\(\int \frac {x^5}{(c+d x)^2 \sqrt {a+b x^2}} \, dx\) [1190]

Optimal result

Integrand size = 22, antiderivative size = 228 \[ \int \frac {x^5}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {\left (3 b c^2-a d^2\right ) \sqrt {a+b x^2}}{b^2 d^4}-\frac {c x \sqrt {a+b x^2}}{b d^3}+\frac {c^5 \sqrt {a+b x^2}}{d^4 \left (b c^2+a d^2\right ) (c+d x)}+\frac {\left (a+b x^2\right )^{3/2}}{3 b^2 d^2}-\frac {c \left (4 b c^2-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2} d^5}-\frac {c^4 \left (4 b c^2+5 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^5 \left (b c^2+a d^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.37 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.10 \[ \int \frac {x^5}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (-2 a^2 d^4 (c+d x)+a b d^2 \left (7 c^3+4 c^2 d x-2 c d^2 x^2+d^3 x^3\right )+b^2 c^2 \left (12 c^3+6 c^2 d x-2 c d^2 x^2+d^3 x^3\right )\right )}{b^2 \left (b c^2+a d^2\right ) (c+d x)}+\frac {6 c^4 \left (4 b c^2+5 a d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+\frac {3 \left (4 b c^3-a c d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}}}{3 d^5} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.92 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.39, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {603, 25, 2185, 2185, 27, 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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {\frac {\int \frac {2 \left (b \left (13 b c^2-2 a d^2\right ) \left (b c^2+a d^2\right ) x^2 d^4+3 a b c^2 \left (2 b c^2+a d^2\right ) d^4+b c \left (b c^2+a d^2\right )^2 x d^3\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^3}-5 c \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{3 b d^4}+\frac {\sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}{3 b d^4}}{a d^2+b c^2}+\frac {c^5 \sqrt {a+b x^2}}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {b \left (13 b c^2-2 a d^2\right ) \left (b c^2+a d^2\right ) x^2 d^4+3 a b c^2 \left (2 b c^2+a d^2\right ) d^4+b c \left (b c^2+a d^2\right )^2 x d^3}{(c+d x) \sqrt {b x^2+a}}dx}{b d^3}-5 c \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{3 b d^4}+\frac {\sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}{3 b d^4}}{a d^2+b c^2}+\frac {c^5 \sqrt {a+b x^2}}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 b^2 c d^5 \left (a c d \left (2 b c^2+a d^2\right )-\left (4 b c^2-a d^2\right ) \left (b c^2+a d^2\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{b d^2}+d^3 \sqrt {a+b x^2} \left (13 b c^2-2 a d^2\right ) \left (a d^2+b c^2\right )}{b d^3}-5 c \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{3 b d^4}+\frac {\sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}{3 b d^4}}{a d^2+b c^2}+\frac {c^5 \sqrt {a+b x^2}}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 b c d^3 \int \frac {a c d \left (2 b c^2+a d^2\right )-\left (4 b c^2-a d^2\right ) \left (b c^2+a d^2\right ) x}{(c+d x) \sqrt {b x^2+a}}dx+d^3 \sqrt {a+b x^2} \left (13 b c^2-2 a d^2\right ) \left (a d^2+b c^2\right )}{b d^3}-5 c \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{3 b d^4}+\frac {\sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}{3 b d^4}}{a d^2+b c^2}+\frac {c^5 \sqrt {a+b x^2}}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {\frac {\frac {3 b c d^3 \left (\frac {b c^3 \left (5 a d^2+4 b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (4 b c^2-a d^2\right ) \left (a d^2+b c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}\right )+d^3 \sqrt {a+b x^2} \left (13 b c^2-2 a d^2\right ) \left (a d^2+b c^2\right )}{b d^3}-5 c \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{3 b d^4}+\frac {\sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}{3 b d^4}}{a d^2+b c^2}+\frac {c^5 \sqrt {a+b x^2}}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {3 b c d^3 \left (\frac {b c^3 \left (5 a d^2+4 b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (4 b c^2-a d^2\right ) \left (a d^2+b c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )+d^3 \sqrt {a+b x^2} \left (13 b c^2-2 a d^2\right ) \left (a d^2+b c^2\right )}{b d^3}-5 c \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{3 b d^4}+\frac {\sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}{3 b d^4}}{a d^2+b c^2}+\frac {c^5 \sqrt {a+b x^2}}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {3 b c d^3 \left (\frac {b c^3 \left (5 a d^2+4 b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (4 b c^2-a d^2\right ) \left (a d^2+b c^2\right )}{\sqrt {b} d}\right )+d^3 \sqrt {a+b x^2} \left (13 b c^2-2 a d^2\right ) \left (a d^2+b c^2\right )}{b d^3}-5 c \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{3 b d^4}+\frac {\sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}{3 b d^4}}{a d^2+b c^2}+\frac {c^5 \sqrt {a+b x^2}}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {\frac {3 b c d^3 \left (-\frac {b c^3 \left (5 a d^2+4 b c^2\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (4 b c^2-a d^2\right ) \left (a d^2+b c^2\right )}{\sqrt {b} d}\right )+d^3 \sqrt {a+b x^2} \left (13 b c^2-2 a d^2\right ) \left (a d^2+b c^2\right )}{b d^3}-5 c \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{3 b d^4}+\frac {\sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}{3 b d^4}}{a d^2+b c^2}+\frac {c^5 \sqrt {a+b x^2}}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {3 b c d^3 \left (-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (4 b c^2-a d^2\right ) \left (a d^2+b c^2\right )}{\sqrt {b} d}-\frac {b c^3 \left (5 a d^2+4 b c^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d \sqrt {a d^2+b c^2}}\right )+d^3 \sqrt {a+b x^2} \left (13 b c^2-2 a d^2\right ) \left (a d^2+b c^2\right )}{b d^3}-5 c \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{3 b d^4}+\frac {\sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}{3 b d^4}}{a d^2+b c^2}+\frac {c^5 \sqrt {a+b x^2}}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

Input:

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

Output:

(c^5*Sqrt[a + b*x^2])/(d^4*(b*c^2 + a*d^2)*(c + d*x)) + (((b*c^2 + a*d^2)* 
(c + d*x)^2*Sqrt[a + b*x^2])/(3*b*d^4) + (-5*c*(b*c^2 + a*d^2)*(c + d*x)*S 
qrt[a + b*x^2] + (d^3*(13*b*c^2 - 2*a*d^2)*(b*c^2 + a*d^2)*Sqrt[a + b*x^2] 
 + 3*b*c*d^3*(-(((4*b*c^2 - a*d^2)*(b*c^2 + a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqr 
t[a + b*x^2]])/(Sqrt[b]*d)) - (b*c^3*(4*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*Sqrt[b*c^2 + a*d^2])))/(b* 
d^3))/(3*b*d^4))/(b*c^2 + a*d^2)
 

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 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 603

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[x^m, c + d*x, x], R = PolynomialRemainde 
r[x^m, c + d*x, x]}, Simp[d*R*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 
1)*(b*c^2 + a*d^2))), x] + Simp[1/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x) 
^(n + 1)*(a + b*x^2)^p*ExpandToSum[(n + 1)*(b*c^2 + a*d^2)*Qx + b*c*R*(n + 
1) - b*d*R*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IGt 
Q[m, 1] && LtQ[n, -1] && 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 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. \(439\) vs. \(2(206)=412\).

Time = 0.42 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.93

method	result	size
risch	\(-\frac {\left (-b \,x^{2} d^{2}+3 b c d x +2 a \,d^{2}-9 b \,c^{2}\right ) \sqrt {b \,x^{2}+a}}{3 b^{2} d^{4}}+\frac {c \left (\frac {\left (a \,d^{2}-4 b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}-\frac {5 c^{3} b \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {c^{4} b \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}\right )}{d^{4} b}\)	\(440\)
default	\(\frac {\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}}{d^{2}}-\frac {4 c^{3} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d^{5} \sqrt {b}}-\frac {2 c \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{d^{3}}+\frac {3 c^{2} \sqrt {b \,x^{2}+a}}{d^{4} b}-\frac {5 c^{4} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{6} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {c^{5} \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{7}}\)	\(477\)

Input:

int(x^5/(d*x+c)^2/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3*(-b*d^2*x^2+3*b*c*d*x+2*a*d^2-9*b*c^2)*(b*x^2+a)^(1/2)/b^2/d^4+c/d^4/ 
b*((a*d^2-4*b*c^2)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)-5*c^3/d^2*b/((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^4/d^3*b*(-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)*l 
n((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. 490 vs. \(2 (207) = 414\).

Time = 25.02 (sec) , antiderivative size = 2025, normalized size of antiderivative = 8.88 \[ \int \frac {x^5}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

integrate(x^5/(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

[-1/6*(3*(4*b^3*c^8 + 7*a*b^2*c^6*d^2 + 2*a^2*b*c^4*d^4 - a^3*c^2*d^6 + (4 
*b^3*c^7*d + 7*a*b^2*c^5*d^3 + 2*a^2*b*c^3*d^5 - a^3*c*d^7)*x)*sqrt(b)*log 
(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 3*(4*b^3*c^7 + 5*a*b^2*c^5* 
d^2 + (4*b^3*c^6*d + 5*a*b^2*c^4*d^3)*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*b^3* 
c^7*d + 19*a*b^2*c^5*d^3 + 5*a^2*b*c^3*d^5 - 2*a^3*c*d^7 + (b^3*c^4*d^4 + 
2*a*b^2*c^2*d^6 + a^2*b*d^8)*x^3 - 2*(b^3*c^5*d^3 + 2*a*b^2*c^3*d^5 + a^2* 
b*c*d^7)*x^2 + 2*(3*b^3*c^6*d^2 + 5*a*b^2*c^4*d^4 + a^2*b*c^2*d^6 - a^3*d^ 
8)*x)*sqrt(b*x^2 + a))/(b^4*c^5*d^5 + 2*a*b^3*c^3*d^7 + a^2*b^2*c*d^9 + (b 
^4*c^4*d^6 + 2*a*b^3*c^2*d^8 + a^2*b^2*d^10)*x), -1/6*(6*(4*b^3*c^7 + 5*a* 
b^2*c^5*d^2 + (4*b^3*c^6*d + 5*a*b^2*c^4*d^3)*x)*sqrt(-b*c^2 - a*d^2)*arct 
an(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + 
 (b^2*c^2 + a*b*d^2)*x^2)) + 3*(4*b^3*c^8 + 7*a*b^2*c^6*d^2 + 2*a^2*b*c^4* 
d^4 - a^3*c^2*d^6 + (4*b^3*c^7*d + 7*a*b^2*c^5*d^3 + 2*a^2*b*c^3*d^5 - a^3 
*c*d^7)*x)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(12 
*b^3*c^7*d + 19*a*b^2*c^5*d^3 + 5*a^2*b*c^3*d^5 - 2*a^3*c*d^7 + (b^3*c^4*d 
^4 + 2*a*b^2*c^2*d^6 + a^2*b*d^8)*x^3 - 2*(b^3*c^5*d^3 + 2*a*b^2*c^3*d^5 + 
 a^2*b*c*d^7)*x^2 + 2*(3*b^3*c^6*d^2 + 5*a*b^2*c^4*d^4 + a^2*b*c^2*d^6 - a 
^3*d^8)*x)*sqrt(b*x^2 + a))/(b^4*c^5*d^5 + 2*a*b^3*c^3*d^7 + a^2*b^2*c*...
 

Sympy [F]

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

integrate(x**5/(d*x+c)**2/(b*x**2+a)**(1/2),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.20 \[ \int \frac {x^5}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} c^{5}}{b c^{2} d^{5} x + a d^{7} x + b c^{3} d^{4} + a c d^{6}} + \frac {\sqrt {b x^{2} + a} x^{2}}{3 \, b d^{2}} - \frac {\sqrt {b x^{2} + a} c x}{b d^{3}} - \frac {4 \, c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{5}} + \frac {a c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}} d^{3}} - \frac {b c^{6} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{8}} + \frac {5 \, c^{4} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{6}} + \frac {3 \, \sqrt {b x^{2} + a} c^{2}}{b d^{4}} - \frac {2 \, \sqrt {b x^{2} + a} a}{3 \, b^{2} d^{2}} \] Input:

integrate(x^5/(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

sqrt(b*x^2 + a)*c^5/(b*c^2*d^5*x + a*d^7*x + b*c^3*d^4 + a*c*d^6) + 1/3*sq 
rt(b*x^2 + a)*x^2/(b*d^2) - sqrt(b*x^2 + a)*c*x/(b*d^3) - 4*c^3*arcsinh(b* 
x/sqrt(a*b))/(sqrt(b)*d^5) + a*c*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*d^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)^(3/2)*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)))/(sqrt(a + b*c^2/d^2)*d^6) + 3*sqrt(b*x^2 
 + a)*c^2/(b*d^4) - 2/3*sqrt(b*x^2 + a)*a/(b^2*d^2)
 

Giac [F(-1)]

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

integrate(x^5/(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

int(x^5/((a + b*x^2)^(1/2)*(c + d*x)^2),x)
 

Output:

int(x^5/((a + b*x^2)^(1/2)*(c + d*x)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1121, normalized size of antiderivative = 4.92 \[ \int \frac {x^5}{(c+d x)^2 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

int(x^5/(d*x+c)^2/(b*x^2+a)^(1/2),x)
 

Output:

(30*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**2*c**5*d**2 + 30*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**2*c**4*d**3*x + 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)*b**3*c 
**7 + 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)*b**3*c**6*d*x - 30*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b** 
2*c**5*d**2 - 30*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**4*d**3*x - 2 
4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**3*c**7 - 24*sqrt(a*d**2 + b*c**2)* 
log(c + d*x)*b**3*c**6*d*x - 4*sqrt(a + b*x**2)*a**3*c*d**7 - 4*sqrt(a + b 
*x**2)*a**3*d**8*x + 10*sqrt(a + b*x**2)*a**2*b*c**3*d**5 + 4*sqrt(a + b*x 
**2)*a**2*b*c**2*d**6*x - 4*sqrt(a + b*x**2)*a**2*b*c*d**7*x**2 + 2*sqrt(a 
 + b*x**2)*a**2*b*d**8*x**3 + 38*sqrt(a + b*x**2)*a*b**2*c**5*d**3 + 20*sq 
rt(a + b*x**2)*a*b**2*c**4*d**4*x - 8*sqrt(a + b*x**2)*a*b**2*c**3*d**5*x* 
*2 + 4*sqrt(a + b*x**2)*a*b**2*c**2*d**6*x**3 + 24*sqrt(a + b*x**2)*b**3*c 
**7*d + 12*sqrt(a + b*x**2)*b**3*c**6*d**2*x - 4*sqrt(a + b*x**2)*b**3*c** 
5*d**3*x**2 + 2*sqrt(a + b*x**2)*b**3*c**4*d**4*x**3 - 3*sqrt(b)*log(sqrt( 
a + b*x**2) - sqrt(b)*x)*a**3*c**2*d**6 - 3*sqrt(b)*log(sqrt(a + b*x**2) - 
 sqrt(b)*x)*a**3*c*d**7*x + 6*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a* 
*2*b*c**4*d**4 + 6*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a**2*b*c**3*d 
**5*x + 21*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a*b**2*c**6*d**2 +...