\(\int \frac {x^5}{(c+d x)^2 (a+b x^2)^{3/2}} \, dx\) [1238]

Optimal result

Integrand size = 22, antiderivative size = 210 \[ \int \frac {x^5}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {a^2 \left (b c^2-a d^2-2 b c d x\right )}{b^2 \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2}}{b^2 d^2}+\frac {c^5 \sqrt {a+b x^2}}{d^2 \left (b c^2+a d^2\right )^2 (c+d x)}-\frac {2 c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2} d^3}-\frac {c^4 \left (2 b c^2+5 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^3 \left (b c^2+a d^2\right )^{5/2}} \] Output:

-a^2*(-2*b*c*d*x-a*d^2+b*c^2)/b^2/(a*d^2+b*c^2)^2/(b*x^2+a)^(1/2)+(b*x^2+a 
)^(1/2)/b^2/d^2+c^5*(b*x^2+a)^(1/2)/d^2/(a*d^2+b*c^2)^2/(d*x+c)-2*c*arctan 
h(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(3/2)/d^3-c^4*(5*a*d^2+2*b*c^2)*arctanh((-b 
*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^3/(a*d^2+b*c^2)^(5/2)
 

Mathematica [A] (verified)

Time = 1.99 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.11 \[ \int \frac {x^5}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {d \left (2 a^3 d^4 (c+d x)+a^2 b d^2 (c+d x)^3+b^3 c^4 x^2 (2 c+d x)+a b^2 c^2 \left (2 c^3+c^2 d x+2 c d^2 x^2+2 d^3 x^3\right )\right )}{b^2 \left (b c^2+a d^2\right )^2 (c+d x) \sqrt {a+b x^2}}-\frac {2 c^4 \left (2 b c^2+5 a d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}}+\frac {2 c \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}}}{d^3} \] Input:

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

Output:

((d*(2*a^3*d^4*(c + d*x) + a^2*b*d^2*(c + d*x)^3 + b^3*c^4*x^2*(2*c + d*x) 
 + a*b^2*c^2*(2*c^3 + c^2*d*x + 2*c*d^2*x^2 + 2*d^3*x^3)))/(b^2*(b*c^2 + a 
*d^2)^2*(c + d*x)*Sqrt[a + b*x^2]) - (2*c^4*(2*b*c^2 + 5*a*d^2)*ArcTan[(Sq 
rt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/(-(b*c^2) - 
a*d^2)^(5/2) + (2*c*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/b^(3/2))/d^3
 

Rubi [A] (verified)

Time = 1.65 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.27, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {601, 2182, 2185, 27, 719, 224, 219, 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.

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

\(\Big \downarrow \) 601

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {\frac {a c \left (\frac {b c^3 \left (5 a d^2+2 b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d \left (a d^2+b c^2\right )}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )}{\sqrt {b} d}\right )}{b d^2}+\frac {a \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b^2 d^2}}{a d^2+b c^2}-\frac {a c^5 \sqrt {a+b x^2}}{d^2 (c+d x) \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (-a d^2+b c^2-2 b c d x\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 488

\(\displaystyle -\frac {-\frac {\frac {a c \left (-\frac {b c^3 \left (5 a d^2+2 b c^2\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d \left (a d^2+b c^2\right )}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )}{\sqrt {b} d}\right )}{b d^2}+\frac {a \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b^2 d^2}}{a d^2+b c^2}-\frac {a c^5 \sqrt {a+b x^2}}{d^2 (c+d x) \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (-a d^2+b c^2-2 b c d x\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

-((a^2*(b*c^2 - a*d^2 - 2*b*c*d*x))/(b^2*(b*c^2 + a*d^2)^2*Sqrt[a + b*x^2] 
)) - (-((a*c^5*Sqrt[a + b*x^2])/(d^2*(b*c^2 + a*d^2)^2*(c + d*x))) - ((a*( 
b*c^2 + a*d^2)*Sqrt[a + b*x^2])/(b^2*d^2) + (a*c*((-2*(b*c^2 + a*d^2)*ArcT 
anh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d) - (b*c^3*(2*b*c^2 + 5*a*d^2) 
*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(d*(b*c^2 + 
 a*d^2)^(3/2))))/(b*d^2))/(b*c^2 + a*d^2))/a
 

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 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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 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 719

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

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(640\) vs. \(2(192)=384\).

Time = 0.53 (sec) , antiderivative size = 641, normalized size of antiderivative = 3.05

method	result	size
risch	\(\frac {\sqrt {b \,x^{2}+a}}{b^{2} d^{2}}-\frac {\frac {2 c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {d^{2} a^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{2 \left (\sqrt {-a b}\, d +b c \right )^{2} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d^{2} a^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{2 \left (\sqrt {-a b}\, d -b c \right )^{2} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {b^{2} c^{5} \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3} \left (\sqrt {-a b}\, d +b c \right ) \left (\sqrt {-a b}\, d -b c \right )}+\frac {b^{3} c^{4} \left (5 a \,d^{2}+3 b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \left (\sqrt {-a b}\, d +b c \right )^{2} \left (\sqrt {-a b}\, d -b c \right )^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{b \,d^{2}}\)	\(641\)
default	\(\frac {\frac {x^{2}}{b \sqrt {b \,x^{2}+a}}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}}{d^{2}}-\frac {4 c^{3} x}{d^{5} \sqrt {b \,x^{2}+a}\, a}-\frac {2 c \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{d^{3}}-\frac {3 c^{2}}{d^{4} b \sqrt {b \,x^{2}+a}}+\frac {5 c^{4} \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{6}}-\frac {c^{5} \left (-\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {3 b c d \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{a \,d^{2}+b \,c^{2}}-\frac {4 b \,d^{2} \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{7}}\)	\(957\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (192) = 384\).

Time = 38.26 (sec) , antiderivative size = 3321, normalized size of antiderivative = 15.81 \[ \int \frac {x^5}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (192) = 384\).

Time = 0.09 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.46 \[ \int \frac {x^5}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {3 \, b^{2} c^{7} x}{\sqrt {b x^{2} + a} a b^{2} c^{4} d^{5} + 2 \, \sqrt {b x^{2} + a} a^{2} b c^{2} d^{7} + \sqrt {b x^{2} + a} a^{3} d^{9}} - \frac {3 \, b c^{6}}{\sqrt {b x^{2} + a} b^{2} c^{4} d^{4} + 2 \, \sqrt {b x^{2} + a} a b c^{2} d^{6} + \sqrt {b x^{2} + a} a^{2} d^{8}} + \frac {7 \, b c^{5} x}{\sqrt {b x^{2} + a} a b c^{2} d^{5} + \sqrt {b x^{2} + a} a^{2} d^{7}} + \frac {c^{5}}{\sqrt {b x^{2} + a} b c^{2} d^{5} x + \sqrt {b x^{2} + a} a d^{7} x + \sqrt {b x^{2} + a} b c^{3} d^{4} + \sqrt {b x^{2} + a} a c d^{6}} + \frac {5 \, c^{4}}{\sqrt {b x^{2} + a} b c^{2} d^{4} + \sqrt {b x^{2} + a} a d^{6}} + \frac {x^{2}}{\sqrt {b x^{2} + a} b d^{2}} - \frac {4 \, c^{3} x}{\sqrt {b x^{2} + a} a d^{5}} + \frac {2 \, c x}{\sqrt {b x^{2} + a} b d^{3}} - \frac {2 \, c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}} d^{3}} - \frac {3 \, b c^{6} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} d^{8}} + \frac {5 \, c^{4} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{6}} - \frac {3 \, c^{2}}{\sqrt {b x^{2} + a} b d^{4}} + \frac {2 \, a}{\sqrt {b x^{2} + a} b^{2} d^{2}} \] Input:

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

Output:

-3*b^2*c^7*x/(sqrt(b*x^2 + a)*a*b^2*c^4*d^5 + 2*sqrt(b*x^2 + a)*a^2*b*c^2* 
d^7 + sqrt(b*x^2 + a)*a^3*d^9) - 3*b*c^6/(sqrt(b*x^2 + a)*b^2*c^4*d^4 + 2* 
sqrt(b*x^2 + a)*a*b*c^2*d^6 + sqrt(b*x^2 + a)*a^2*d^8) + 7*b*c^5*x/(sqrt(b 
*x^2 + a)*a*b*c^2*d^5 + sqrt(b*x^2 + a)*a^2*d^7) + c^5/(sqrt(b*x^2 + a)*b* 
c^2*d^5*x + sqrt(b*x^2 + a)*a*d^7*x + sqrt(b*x^2 + a)*b*c^3*d^4 + sqrt(b*x 
^2 + a)*a*c*d^6) + 5*c^4/(sqrt(b*x^2 + a)*b*c^2*d^4 + sqrt(b*x^2 + a)*a*d^ 
6) + x^2/(sqrt(b*x^2 + a)*b*d^2) - 4*c^3*x/(sqrt(b*x^2 + a)*a*d^5) + 2*c*x 
/(sqrt(b*x^2 + a)*b*d^3) - 2*c*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*d^3) - 3*b* 
c^6*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c))) 
/((a + b*c^2/d^2)^(5/2)*d^8) + 5*c^4*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c) 
) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(3/2)*d^6) - 3*c^2/(sqr 
t(b*x^2 + a)*b*d^4) + 2*a/(sqrt(b*x^2 + a)*b^2*d^2)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 2174, normalized size of antiderivative = 10.35 \[ \int \frac {x^5}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(5*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**2*c**5*d**2 + 5*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**2*c**4*d**3*x + 2*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*b 
**3*c**7 + 2*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*b**3*c**6*d*x + 5*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*b**3*c**5*d**2*x**2 + 5*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*b**3*c**4*d**3*x**3 + 2*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)*b**4*c**7*x**2 + 2*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)*b**4*c**6*d*x* 
*3 - 5*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b**2*c**5*d**2 - 5*sqrt(a*d 
**2 + b*c**2)*log(c + d*x)*a**2*b**2*c**4*d**3*x - 2*sqrt(a*d**2 + b*c**2) 
*log(c + d*x)*a*b**3*c**7 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**3*c* 
*6*d*x - 5*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**3*c**5*d**2*x**2 - 5*sq 
rt(a*d**2 + b*c**2)*log(c + d*x)*a*b**3*c**4*d**3*x**3 - 2*sqrt(a*d**2 + b 
*c**2)*log(c + d*x)*b**4*c**7*x**2 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)* 
b**4*c**6*d*x**3 + 2*sqrt(a + b*x**2)*a**4*c*d**7 + 2*sqrt(a + b*x**2)*a** 
4*d**8*x + 3*sqrt(a + b*x**2)*a**3*b*c**3*d**5 + 5*sqrt(a + b*x**2)*a**3*b 
*c**2*d**6*x + 3*sqrt(a + b*x**2)*a**3*b*c*d**7*x**2 + sqrt(a + b*x**2)...