2 a b c^2}{d}-8 a^2 d}{2 \sqrt {e x} (c+d x)^{3/2}}dx}{c e}-\frac {2 \left (4 a^2+\frac {2 a b c^2}{d^2}+\frac {b^2 c^4}{d^4}\right )}{c e \sqrt {e x} \sqrt {c+d x}}}{3 c}+\frac {2 \left (a d^2+b c^2\right )^2}{3 c d^4 e \sqrt {e x} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\frac {b^2 c^4}{d^3}-\frac {3 b^2 x c^3}{d^2}+\frac {3 b^2 x^2 c^2}{d}+\frac {2 a b c^2}{d}-8 a^2 d}{\sqrt {e x} (c+d x)^{3/2}}dx}{c e}-\frac {2 \left (4 a^2+\frac {2 a b c^2}{d^2}+\frac {b^2 c^4}{d^4}\right )}{c e \sqrt {e x} \sqrt {c+d x}}}{3 c}+\frac {2 \left (a d^2+b c^2\right )^2}{3 c d^4 e \sqrt {e x} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 1193

\(\displaystyle \frac {\frac {\frac {2 \int -\frac {3 b^2 c^3 e (2 c-d x)}{2 d^3 \sqrt {e x} \sqrt {c+d x}}dx}{c e}+\frac {2 \sqrt {e x} \left (-8 a^2 d+\frac {2 a b c^2}{d}+\frac {7 b^2 c^4}{d^3}\right )}{c e \sqrt {c+d x}}}{c e}-\frac {2 \left (4 a^2+\frac {2 a b c^2}{d^2}+\frac {b^2 c^4}{d^4}\right )}{c e \sqrt {e x} \sqrt {c+d x}}}{3 c}+\frac {2 \left (a d^2+b c^2\right )^2}{3 c d^4 e \sqrt {e x} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {2 \sqrt {e x} \left (-8 a^2 d+\frac {2 a b c^2}{d}+\frac {7 b^2 c^4}{d^3}\right )}{c e \sqrt {c+d x}}-\frac {3 b^2 c^2 \int \frac {2 c-d x}{\sqrt {e x} \sqrt {c+d x}}dx}{d^3}}{c e}-\frac {2 \left (4 a^2+\frac {2 a b c^2}{d^2}+\frac {b^2 c^4}{d^4}\right )}{c e \sqrt {e x} \sqrt {c+d x}}}{3 c}+\frac {2 \left (a d^2+b c^2\right )^2}{3 c d^4 e \sqrt {e x} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {\frac {2 \sqrt {e x} \left (-8 a^2 d+\frac {2 a b c^2}{d}+\frac {7 b^2 c^4}{d^3}\right )}{c e \sqrt {c+d x}}-\frac {3 b^2 c^2 \left (\frac {5}{2} c \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx-\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )}{d^3}}{c e}-\frac {2 \left (4 a^2+\frac {2 a b c^2}{d^2}+\frac {b^2 c^4}{d^4}\right )}{c e \sqrt {e x} \sqrt {c+d x}}}{3 c}+\frac {2 \left (a d^2+b c^2\right )^2}{3 c d^4 e \sqrt {e x} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\frac {\frac {2 \sqrt {e x} \left (-8 a^2 d+\frac {2 a b c^2}{d}+\frac {7 b^2 c^4}{d^3}\right )}{c e \sqrt {c+d x}}-\frac {3 b^2 c^2 \left (5 c \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}-\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )}{d^3}}{c e}-\frac {2 \left (4 a^2+\frac {2 a b c^2}{d^2}+\frac {b^2 c^4}{d^4}\right )}{c e \sqrt {e x} \sqrt {c+d x}}}{3 c}+\frac {2 \left (a d^2+b c^2\right )^2}{3 c d^4 e \sqrt {e x} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {2 \sqrt {e x} \left (-8 a^2 d+\frac {2 a b c^2}{d}+\frac {7 b^2 c^4}{d^3}\right )}{c e \sqrt {c+d x}}-\frac {3 b^2 c^2 \left (\frac {5 c \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{\sqrt {d} \sqrt {e}}-\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )}{d^3}}{c e}-\frac {2 \left (4 a^2+\frac {2 a b c^2}{d^2}+\frac {b^2 c^4}{d^4}\right )}{c e \sqrt {e x} \sqrt {c+d x}}}{3 c}+\frac {2 \left (a d^2+b c^2\right )^2}{3 c d^4 e \sqrt {e x} (c+d x)^{3/2}}\)

Input:

Int[(a + b*x^2)^2/((e*x)^(3/2)*(c + d*x)^(5/2)),x]
 

Output:

(2*(b*c^2 + a*d^2)^2)/(3*c*d^4*e*Sqrt[e*x]*(c + d*x)^(3/2)) + ((-2*(4*a^2 
+ (b^2*c^4)/d^4 + (2*a*b*c^2)/d^2))/(c*e*Sqrt[e*x]*Sqrt[c + d*x]) + ((2*(( 
7*b^2*c^4)/d^3 + (2*a*b*c^2)/d - 8*a^2*d)*Sqrt[e*x])/(c*e*Sqrt[c + d*x]) - 
 (3*b^2*c^2*(-((Sqrt[e*x]*Sqrt[c + d*x])/e) + (5*c*ArcTanh[(Sqrt[d]*Sqrt[e 
*x])/(Sqrt[e]*Sqrt[c + d*x])])/(Sqrt[d]*Sqrt[e])))/d^3)/(c*e))/(3*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 65

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2   Sub 
st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d 
}, x] &&  !GtQ[c, 0]
 

rule 90

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

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 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 = 312, normalized size of antiderivative = 1.49

method	result	size
risch	\(-\frac {\sqrt {d x +c}\, \left (-c^{3} b^{2} x +2 a^{2} d^{3}\right )}{c^{3} d^{3} e \sqrt {e x}}-\frac {\left (\frac {2 \left (2 a^{2} d^{4}-4 b \,c^{2} d^{2} a -6 b^{2} c^{4}\right ) \sqrt {d e \left (x +\frac {c}{d}\right )^{2}-c e \left (x +\frac {c}{d}\right )}}{d c e \left (x +\frac {c}{d}\right )}+\frac {5 c^{3} b^{2} \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right )}{\sqrt {d e}}+\frac {2 c \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \left (\frac {2 \sqrt {d e \left (x +\frac {c}{d}\right )^{2}-c e \left (x +\frac {c}{d}\right )}}{3 c e \left (x +\frac {c}{d}\right )^{2}}+\frac {4 d \sqrt {d e \left (x +\frac {c}{d}\right )^{2}-c e \left (x +\frac {c}{d}\right )}}{3 e \,c^{2} \left (x +\frac {c}{d}\right )}\right )}{d^{2}}\right ) \sqrt {\left (d x +c \right ) e x}}{2 c^{2} d^{3} e \sqrt {e x}\, \sqrt {d x +c}}\)	\(312\)
default	\(-\frac {15 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{4} d^{2} e \,x^{3}+30 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{5} d e \,x^{2}-6 b^{2} c^{3} d^{2} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+15 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{6} e x +32 a^{2} d^{5} x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-8 a b \,c^{2} d^{3} x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-40 b^{2} c^{4} d \,x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+48 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c \,d^{4} x -30 b^{2} c^{5} x \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+12 a^{2} c^{2} d^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}}{6 c^{3} e \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {e x}\, d^{3} \left (d x +c \right )^{\frac {3}{2}}}\)	\(365\)

Input:

int((b*x^2+a)^2/(e*x)^(3/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-(d*x+c)^(1/2)*(-b^2*c^3*x+2*a^2*d^3)/c^3/d^3/e/(e*x)^(1/2)-1/2/c^2/d^3*(2 
*(2*a^2*d^4-4*a*b*c^2*d^2-6*b^2*c^4)/d/c/e/(x+c/d)*(d*e*(x+c/d)^2-c*e*(x+c 
/d))^(1/2)+5*c^3*b^2*ln((1/2*c*e+d*e*x)/(d*e)^(1/2)+(d*e*x^2+c*e*x)^(1/2)) 
/(d*e)^(1/2)+2*c*(a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)/d^2*(2/3/c/e/(x+c/d)^2*(d 
*e*(x+c/d)^2-c*e*(x+c/d))^(1/2)+4/3*d/e/c^2/(x+c/d)*(d*e*(x+c/d)^2-c*e*(x+ 
c/d))^(1/2)))/e*((d*x+c)*e*x)^(1/2)/(e*x)^(1/2)/(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.02 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} (c+d x)^{5/2}} \, dx=\left [\frac {15 \, {\left (b^{2} c^{4} d^{2} x^{3} + 2 \, b^{2} c^{5} d x^{2} + b^{2} c^{6} x\right )} \sqrt {d e} \log \left (2 \, d e x + c e - 2 \, \sqrt {d e} \sqrt {d x + c} \sqrt {e x}\right ) + 2 \, {\left (3 \, b^{2} c^{3} d^{3} x^{3} - 6 \, a^{2} c^{2} d^{4} + 4 \, {\left (5 \, b^{2} c^{4} d^{2} + a b c^{2} d^{4} - 4 \, a^{2} d^{6}\right )} x^{2} + 3 \, {\left (5 \, b^{2} c^{5} d - 8 \, a^{2} c d^{5}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{6 \, {\left (c^{3} d^{6} e^{2} x^{3} + 2 \, c^{4} d^{5} e^{2} x^{2} + c^{5} d^{4} e^{2} x\right )}}, \frac {15 \, {\left (b^{2} c^{4} d^{2} x^{3} + 2 \, b^{2} c^{5} d x^{2} + b^{2} c^{6} x\right )} \sqrt {-d e} \arctan \left (\frac {\sqrt {-d e} \sqrt {d x + c} \sqrt {e x}}{d e x + c e}\right ) + {\left (3 \, b^{2} c^{3} d^{3} x^{3} - 6 \, a^{2} c^{2} d^{4} + 4 \, {\left (5 \, b^{2} c^{4} d^{2} + a b c^{2} d^{4} - 4 \, a^{2} d^{6}\right )} x^{2} + 3 \, {\left (5 \, b^{2} c^{5} d - 8 \, a^{2} c d^{5}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{3 \, {\left (c^{3} d^{6} e^{2} x^{3} + 2 \, c^{4} d^{5} e^{2} x^{2} + c^{5} d^{4} e^{2} x\right )}}\right ] \] Input:

integrate((b*x^2+a)^2/(e*x)^(3/2)/(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

[1/6*(15*(b^2*c^4*d^2*x^3 + 2*b^2*c^5*d*x^2 + b^2*c^6*x)*sqrt(d*e)*log(2*d 
*e*x + c*e - 2*sqrt(d*e)*sqrt(d*x + c)*sqrt(e*x)) + 2*(3*b^2*c^3*d^3*x^3 - 
 6*a^2*c^2*d^4 + 4*(5*b^2*c^4*d^2 + a*b*c^2*d^4 - 4*a^2*d^6)*x^2 + 3*(5*b^ 
2*c^5*d - 8*a^2*c*d^5)*x)*sqrt(d*x + c)*sqrt(e*x))/(c^3*d^6*e^2*x^3 + 2*c^ 
4*d^5*e^2*x^2 + c^5*d^4*e^2*x), 1/3*(15*(b^2*c^4*d^2*x^3 + 2*b^2*c^5*d*x^2 
 + b^2*c^6*x)*sqrt(-d*e)*arctan(sqrt(-d*e)*sqrt(d*x + c)*sqrt(e*x)/(d*e*x 
+ c*e)) + (3*b^2*c^3*d^3*x^3 - 6*a^2*c^2*d^4 + 4*(5*b^2*c^4*d^2 + a*b*c^2* 
d^4 - 4*a^2*d^6)*x^2 + 3*(5*b^2*c^5*d - 8*a^2*c*d^5)*x)*sqrt(d*x + c)*sqrt 
(e*x))/(c^3*d^6*e^2*x^3 + 2*c^4*d^5*e^2*x^2 + c^5*d^4*e^2*x)]
 

Sympy [F]

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

integrate((b*x**2+a)**2/(e*x)**(3/2)/(d*x+c)**(5/2),x)
 

Output:

Integral((a + b*x**2)**2/((e*x)**(3/2)*(c + d*x)**(5/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x^2+a)^2/(e*x)^(3/2)/(d*x+c)^(5/2),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. 455 vs. \(2 (177) = 354\).

Time = 0.24 (sec) , antiderivative size = 455, normalized size of antiderivative = 2.18 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {\frac {6 \, \sqrt {d x + c} {\left (\frac {{\left (d x + c\right )} b^{2}}{d^{2} {\left | d \right |}} - \frac {b^{2} c^{4} d^{6} + 2 \, a^{2} d^{10}}{c^{3} d^{8} {\left | d \right |}}\right )}}{\sqrt {{\left (d x + c\right )} d e - c d e}} + \frac {15 \, b^{2} c \log \left ({\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2}\right )}{\sqrt {d e} d^{2} {\left | d \right |}} + \frac {8 \, {\left (7 \, b^{2} c^{6} d^{2} e^{3} + 2 \, a b c^{4} d^{4} e^{3} - 5 \, a^{2} c^{2} d^{6} e^{3} + 12 \, {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2} b^{2} c^{5} d e^{2} - 12 \, {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2} a^{2} c d^{5} e^{2} + 9 \, {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{4} b^{2} c^{4} e + 6 \, {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{4} a b c^{2} d^{2} e - 3 \, {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{4} a^{2} d^{4} e\right )}}{{\left (c d e + {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2}\right )}^{3} \sqrt {d e} c^{2} d {\left | d \right |}}}{6 \, e} \] Input:

integrate((b*x^2+a)^2/(e*x)^(3/2)/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

1/6*(6*sqrt(d*x + c)*((d*x + c)*b^2/(d^2*abs(d)) - (b^2*c^4*d^6 + 2*a^2*d^ 
10)/(c^3*d^8*abs(d)))/sqrt((d*x + c)*d*e - c*d*e) + 15*b^2*c*log((sqrt(d*e 
)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2)/(sqrt(d*e)*d^2*abs(d)) + 
 8*(7*b^2*c^6*d^2*e^3 + 2*a*b*c^4*d^4*e^3 - 5*a^2*c^2*d^6*e^3 + 12*(sqrt(d 
*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2*b^2*c^5*d*e^2 - 12*(sqr 
t(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2*a^2*c*d^5*e^2 + 9*(s 
qrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*b^2*c^4*e + 6*(sqr 
t(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*a*b*c^2*d^2*e - 3*(s 
qrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*a^2*d^4*e)/((c*d*e 
 + (sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2)^3*sqrt(d*e)* 
c^2*d*abs(d)))/e
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} (c+d x)^{5/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int((a + b*x^2)^2/((e*x)^(3/2)*(c + d*x)^(5/2)),x)
 

Output:

int((a + b*x^2)^2/((e*x)^(3/2)*(c + d*x)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {\sqrt {e}\, \left (-120 \sqrt {d}\, \sqrt {d x +c}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{5} x -120 \sqrt {d}\, \sqrt {d x +c}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{4} d \,x^{2}+128 \sqrt {d}\, \sqrt {d x +c}\, a^{2} c \,d^{4} x +128 \sqrt {d}\, \sqrt {d x +c}\, a^{2} d^{5} x^{2}+64 \sqrt {d}\, \sqrt {d x +c}\, a b \,c^{3} d^{2} x +64 \sqrt {d}\, \sqrt {d x +c}\, a b \,c^{2} d^{3} x^{2}+35 \sqrt {d}\, \sqrt {d x +c}\, b^{2} c^{5} x +35 \sqrt {d}\, \sqrt {d x +c}\, b^{2} c^{4} d \,x^{2}-48 \sqrt {x}\, a^{2} c^{2} d^{4}-192 \sqrt {x}\, a^{2} c \,d^{5} x -128 \sqrt {x}\, a^{2} d^{6} x^{2}+32 \sqrt {x}\, a b \,c^{2} d^{4} x^{2}+120 \sqrt {x}\, b^{2} c^{5} d x +160 \sqrt {x}\, b^{2} c^{4} d^{2} x^{2}+24 \sqrt {x}\, b^{2} c^{3} d^{3} x^{3}\right )}{24 \sqrt {d x +c}\, c^{3} d^{4} e^{2} x \left (d x +c \right )} \] Input:

int((b*x^2+a)^2/(e*x)^(3/2)/(d*x+c)^(5/2),x)
 

Output:

(sqrt(e)*( - 120*sqrt(d)*sqrt(c + d*x)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d 
))/sqrt(c))*b**2*c**5*x - 120*sqrt(d)*sqrt(c + d*x)*log((sqrt(c + d*x) + s 
qrt(x)*sqrt(d))/sqrt(c))*b**2*c**4*d*x**2 + 128*sqrt(d)*sqrt(c + d*x)*a**2 
*c*d**4*x + 128*sqrt(d)*sqrt(c + d*x)*a**2*d**5*x**2 + 64*sqrt(d)*sqrt(c + 
 d*x)*a*b*c**3*d**2*x + 64*sqrt(d)*sqrt(c + d*x)*a*b*c**2*d**3*x**2 + 35*s 
qrt(d)*sqrt(c + d*x)*b**2*c**5*x + 35*sqrt(d)*sqrt(c + d*x)*b**2*c**4*d*x* 
*2 - 48*sqrt(x)*a**2*c**2*d**4 - 192*sqrt(x)*a**2*c*d**5*x - 128*sqrt(x)*a 
**2*d**6*x**2 + 32*sqrt(x)*a*b*c**2*d**4*x**2 + 120*sqrt(x)*b**2*c**5*d*x 
+ 160*sqrt(x)*b**2*c**4*d**2*x**2 + 24*sqrt(x)*b**2*c**3*d**3*x**3))/(24*s 
qrt(c + d*x)*c**3*d**4*e**2*x*(c + d*x))