\(\int \frac {x^2}{(c+d x)^3 (a+b x^2)^{5/2}} \, dx\) [1301]

Optimal result

Integrand size = 22, antiderivative size = 335 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {-a d \left (3 b c^2-a d^2\right )-b c \left (b c^2-3 a d^2\right ) x}{3 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^{3/2}}+\frac {3 a d \left (3 b^2 c^4-8 a b c^2 d^2+a^2 d^4\right )+b c \left (b^2 c^4-20 a b c^2 d^2+15 a^2 d^4\right ) x}{3 a \left (b c^2+a d^2\right )^4 \sqrt {a+b x^2}}-\frac {c^2 d^3 \sqrt {a+b x^2}}{2 \left (b c^2+a d^2\right )^3 (c+d x)^2}-\frac {c d^3 \left (7 b c^2-4 a d^2\right ) \sqrt {a+b x^2}}{2 \left (b c^2+a d^2\right )^4 (c+d x)}-\frac {d^2 \left (12 b^2 c^4-21 a b c^2 d^2+2 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 \left (b c^2+a d^2\right )^{9/2}} \] Output:

1/3*(-a*d*(-a*d^2+3*b*c^2)-b*c*(-3*a*d^2+b*c^2)*x)/(a*d^2+b*c^2)^3/(b*x^2+ 
a)^(3/2)+1/3*(3*a*d*(a^2*d^4-8*a*b*c^2*d^2+3*b^2*c^4)+b*c*(15*a^2*d^4-20*a 
*b*c^2*d^2+b^2*c^4)*x)/a/(a*d^2+b*c^2)^4/(b*x^2+a)^(1/2)-1/2*c^2*d^3*(b*x^ 
2+a)^(1/2)/(a*d^2+b*c^2)^3/(d*x+c)^2-1/2*c*d^3*(-4*a*d^2+7*b*c^2)*(b*x^2+a 
)^(1/2)/(a*d^2+b*c^2)^4/(d*x+c)-1/2*d^2*(2*a^2*d^4-21*a*b*c^2*d^2+12*b^2*c 
^4)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/(a*d^2+b*c^2 
)^(9/2)
 

Mathematica [A] (verified)

Time = 10.79 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.01 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {1}{6} \left (\frac {\sqrt {a+b x^2} \left (-\frac {3 c^2 d^3 \left (b c^2+a d^2\right )}{(c+d x)^2}+\frac {3 \left (-7 b c^3 d^3+4 a c d^5\right )}{c+d x}+\frac {2 \left (3 a^3 d^5+b^3 c^5 x+a b^2 c^3 d (9 c-20 d x)-3 a^2 b c d^3 (8 c-5 d x)\right )}{a \left (a+b x^2\right )}-\frac {2 \left (b c^2+a d^2\right ) \left (-a^2 d^3+b^2 c^3 x+3 a b c d (c-d x)\right )}{\left (a+b x^2\right )^2}\right )}{\left (b c^2+a d^2\right )^4}+\frac {3 d^2 \left (12 b^2 c^4-21 a b c^2 d^2+2 a^2 d^4\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{9/2}}-\frac {3 d^2 \left (12 b^2 c^4-21 a b c^2 d^2+2 a^2 d^4\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{9/2}}\right ) \] Input:

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

Output:

((Sqrt[a + b*x^2]*((-3*c^2*d^3*(b*c^2 + a*d^2))/(c + d*x)^2 + (3*(-7*b*c^3 
*d^3 + 4*a*c*d^5))/(c + d*x) + (2*(3*a^3*d^5 + b^3*c^5*x + a*b^2*c^3*d*(9* 
c - 20*d*x) - 3*a^2*b*c*d^3*(8*c - 5*d*x)))/(a*(a + b*x^2)) - (2*(b*c^2 + 
a*d^2)*(-(a^2*d^3) + b^2*c^3*x + 3*a*b*c*d*(c - d*x)))/(a + b*x^2)^2))/(b* 
c^2 + a*d^2)^4 + (3*d^2*(12*b^2*c^4 - 21*a*b*c^2*d^2 + 2*a^2*d^4)*Log[c + 
d*x])/(b*c^2 + a*d^2)^(9/2) - (3*d^2*(12*b^2*c^4 - 21*a*b*c^2*d^2 + 2*a^2* 
d^4)*Log[a*d - b*c*x + Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2]])/(b*c^2 + a*d^ 
2)^(9/2))/6
 

Rubi [A] (verified)

Time = 2.93 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {601, 25, 2178, 27, 2182, 25, 27, 679, 488, 219}

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.

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

\(\Big \downarrow \) 601

\(\displaystyle -\frac {\int -\frac {\frac {a b \left (b c^2-3 a d^2\right ) c^4}{\left (b c^2+a d^2\right )^3}-\frac {6 a b d x c^3}{\left (b c^2+a d^2\right )^2}-\frac {2 a b d^3 \left (b c^2-3 a d^2\right ) x^3 c}{\left (b c^2+a d^2\right )^3}-\frac {3 a d^2 \left (2 b^2 c^4-3 a b d^2 c^2-a^2 d^4\right ) x^2}{\left (b c^2+a d^2\right )^3}}{(c+d x)^3 \left (b x^2+a\right )^{3/2}}dx}{3 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\frac {a b \left (b c^2-3 a d^2\right ) c^4}{\left (b c^2+a d^2\right )^3}-\frac {6 a b d x c^3}{\left (b c^2+a d^2\right )^2}-\frac {2 a b d^3 \left (b c^2-3 a d^2\right ) x^3 c}{\left (b c^2+a d^2\right )^3}-\frac {3 a d^2 \left (2 b^2 c^4-3 a b d^2 c^2-a^2 d^4\right ) x^2}{\left (b c^2+a d^2\right )^3}}{(c+d x)^3 \left (b x^2+a\right )^{3/2}}dx}{3 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2178

\(\displaystyle \frac {\frac {b c x \left (15 a^2 d^4-20 a b c^2 d^2+b^2 c^4\right )+3 a d \left (a^2 d^4-8 a b c^2 d^2+3 b^2 c^4\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {\int -\frac {3 \left (\frac {6 a^2 b^2 d^2 \left (b c^2-a d^2\right ) c^4}{\left (b c^2+a d^2\right )^4}+\frac {8 a^2 b^2 d^3 \left (b c^2-2 a d^2\right ) x c^3}{\left (b c^2+a d^2\right )^4}+\frac {a^2 b d^4 \left (3 b^2 c^4-8 a b d^2 c^2+a^2 d^4\right ) x^2}{\left (b c^2+a d^2\right )^4}\right )}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a b}}{3 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \int \frac {\frac {6 a^2 b^2 d^2 \left (b c^2-a d^2\right ) c^4}{\left (b c^2+a d^2\right )^4}+\frac {8 a^2 b^2 d^3 \left (b c^2-2 a d^2\right ) x c^3}{\left (b c^2+a d^2\right )^4}+\frac {a^2 b d^4 \left (3 b^2 c^4-8 a b d^2 c^2+a^2 d^4\right ) x^2}{\left (b c^2+a d^2\right )^4}}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a b}+\frac {b c x \left (15 a^2 d^4-20 a b c^2 d^2+b^2 c^4\right )+3 a d \left (a^2 d^4-8 a b c^2 d^2+3 b^2 c^4\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {\frac {3 \left (-\frac {\int -\frac {a^2 b d^2 \left (2 c \left (6 b^2 c^4-7 a b d^2 c^2-a^2 d^4\right )+d \left (5 b^2 c^4-17 a b d^2 c^2+2 a^2 d^4\right ) x\right )}{\left (b c^2+a d^2\right )^3 (c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {a^2 b c^2 d^3 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}+\frac {b c x \left (15 a^2 d^4-20 a b c^2 d^2+b^2 c^4\right )+3 a d \left (a^2 d^4-8 a b c^2 d^2+3 b^2 c^4\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {a^2 b d^2 \left (2 c \left (6 b^2 c^4-7 a b d^2 c^2-a^2 d^4\right )+d \left (5 b^2 c^4-17 a b d^2 c^2+2 a^2 d^4\right ) x\right )}{\left (b c^2+a d^2\right )^3 (c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {a^2 b c^2 d^3 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}+\frac {b c x \left (15 a^2 d^4-20 a b c^2 d^2+b^2 c^4\right )+3 a d \left (a^2 d^4-8 a b c^2 d^2+3 b^2 c^4\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \left (\frac {a^2 b d^2 \int \frac {2 c \left (6 b^2 c^4-7 a b d^2 c^2-a^2 d^4\right )+d \left (5 b^2 c^4-17 a b d^2 c^2+2 a^2 d^4\right ) x}{(c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )^4}-\frac {a^2 b c^2 d^3 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}+\frac {b c x \left (15 a^2 d^4-20 a b c^2 d^2+b^2 c^4\right )+3 a d \left (a^2 d^4-8 a b c^2 d^2+3 b^2 c^4\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {\frac {3 \left (\frac {a^2 b d^2 \left (\left (2 a^2 d^4-21 a b c^2 d^2+12 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx-\frac {c d \sqrt {a+b x^2} \left (7 b c^2-4 a d^2\right )}{c+d x}\right )}{2 \left (a d^2+b c^2\right )^4}-\frac {a^2 b c^2 d^3 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}+\frac {b c x \left (15 a^2 d^4-20 a b c^2 d^2+b^2 c^4\right )+3 a d \left (a^2 d^4-8 a b c^2 d^2+3 b^2 c^4\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {3 \left (\frac {a^2 b d^2 \left (-\left (2 a^2 d^4-21 a b c^2 d^2+12 b^2 c^4\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}-\frac {c d \sqrt {a+b x^2} \left (7 b c^2-4 a d^2\right )}{c+d x}\right )}{2 \left (a d^2+b c^2\right )^4}-\frac {a^2 b c^2 d^3 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}+\frac {b c x \left (15 a^2 d^4-20 a b c^2 d^2+b^2 c^4\right )+3 a d \left (a^2 d^4-8 a b c^2 d^2+3 b^2 c^4\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {3 \left (\frac {a^2 b d^2 \left (-\frac {\left (2 a^2 d^4-21 a b c^2 d^2+12 b^2 c^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a d^2+b c^2}}-\frac {c d \sqrt {a+b x^2} \left (7 b c^2-4 a d^2\right )}{c+d x}\right )}{2 \left (a d^2+b c^2\right )^4}-\frac {a^2 b c^2 d^3 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}+\frac {b c x \left (15 a^2 d^4-20 a b c^2 d^2+b^2 c^4\right )+3 a d \left (a^2 d^4-8 a b c^2 d^2+3 b^2 c^4\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

Input:

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

Output:

-1/3*(a*d*(3*b*c^2 - a*d^2) + b*c*(b*c^2 - 3*a*d^2)*x)/((b*c^2 + a*d^2)^3* 
(a + b*x^2)^(3/2)) + ((3*a*d*(3*b^2*c^4 - 8*a*b*c^2*d^2 + a^2*d^4) + b*c*( 
b^2*c^4 - 20*a*b*c^2*d^2 + 15*a^2*d^4)*x)/((b*c^2 + a*d^2)^4*Sqrt[a + b*x^ 
2]) + (3*(-1/2*(a^2*b*c^2*d^3*Sqrt[a + b*x^2])/((b*c^2 + a*d^2)^3*(c + d*x 
)^2) + (a^2*b*d^2*(-((c*d*(7*b*c^2 - 4*a*d^2)*Sqrt[a + b*x^2])/(c + d*x)) 
- ((12*b^2*c^4 - 21*a*b*c^2*d^2 + 2*a^2*d^4)*ArcTanh[(a*d - b*c*x)/(Sqrt[b 
*c^2 + a*d^2]*Sqrt[a + b*x^2])])/Sqrt[b*c^2 + a*d^2]))/(2*(b*c^2 + a*d^2)^ 
4)))/(a*b))/(3*a)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 601

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c 
+ d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* 
(2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] 
 && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
 

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 2178

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po 
lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia 
lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + 
b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x 
)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 
2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

rule 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3106\) vs. \(2(311)=622\).

Time = 0.41 (sec) , antiderivative size = 3107, normalized size of antiderivative = 9.27

method	result	size
default	\(\text {Expression too large to display}\)	\(3107\)

Input:

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

Output:

1/d^3*(1/3/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^ 
2)^(3/2)+b*c*d/(a*d^2+b*c^2)*(2/3*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2) 
/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+ 
16/3*b/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)^2*(2*b*(x+c/d)-2*b*c/d)/(b*(x 
+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))+1/(a*d^2+b*c^2)*d^2*(1/( 
a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+2*b 
*c*d/(a*d^2+b*c^2)*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/ 
d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-1/(a*d^2+b*c^2) 
*d^2/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*( 
(a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^( 
1/2))/(x+c/d))))+c^2/d^5*(-1/2/(a*d^2+b*c^2)*d^2/(x+c/d)^2/(b*(x+c/d)^2-2* 
b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+7/2*b*c*d/(a*d^2+b*c^2)*(-1/(a*d^2+ 
b*c^2)*d^2/(x+c/d)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+5 
*b*c*d/(a*d^2+b*c^2)*(1/3/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+( 
a*d^2+b*c^2)/d^2)^(3/2)+b*c*d/(a*d^2+b*c^2)*(2/3*(2*b*(x+c/d)-2*b*c/d)/(4* 
b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c 
^2)/d^2)^(3/2)+16/3*b/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)^2*(2*b*(x+c/d) 
-2*b*c/d)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))+1/(a*d^2+ 
b*c^2)*d^2*(1/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2) 
/d^2)^(1/2)+2*b*c*d/(a*d^2+b*c^2)*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1344 vs. \(2 (310) = 620\).

Time = 1.45 (sec) , antiderivative size = 2714, normalized size of antiderivative = 8.10 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/12*(3*(12*a^3*b^2*c^6*d^2 - 21*a^4*b*c^4*d^4 + 2*a^5*c^2*d^6 + (12*a*b^ 
4*c^4*d^4 - 21*a^2*b^3*c^2*d^6 + 2*a^3*b^2*d^8)*x^6 + 2*(12*a*b^4*c^5*d^3 
- 21*a^2*b^3*c^3*d^5 + 2*a^3*b^2*c*d^7)*x^5 + (12*a*b^4*c^6*d^2 + 3*a^2*b^ 
3*c^4*d^4 - 40*a^3*b^2*c^2*d^6 + 4*a^4*b*d^8)*x^4 + 4*(12*a^2*b^3*c^5*d^3 
- 21*a^3*b^2*c^3*d^5 + 2*a^4*b*c*d^7)*x^3 + (24*a^2*b^3*c^6*d^2 - 30*a^3*b 
^2*c^4*d^4 - 17*a^4*b*c^2*d^6 + 2*a^5*d^8)*x^2 + 2*(12*a^3*b^2*c^5*d^3 - 2 
1*a^4*b*c^3*d^5 + 2*a^5*c*d^7)*x)*sqrt(b*c^2 + a*d^2)*log((2*a*b*c*d*x - a 
*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 - 2*sqrt(b*c^2 + a*d^2)*(b* 
c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) + 2*(12*a^2*b^3*c^8 
*d - 64*a^3*b^2*c^6*d^3 - 59*a^4*b*c^4*d^5 + 17*a^5*c^2*d^7 + (2*b^5*c^7*d 
^2 - 59*a*b^4*c^5*d^4 - 19*a^2*b^3*c^3*d^6 + 42*a^3*b^2*c*d^8)*x^5 + (4*b^ 
5*c^8*d - 82*a*b^4*c^6*d^3 - 65*a^2*b^3*c^4*d^5 + 27*a^3*b^2*c^2*d^7 + 6*a 
^4*b*d^9)*x^4 + 2*(b^5*c^9 - a*b^4*c^7*d^2 - 74*a^2*b^3*c^5*d^4 - 36*a^3*b 
^2*c^3*d^6 + 36*a^4*b*c*d^8)*x^3 + 2*(9*a*b^4*c^8*d - 69*a^2*b^3*c^6*d^3 - 
 56*a^3*b^2*c^4*d^5 + 26*a^4*b*c^2*d^7 + 4*a^5*d^9)*x^2 - (12*a^2*b^3*c^7* 
d^2 + 101*a^3*b^2*c^5*d^4 + 61*a^4*b*c^3*d^6 - 28*a^5*c*d^8)*x)*sqrt(b*x^2 
 + a))/(a^3*b^5*c^12 + 5*a^4*b^4*c^10*d^2 + 10*a^5*b^3*c^8*d^4 + 10*a^6*b^ 
2*c^6*d^6 + 5*a^7*b*c^4*d^8 + a^8*c^2*d^10 + (a*b^7*c^10*d^2 + 5*a^2*b^6*c 
^8*d^4 + 10*a^3*b^5*c^6*d^6 + 10*a^4*b^4*c^4*d^8 + 5*a^5*b^3*c^2*d^10 + a^ 
6*b^2*d^12)*x^6 + 2*(a*b^7*c^11*d + 5*a^2*b^6*c^9*d^3 + 10*a^3*b^5*c^7*...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1573 vs. \(2 (310) = 620\).

Time = 0.16 (sec) , antiderivative size = 1573, normalized size of antiderivative = 4.70 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

35/2*b^3*c^5*x/(sqrt(b*x^2 + a)*a*b^4*c^8 + 4*sqrt(b*x^2 + a)*a^2*b^3*c^6* 
d^2 + 6*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^4 + 4*sqrt(b*x^2 + a)*a^4*b*c^2*d^6 
+ sqrt(b*x^2 + a)*a^5*d^8) + 35/6*b^3*c^5*x/((b*x^2 + a)^(3/2)*a*b^3*c^6*d 
^2 + 3*(b*x^2 + a)^(3/2)*a^2*b^2*c^4*d^4 + 3*(b*x^2 + a)^(3/2)*a^3*b*c^2*d 
^6 + (b*x^2 + a)^(3/2)*a^4*d^8) + 35/3*b^3*c^5*x/(sqrt(b*x^2 + a)*a^2*b^3* 
c^6*d^2 + 3*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^4 + 3*sqrt(b*x^2 + a)*a^4*b*c^2* 
d^6 + sqrt(b*x^2 + a)*a^5*d^8) + 35/2*b^2*c^4/(sqrt(b*x^2 + a)*b^4*c^8/d + 
 4*sqrt(b*x^2 + a)*a*b^3*c^6*d + 6*sqrt(b*x^2 + a)*a^2*b^2*c^4*d^3 + 4*sqr 
t(b*x^2 + a)*a^3*b*c^2*d^5 + sqrt(b*x^2 + a)*a^4*d^7) + 35/6*b^2*c^4/((b*x 
^2 + a)^(3/2)*b^3*c^6*d + 3*(b*x^2 + a)^(3/2)*a*b^2*c^4*d^3 + 3*(b*x^2 + a 
)^(3/2)*a^2*b*c^2*d^5 + (b*x^2 + a)^(3/2)*a^3*d^7) - 25/2*b^2*c^3*x/(sqrt( 
b*x^2 + a)*a*b^3*c^6 + 3*sqrt(b*x^2 + a)*a^2*b^2*c^4*d^2 + 3*sqrt(b*x^2 + 
a)*a^3*b*c^2*d^4 + sqrt(b*x^2 + a)*a^4*d^6) - 53/6*b^2*c^3*x/((b*x^2 + a)^ 
(3/2)*a*b^2*c^4*d^2 + 2*(b*x^2 + a)^(3/2)*a^2*b*c^2*d^4 + (b*x^2 + a)^(3/2 
)*a^3*d^6) - 53/3*b^2*c^3*x/(sqrt(b*x^2 + a)*a^2*b^2*c^4*d^2 + 2*sqrt(b*x^ 
2 + a)*a^3*b*c^2*d^4 + sqrt(b*x^2 + a)*a^4*d^6) - 7/2*b*c^3/((b*x^2 + a)^( 
3/2)*b^2*c^4*d^2*x + 2*(b*x^2 + a)^(3/2)*a*b*c^2*d^4*x + (b*x^2 + a)^(3/2) 
*a^2*d^6*x + (b*x^2 + a)^(3/2)*b^2*c^5*d + 2*(b*x^2 + a)^(3/2)*a*b*c^3*d^3 
 + (b*x^2 + a)^(3/2)*a^2*c*d^5) - 25/2*b*c^2/(sqrt(b*x^2 + a)*b^3*c^6/d + 
3*sqrt(b*x^2 + a)*a*b^2*c^4*d + 3*sqrt(b*x^2 + a)*a^2*b*c^2*d^3 + sqrt(...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2232 vs. \(2 (310) = 620\).

Time = 0.28 (sec) , antiderivative size = 2232, normalized size of antiderivative = 6.66 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-(12*b^2*c^4*d^2 - 21*a*b*c^2*d^4 + 2*a^2*d^6)*arctan(((sqrt(b)*x - sqrt(b 
*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b^4*c^8 + 4*a*b^3*c^6*d^ 
2 + 6*a^2*b^2*c^4*d^4 + 4*a^3*b*c^2*d^6 + a^4*d^8)*sqrt(-b*c^2 - a*d^2)) + 
 1/3*((((b^17*c^29 - 8*a*b^16*c^27*d^2 - 159*a^2*b^15*c^25*d^4 - 920*a^3*b 
^14*c^23*d^6 - 2915*a^4*b^13*c^21*d^8 - 5808*a^5*b^12*c^19*d^10 - 7491*a^6 
*b^11*c^17*d^12 - 5808*a^7*b^10*c^15*d^14 - 1485*a^8*b^9*c^13*d^16 + 2200* 
a^9*b^8*c^11*d^18 + 3091*a^10*b^7*c^9*d^20 + 1992*a^11*b^6*c^7*d^22 + 751* 
a^12*b^5*c^5*d^24 + 160*a^13*b^4*c^3*d^26 + 15*a^14*b^3*c*d^28)*x/(a*b^17* 
c^32 + 16*a^2*b^16*c^30*d^2 + 120*a^3*b^15*c^28*d^4 + 560*a^4*b^14*c^26*d^ 
6 + 1820*a^5*b^13*c^24*d^8 + 4368*a^6*b^12*c^22*d^10 + 8008*a^7*b^11*c^20* 
d^12 + 11440*a^8*b^10*c^18*d^14 + 12870*a^9*b^9*c^16*d^16 + 11440*a^10*b^8 
*c^14*d^18 + 8008*a^11*b^7*c^12*d^20 + 4368*a^12*b^6*c^10*d^22 + 1820*a^13 
*b^5*c^8*d^24 + 560*a^14*b^4*c^6*d^26 + 120*a^15*b^3*c^4*d^28 + 16*a^16*b^ 
2*c^2*d^30 + a^17*b*d^32) + 3*(3*a*b^16*c^28*d + 28*a^2*b^15*c^26*d^3 + 10 
3*a^3*b^14*c^24*d^5 + 144*a^4*b^13*c^22*d^7 - 209*a^5*b^12*c^20*d^9 - 1364 
*a^6*b^11*c^18*d^11 - 3069*a^7*b^10*c^16*d^13 - 4224*a^8*b^9*c^14*d^15 - 3 
927*a^9*b^8*c^12*d^17 - 2508*a^10*b^7*c^10*d^19 - 1067*a^11*b^6*c^8*d^21 - 
 272*a^12*b^5*c^6*d^23 - 27*a^13*b^4*c^4*d^25 + 4*a^14*b^3*c^2*d^27 + a^15 
*b^2*d^29)/(a*b^17*c^32 + 16*a^2*b^16*c^30*d^2 + 120*a^3*b^15*c^28*d^4 + 5 
60*a^4*b^14*c^26*d^6 + 1820*a^5*b^13*c^24*d^8 + 4368*a^6*b^12*c^22*d^10...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 3326, normalized size of antiderivative = 9.93 \[ \int \frac {x^2}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(6*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d 
+ b*c*x)*a**5*c**2*d**6 + 12*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sq 
rt(a*d**2 + b*c**2) - a*d + b*c*x)*a**5*c*d**7*x + 6*sqrt(a*d**2 + b*c**2) 
*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**5*d**8*x**2 
- 63*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a* 
d + b*c*x)*a**4*b*c**4*d**4 - 126*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x** 
2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**4*b*c**3*d**5*x - 51*sqrt(a*d** 
2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**4 
*b*c**2*d**6*x**2 + 24*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d 
**2 + b*c**2) - a*d + b*c*x)*a**4*b*c*d**7*x**3 + 12*sqrt(a*d**2 + b*c**2) 
*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**4*b*d**8*x** 
4 + 36*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - 
a*d + b*c*x)*a**3*b**2*c**6*d**2 + 72*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b 
*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*b**2*c**5*d**3*x - 90*sqr 
t(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c* 
x)*a**3*b**2*c**4*d**4*x**2 - 252*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x** 
2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*b**2*c**3*d**5*x**3 - 120*sqr 
t(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c* 
x)*a**3*b**2*c**2*d**6*x**4 + 12*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2 
)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*b**2*c*d**7*x**5 + 6*sqrt(a...