\(\int \frac {x^5}{(c+d x)^3 (a+b x^2)^{5/2}} \, dx\) [1298]

Optimal result

Integrand size = 22, antiderivative size = 311 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d \left (3 b c^2-a d^2\right ) x\right )}{3 b \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^{3/2}}+\frac {a \left (6 b c^3 \left (b c^2-5 a d^2\right )-d \left (21 b^2 c^4-16 a b c^2 d^2-a^2 d^4\right ) x\right )}{3 b \left (b c^2+a d^2\right )^4 \sqrt {a+b x^2}}+\frac {c^5 \sqrt {a+b x^2}}{2 \left (b c^2+a d^2\right )^3 (c+d x)^2}+\frac {c^4 \left (b c^2-10 a d^2\right ) \sqrt {a+b x^2}}{2 \left (b c^2+a d^2\right )^4 (c+d x)}-\frac {5 a c^3 d \left (3 b c^2-4 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 \left (b c^2+a d^2\right )^{9/2}} \] Output:

-1/3*a^2*(c*(-3*a*d^2+b*c^2)-d*(-a*d^2+3*b*c^2)*x)/b/(a*d^2+b*c^2)^3/(b*x^ 
2+a)^(3/2)+1/3*a*(6*b*c^3*(-5*a*d^2+b*c^2)-d*(-a^2*d^4-16*a*b*c^2*d^2+21*b 
^2*c^4)*x)/b/(a*d^2+b*c^2)^4/(b*x^2+a)^(1/2)+1/2*c^5*(b*x^2+a)^(1/2)/(a*d^ 
2+b*c^2)^3/(d*x+c)^2+1/2*c^4*(-10*a*d^2+b*c^2)*(b*x^2+a)^(1/2)/(a*d^2+b*c^ 
2)^4/(d*x+c)-5/2*a*c^3*d*(-4*a*d^2+3*b*c^2)*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 = 10.80 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.97 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {1}{6} \left (\frac {\sqrt {a+b x^2} \left (\frac {3 c^5 \left (b c^2+a d^2\right )}{(c+d x)^2}+\frac {3 b c^6-30 a c^4 d^2}{c+d x}-\frac {2 a^2 \left (b c^2+a d^2\right ) \left (b c^2 (c-3 d x)+a d^2 (-3 c+d x)\right )}{b \left (a+b x^2\right )^2}+\frac {2 a \left (a^2 d^5 x+3 b^2 c^4 (2 c-7 d x)+2 a b c^2 d^2 (-15 c+8 d x)\right )}{b \left (a+b x^2\right )}\right )}{\left (b c^2+a d^2\right )^4}+\frac {15 a c^3 d \left (3 b c^2-4 a d^2\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{9/2}}+\frac {15 a c^3 d \left (-3 b c^2+4 a d^2\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{9/2}}\right ) \] Input:

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

Output:

((Sqrt[a + b*x^2]*((3*c^5*(b*c^2 + a*d^2))/(c + d*x)^2 + (3*b*c^6 - 30*a*c 
^4*d^2)/(c + d*x) - (2*a^2*(b*c^2 + a*d^2)*(b*c^2*(c - 3*d*x) + a*d^2*(-3* 
c + d*x)))/(b*(a + b*x^2)^2) + (2*a*(a^2*d^5*x + 3*b^2*c^4*(2*c - 7*d*x) + 
 2*a*b*c^2*d^2*(-15*c + 8*d*x)))/(b*(a + b*x^2))))/(b*c^2 + a*d^2)^4 + (15 
*a*c^3*d*(3*b*c^2 - 4*a*d^2)*Log[c + d*x])/(b*c^2 + a*d^2)^(9/2) + (15*a*c 
^3*d*(-3*b*c^2 + 4*a*d^2)*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 = 3.17 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {601, 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.

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

\(\Big \downarrow \) 601

\(\displaystyle -\frac {\int \frac {\frac {a^3 d \left (3 b c^2-a d^2\right ) c^3}{b \left (b c^2+a d^2\right )^3}+\frac {3 a^2 \left (b c^2-a d^2\right ) x c^2}{b \left (b c^2+a d^2\right )^2}-\frac {3 a^3 d^3 \left (5 b c^2+a d^2\right ) x^2 c}{b \left (b c^2+a d^2\right )^3}-\frac {a \left (3 b^3 c^6+9 a b^2 d^2 c^4+15 a^2 b d^4 c^2+a^3 d^6\right ) x^3}{b \left (b c^2+a d^2\right )^3}}{(c+d x)^3 \left (b x^2+a\right )^{3/2}}dx}{3 a}-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2178

\(\displaystyle -\frac {-\frac {\int \frac {3 \left (\frac {6 a^3 b d \left (b c^2-a d^2\right ) c^5}{\left (b c^2+a d^2\right )^4}+\frac {a^2 b \left (b^2 c^4+10 a b d^2 c^2-15 a^2 d^4\right ) x c^4}{\left (b c^2+a d^2\right )^4}+\frac {2 a^3 b d^3 \left (b c^2-5 a d^2\right ) x^2 c^3}{\left (b c^2+a d^2\right )^4}\right )}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a b}-\frac {a^2 \left (6 b c^3 \left (b c^2-5 a d^2\right )-d x \left (-a^2 d^4-16 a b c^2 d^2+21 b^2 c^4\right )\right )}{b \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 \int \frac {\frac {6 a^3 b d \left (b c^2-a d^2\right ) c^5}{\left (b c^2+a d^2\right )^4}+\frac {a^2 b \left (b^2 c^4+10 a b d^2 c^2-15 a^2 d^4\right ) x c^4}{\left (b c^2+a d^2\right )^4}+\frac {2 a^3 b d^3 \left (b c^2-5 a d^2\right ) x^2 c^3}{\left (b c^2+a d^2\right )^4}}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a b}-\frac {a^2 \left (6 b c^3 \left (b c^2-5 a d^2\right )-d x \left (-a^2 d^4-16 a b c^2 d^2+21 b^2 c^4\right )\right )}{b \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2182

\(\displaystyle -\frac {-\frac {3 \left (\frac {a^2 b c^5 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}-\frac {\int -\frac {a^2 b c^3 \left (2 a c d \left (7 b c^2-5 a d^2\right )+\left (b^2 c^4+5 a b d^2 c^2-20 a^2 d^4\right ) x\right )}{\left (b c^2+a d^2\right )^3 (c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}\right )}{a b}-\frac {a^2 \left (6 b c^3 \left (b c^2-5 a d^2\right )-d x \left (-a^2 d^4-16 a b c^2 d^2+21 b^2 c^4\right )\right )}{b \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {3 \left (\frac {\int \frac {a^2 b c^3 \left (2 a c d \left (7 b c^2-5 a d^2\right )+\left (b^2 c^4+5 a b d^2 c^2-20 a^2 d^4\right ) x\right )}{\left (b c^2+a d^2\right )^3 (c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}+\frac {a^2 b c^5 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a^2 \left (6 b c^3 \left (b c^2-5 a d^2\right )-d x \left (-a^2 d^4-16 a b c^2 d^2+21 b^2 c^4\right )\right )}{b \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 \left (\frac {a^2 b c^3 \int \frac {2 a c d \left (7 b c^2-5 a d^2\right )+\left (b^2 c^4+5 a b d^2 c^2-20 a^2 d^4\right ) x}{(c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )^4}+\frac {a^2 b c^5 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a^2 \left (6 b c^3 \left (b c^2-5 a d^2\right )-d x \left (-a^2 d^4-16 a b c^2 d^2+21 b^2 c^4\right )\right )}{b \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 679

\(\displaystyle -\frac {-\frac {3 \left (\frac {a^2 b c^3 \left (5 a d \left (3 b c^2-4 a d^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx+\frac {c \sqrt {a+b x^2} \left (b c^2-10 a d^2\right )}{c+d x}\right )}{2 \left (a d^2+b c^2\right )^4}+\frac {a^2 b c^5 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a^2 \left (6 b c^3 \left (b c^2-5 a d^2\right )-d x \left (-a^2 d^4-16 a b c^2 d^2+21 b^2 c^4\right )\right )}{b \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 488

\(\displaystyle -\frac {-\frac {3 \left (\frac {a^2 b c^3 \left (\frac {c \sqrt {a+b x^2} \left (b c^2-10 a d^2\right )}{c+d x}-5 a d \left (3 b c^2-4 a d^2\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}\right )}{2 \left (a d^2+b c^2\right )^4}+\frac {a^2 b c^5 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a^2 \left (6 b c^3 \left (b c^2-5 a d^2\right )-d x \left (-a^2 d^4-16 a b c^2 d^2+21 b^2 c^4\right )\right )}{b \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {3 \left (\frac {a^2 b c^3 \left (\frac {c \sqrt {a+b x^2} \left (b c^2-10 a d^2\right )}{c+d x}-\frac {5 a d \left (3 b c^2-4 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a d^2+b c^2}}\right )}{2 \left (a d^2+b c^2\right )^4}+\frac {a^2 b c^5 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a^2 \left (6 b c^3 \left (b c^2-5 a d^2\right )-d x \left (-a^2 d^4-16 a b c^2 d^2+21 b^2 c^4\right )\right )}{b \sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}-\frac {a^2 \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

Input:

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

Output:

-1/3*(a^2*(c*(b*c^2 - 3*a*d^2) - d*(3*b*c^2 - a*d^2)*x))/(b*(b*c^2 + a*d^2 
)^3*(a + b*x^2)^(3/2)) - (-((a^2*(6*b*c^3*(b*c^2 - 5*a*d^2) - d*(21*b^2*c^ 
4 - 16*a*b*c^2*d^2 - a^2*d^4)*x))/(b*(b*c^2 + a*d^2)^4*Sqrt[a + b*x^2])) - 
 (3*((a^2*b*c^5*Sqrt[a + b*x^2])/(2*(b*c^2 + a*d^2)^3*(c + d*x)^2) + (a^2* 
b*c^3*((c*(b*c^2 - 10*a*d^2)*Sqrt[a + b*x^2])/(c + d*x) - (5*a*d*(3*b*c^2 
- 4*a*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/S 
qrt[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. \(3226\) vs. \(2(287)=574\).

Time = 0.49 (sec) , antiderivative size = 3227, normalized size of antiderivative = 10.38

method	result	size
default	\(\text {Expression too large to display}\)	\(3227\)

Input:

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

Output:

1/d^3*(-1/2*x/b/(b*x^2+a)^(3/2)+1/2*a/b*(1/3*x/a/(b*x^2+a)^(3/2)+2/3/a^2/( 
b*x^2+a)^(1/2)*x))+6*c^2/d^5*(1/3*x/a/(b*x^2+a)^(3/2)+2/3/a^2/(b*x^2+a)^(1 
/2)*x)+c/d^4/b/(b*x^2+a)^(3/2)-10/d^6*c^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))))+5/d^7*c^4*(-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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1269 vs. \(2 (288) = 576\).

Time = 0.83 (sec) , antiderivative size = 2564, normalized size of antiderivative = 8.24 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/12*(15*(3*a^3*b^2*c^7*d - 4*a^4*b*c^5*d^3 + (3*a*b^4*c^5*d^3 - 4*a^2*b 
^3*c^3*d^5)*x^6 + 2*(3*a*b^4*c^6*d^2 - 4*a^2*b^3*c^4*d^4)*x^5 + (3*a*b^4*c 
^7*d + 2*a^2*b^3*c^5*d^3 - 8*a^3*b^2*c^3*d^5)*x^4 + 4*(3*a^2*b^3*c^6*d^2 - 
 4*a^3*b^2*c^4*d^4)*x^3 + (6*a^2*b^3*c^7*d - 5*a^3*b^2*c^5*d^3 - 4*a^4*b*c 
^3*d^5)*x^2 + 2*(3*a^3*b^2*c^6*d^2 - 4*a^4*b*c^4*d^4)*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*(16*a^2*b^3*c^9 - 67*a^3*b^2*c^7*d^2 - 77*a^4*b*c^5*d^4 + 6*a^5*c^ 
3*d^6 + (3*b^5*c^8*d - 69*a*b^4*c^6*d^3 - 40*a^2*b^3*c^4*d^5 + 34*a^3*b^2* 
c^2*d^7 + 2*a^4*b*d^9)*x^5 + (6*b^5*c^9 - 93*a*b^4*c^7*d^2 - 95*a^2*b^3*c^ 
5*d^4 + 8*a^3*b^2*c^3*d^6 + 4*a^4*b*c*d^8)*x^4 - 2*(6*a*b^4*c^8*d + 98*a^2 
*b^3*c^6*d^3 + 73*a^3*b^2*c^4*d^5 - 19*a^4*b*c^2*d^7)*x^3 + 2*(12*a*b^4*c^ 
9 - 76*a^2*b^3*c^7*d^2 - 80*a^3*b^2*c^5*d^4 + 11*a^4*b*c^3*d^6 + 3*a^5*c*d 
^8)*x^2 - (13*a^2*b^3*c^8*d + 119*a^3*b^2*c^6*d^3 + 94*a^4*b*c^4*d^5 - 12* 
a^5*c^2*d^7)*x)*sqrt(b*x^2 + a))/(a^2*b^6*c^12 + 5*a^3*b^5*c^10*d^2 + 10*a 
^4*b^4*c^8*d^4 + 10*a^5*b^3*c^6*d^6 + 5*a^6*b^2*c^4*d^8 + a^7*b*c^2*d^10 + 
 (b^8*c^10*d^2 + 5*a*b^7*c^8*d^4 + 10*a^2*b^6*c^6*d^6 + 10*a^3*b^5*c^4*d^8 
 + 5*a^4*b^4*c^2*d^10 + a^5*b^3*d^12)*x^6 + 2*(b^8*c^11*d + 5*a*b^7*c^9*d^ 
3 + 10*a^2*b^6*c^7*d^5 + 10*a^3*b^5*c^5*d^7 + 5*a^4*b^4*c^3*d^9 + a^5*b^3* 
c*d^11)*x^5 + (b^8*c^12 + 7*a*b^7*c^10*d^2 + 20*a^2*b^6*c^8*d^4 + 30*a^...
 

Sympy [F]

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

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

Output:

Integral(x**5/((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. 1719 vs. \(2 (288) = 576\).

Time = 0.21 (sec) , antiderivative size = 1719, normalized size of antiderivative = 5.53 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-35/2*b^3*c^8*x/(sqrt(b*x^2 + a)*a*b^4*c^8*d^3 + 4*sqrt(b*x^2 + a)*a^2*b^3 
*c^6*d^5 + 6*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^7 + 4*sqrt(b*x^2 + a)*a^4*b*c^2 
*d^9 + sqrt(b*x^2 + a)*a^5*d^11) - 35/6*b^3*c^8*x/((b*x^2 + a)^(3/2)*a*b^3 
*c^6*d^5 + 3*(b*x^2 + a)^(3/2)*a^2*b^2*c^4*d^7 + 3*(b*x^2 + a)^(3/2)*a^3*b 
*c^2*d^9 + (b*x^2 + a)^(3/2)*a^4*d^11) - 35/3*b^3*c^8*x/(sqrt(b*x^2 + a)*a 
^2*b^3*c^6*d^5 + 3*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^7 + 3*sqrt(b*x^2 + a)*a^4 
*b*c^2*d^9 + sqrt(b*x^2 + a)*a^5*d^11) - 35/2*b^2*c^7/(sqrt(b*x^2 + a)*b^4 
*c^8*d^2 + 4*sqrt(b*x^2 + a)*a*b^3*c^6*d^4 + 6*sqrt(b*x^2 + a)*a^2*b^2*c^4 
*d^6 + 4*sqrt(b*x^2 + a)*a^3*b*c^2*d^8 + sqrt(b*x^2 + a)*a^4*d^10) - 35/6* 
b^2*c^7/((b*x^2 + a)^(3/2)*b^3*c^6*d^4 + 3*(b*x^2 + a)^(3/2)*a*b^2*c^4*d^6 
 + 3*(b*x^2 + a)^(3/2)*a^2*b*c^2*d^8 + (b*x^2 + a)^(3/2)*a^3*d^10) + 55/2* 
b^2*c^6*x/(sqrt(b*x^2 + a)*a*b^3*c^6*d^3 + 3*sqrt(b*x^2 + a)*a^2*b^2*c^4*d 
^5 + 3*sqrt(b*x^2 + a)*a^3*b*c^2*d^7 + sqrt(b*x^2 + a)*a^4*d^9) + 83/6*b^2 
*c^6*x/((b*x^2 + a)^(3/2)*a*b^2*c^4*d^5 + 2*(b*x^2 + a)^(3/2)*a^2*b*c^2*d^ 
7 + (b*x^2 + a)^(3/2)*a^3*d^9) + 83/3*b^2*c^6*x/(sqrt(b*x^2 + a)*a^2*b^2*c 
^4*d^5 + 2*sqrt(b*x^2 + a)*a^3*b*c^2*d^7 + sqrt(b*x^2 + a)*a^4*d^9) + 7/2* 
b*c^6/((b*x^2 + a)^(3/2)*b^2*c^4*d^5*x + 2*(b*x^2 + a)^(3/2)*a*b*c^2*d^7*x 
 + (b*x^2 + a)^(3/2)*a^2*d^9*x + (b*x^2 + a)^(3/2)*b^2*c^5*d^4 + 2*(b*x^2 
+ a)^(3/2)*a*b*c^3*d^6 + (b*x^2 + a)^(3/2)*a^2*c*d^8) + 55/2*b*c^5/(sqrt(b 
*x^2 + a)*b^3*c^6*d^2 + 3*sqrt(b*x^2 + a)*a*b^2*c^4*d^4 + 3*sqrt(b*x^2 ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2182 vs. \(2 (288) = 576\).

Time = 0.29 (sec) , antiderivative size = 2182, normalized size of antiderivative = 7.02 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

5*(3*a*b*c^5*d - 4*a^2*c^3*d^3)*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*((((21*a^ 
2*b^15*c^28*d + 236*a^3*b^14*c^26*d^3 + 1193*a^4*b^13*c^24*d^5 + 3552*a^5* 
b^12*c^22*d^7 + 6809*a^6*b^11*c^20*d^9 + 8492*a^7*b^10*c^18*d^11 + 6237*a^ 
8*b^9*c^16*d^13 + 1056*a^9*b^8*c^14*d^15 - 3201*a^10*b^7*c^12*d^17 - 4092* 
a^11*b^6*c^10*d^19 - 2629*a^12*b^5*c^8*d^21 - 1024*a^13*b^4*c^6*d^23 - 237 
*a^14*b^3*c^4*d^25 - 28*a^15*b^2*c^2*d^27 - a^16*b*d^29)*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 + 182 
0*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) - 6*(a^2*b^15*c^29 + 7*a^3*b^14*c^27*d^2 + 6*a^4*b^13*c 
^25*d^4 - 110*a^5*b^12*c^23*d^6 - 605*a^6*b^11*c^21*d^8 - 1683*a^7*b^10*c^ 
19*d^10 - 3036*a^8*b^9*c^17*d^12 - 3828*a^9*b^8*c^15*d^14 - 3465*a^10*b^7* 
c^13*d^16 - 2255*a^11*b^6*c^11*d^18 - 1034*a^12*b^5*c^9*d^20 - 318*a^13*b^ 
4*c^7*d^22 - 59*a^14*b^3*c^5*d^24 - 5*a^15*b^2*c^3*d^26)/(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 +...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 2731, normalized size of antiderivative = 8.78 \[ \int \frac {x^5}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(60*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**5*d**3 + 120*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*x + 60*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**2 - 45*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**7*d - 90*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*x + 75*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**2 + 240*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**4*d**4*x**3 + 120*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**4 - 90*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**2*b**3*c**7*d*x**2 - 180*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**2*b**3*c**6*d**2*x**3 - 30*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**2*b**3*c**5*d**3*x**4 + 120*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**2*b**3*c**4*d**4*x**5 + 6 
0*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**2*b**3*c**3*d**5*x**6 - 45*sqrt(a*d**2 + b*c**2)*log( - s...