\(\int \frac {x^5}{(c+d x)^3 \sqrt {a+b x^2}} \, dx\) [1199]

Optimal result

Integrand size = 22, antiderivative size = 271 \[ \int \frac {x^5}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=-\frac {3 c \sqrt {a+b x^2}}{b d^4}+\frac {x \sqrt {a+b x^2}}{2 b d^3}+\frac {c^5 \sqrt {a+b x^2}}{2 d^4 \left (b c^2+a d^2\right ) (c+d x)^2}-\frac {c^4 \left (7 b c^2+10 a d^2\right ) \sqrt {a+b x^2}}{2 d^4 \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {\left (12 b c^2-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2} d^5}+\frac {c^3 \left (12 b^2 c^4+29 a b c^2 d^2+20 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^5 \left (b c^2+a d^2\right )^{5/2}} \] Output:

-3*c*(b*x^2+a)^(1/2)/b/d^4+1/2*x*(b*x^2+a)^(1/2)/b/d^3+1/2*c^5*(b*x^2+a)^( 
1/2)/d^4/(a*d^2+b*c^2)/(d*x+c)^2-1/2*c^4*(10*a*d^2+7*b*c^2)*(b*x^2+a)^(1/2 
)/d^4/(a*d^2+b*c^2)^2/(d*x+c)+1/2*(-a*d^2+12*b*c^2)*arctanh(b^(1/2)*x/(b*x 
^2+a)^(1/2))/b^(3/2)/d^5+1/2*c^3*(20*a^2*d^4+29*a*b*c^2*d^2+12*b^2*c^4)*ar 
ctanh((-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)^ 
(5/2)
 

Mathematica [A] (verified)

Time = 10.43 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.14 \[ \int \frac {x^5}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (b c^5 \left (b c^2+a d^2\right )-b c^4 \left (7 b c^2+10 a d^2\right ) (c+d x)-6 c \left (b c^2+a d^2\right )^2 (c+d x)^2+d \left (b c^2+a d^2\right )^2 x (c+d x)^2\right )}{b \left (b c^2+a d^2\right )^2 (c+d x)^2}-\frac {c^3 \left (12 b^2 c^4+29 a b c^2 d^2+20 a^2 d^4\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{5/2}}+\frac {\left (12 b c^2-a d^2\right ) \log \left (b x+\sqrt {b} \sqrt {a+b x^2}\right )}{b^{3/2}}+\frac {c^3 \left (12 b^2 c^4+29 a b c^2 d^2+20 a^2 d^4\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{5/2}}}{2 d^5} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(b*c^5*(b*c^2 + a*d^2) - b*c^4*(7*b*c^2 + 10*a*d^2)*(c 
 + d*x) - 6*c*(b*c^2 + a*d^2)^2*(c + d*x)^2 + d*(b*c^2 + a*d^2)^2*x*(c + d 
*x)^2))/(b*(b*c^2 + a*d^2)^2*(c + d*x)^2) - (c^3*(12*b^2*c^4 + 29*a*b*c^2* 
d^2 + 20*a^2*d^4)*Log[c + d*x])/(b*c^2 + a*d^2)^(5/2) + ((12*b*c^2 - a*d^2 
)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/b^(3/2) + (c^3*(12*b^2*c^4 + 29*a*b* 
c^2*d^2 + 20*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)^(5/2))/(2*d^5)
 

Rubi [A] (verified)

Time = 2.11 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.30, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {603, 25, 2182, 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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2182

\(\displaystyle \frac {-\frac {\int \frac {\frac {a \left (5 b c^2+8 a d^2\right ) c^3}{d^3}-\frac {6 \left (b c^2+a d^2\right )^2 x c^2}{d^4}+\frac {4 \left (b c^2+a d^2\right )^2 x^2 c}{d^3}-\frac {2 \left (b c^2+a d^2\right )^2 x^3}{d^2}}{(c+d x) \sqrt {b x^2+a}}dx}{a d^2+b c^2}-\frac {c^4 \sqrt {a+b x^2} \left (10 a d^2+7 b c^2\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^5 \sqrt {a+b x^2}}{2 d^4 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {-\frac {\frac {\int \frac {2 \left (7 b c x^2 \left (b c^2+a d^2\right )^2-\frac {\left (5 b c^2-a d^2\right ) x \left (b c^2+a d^2\right )^2}{d}+a c \left (6 b^2 c^4+10 a b d^2 c^2+a^2 d^4\right )\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^3}-\frac {\sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )^2}{b d^4}}{a d^2+b c^2}-\frac {c^4 \sqrt {a+b x^2} \left (10 a d^2+7 b c^2\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^5 \sqrt {a+b x^2}}{2 d^4 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\frac {\int \frac {7 b c x^2 \left (b c^2+a d^2\right )^2-\frac {\left (5 b c^2-a d^2\right ) x \left (b c^2+a d^2\right )^2}{d}+a c \left (6 b^2 c^4+10 a b d^2 c^2+a^2 d^4\right )}{(c+d x) \sqrt {b x^2+a}}dx}{b d^3}-\frac {\sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )^2}{b d^4}}{a d^2+b c^2}-\frac {c^4 \sqrt {a+b x^2} \left (10 a d^2+7 b c^2\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^5 \sqrt {a+b x^2}}{2 d^4 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {b d \left (a c d \left (6 b^2 c^4+10 a b d^2 c^2+a^2 d^4\right )-\left (12 b c^2-a d^2\right ) \left (b c^2+a d^2\right )^2 x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{b d^2}+\frac {7 c \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{d}}{b d^3}-\frac {\sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )^2}{b d^4}}{a d^2+b c^2}-\frac {c^4 \sqrt {a+b x^2} \left (10 a d^2+7 b c^2\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^5 \sqrt {a+b x^2}}{2 d^4 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {a c d \left (6 b^2 c^4+10 a b d^2 c^2+a^2 d^4\right )-\left (12 b c^2-a d^2\right ) \left (b c^2+a d^2\right )^2 x}{(c+d x) \sqrt {b x^2+a}}dx}{d}+\frac {7 c \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{d}}{b d^3}-\frac {\sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )^2}{b d^4}}{a d^2+b c^2}-\frac {c^4 \sqrt {a+b x^2} \left (10 a d^2+7 b c^2\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^5 \sqrt {a+b x^2}}{2 d^4 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {-\frac {\frac {\frac {\frac {b c^3 \left (20 a^2 d^4+29 a b c^2 d^2+12 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (12 b c^2-a d^2\right ) \left (a d^2+b c^2\right )^2 \int \frac {1}{\sqrt {b x^2+a}}dx}{d}}{d}+\frac {7 c \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{d}}{b d^3}-\frac {\sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )^2}{b d^4}}{a d^2+b c^2}-\frac {c^4 \sqrt {a+b x^2} \left (10 a d^2+7 b c^2\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^5 \sqrt {a+b x^2}}{2 d^4 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {-\frac {\frac {\frac {\frac {b c^3 \left (20 a^2 d^4+29 a b c^2 d^2+12 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (12 b c^2-a d^2\right ) \left (a d^2+b c^2\right )^2 \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}}{d}+\frac {7 c \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{d}}{b d^3}-\frac {\sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )^2}{b d^4}}{a d^2+b c^2}-\frac {c^4 \sqrt {a+b x^2} \left (10 a d^2+7 b c^2\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^5 \sqrt {a+b x^2}}{2 d^4 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\frac {\frac {\frac {b c^3 \left (20 a^2 d^4+29 a b c^2 d^2+12 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 b c^2-a d^2\right ) \left (a d^2+b c^2\right )^2}{\sqrt {b} d}}{d}+\frac {7 c \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{d}}{b d^3}-\frac {\sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )^2}{b d^4}}{a d^2+b c^2}-\frac {c^4 \sqrt {a+b x^2} \left (10 a d^2+7 b c^2\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^5 \sqrt {a+b x^2}}{2 d^4 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {-\frac {\frac {\frac {-\frac {b c^3 \left (20 a^2 d^4+29 a b c^2 d^2+12 b^2 c^4\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 b c^2-a d^2\right ) \left (a d^2+b c^2\right )^2}{\sqrt {b} d}}{d}+\frac {7 c \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{d}}{b d^3}-\frac {\sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )^2}{b d^4}}{a d^2+b c^2}-\frac {c^4 \sqrt {a+b x^2} \left (10 a d^2+7 b c^2\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^5 \sqrt {a+b x^2}}{2 d^4 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\frac {\frac {-\frac {b c^3 \left (20 a^2 d^4+29 a b c^2 d^2+12 b^2 c^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d \sqrt {a d^2+b c^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 b c^2-a d^2\right ) \left (a d^2+b c^2\right )^2}{\sqrt {b} d}}{d}+\frac {7 c \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}{d}}{b d^3}-\frac {\sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )^2}{b d^4}}{a d^2+b c^2}-\frac {c^4 \sqrt {a+b x^2} \left (10 a d^2+7 b c^2\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^5 \sqrt {a+b x^2}}{2 d^4 (c+d x)^2 \left (a d^2+b c^2\right )}\)

Input:

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

Output:

(c^5*Sqrt[a + b*x^2])/(2*d^4*(b*c^2 + a*d^2)*(c + d*x)^2) + (-((c^4*(7*b*c 
^2 + 10*a*d^2)*Sqrt[a + b*x^2])/(d^4*(b*c^2 + a*d^2)*(c + d*x))) - (-(((b* 
c^2 + a*d^2)^2*(c + d*x)*Sqrt[a + b*x^2])/(b*d^4)) + ((7*c*(b*c^2 + a*d^2) 
^2*Sqrt[a + b*x^2])/d + (-(((12*b*c^2 - a*d^2)*(b*c^2 + a*d^2)^2*ArcTanh[( 
Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - (b*c^3*(12*b^2*c^4 + 29*a*b*c^ 
2*d^2 + 20*a^2*d^4)*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]))/d)/(b*d^3))/(b*c^2 + a*d^2))/(2*(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 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. \(715\) vs. \(2(241)=482\).

Time = 0.45 (sec) , antiderivative size = 716, normalized size of antiderivative = 2.64

method	result	size
risch	\(-\frac {\left (-d x +6 c \right ) \sqrt {b \,x^{2}+a}}{2 b \,d^{4}}-\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a}{2 b^{\frac {3}{2}} d^{3}}+\frac {6 c^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d^{5} \sqrt {b}}+\frac {10 c^{3} \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 {5 c^{4} \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}}}}{d^{5} \left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {11 b \,c^{5} \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 )}{2 d^{6} \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {c^{5} \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}}}}{2 d^{6} \left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {3 b \,c^{6} \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}}}}{2 d^{5} \left (a \,d^{2}+b \,c^{2}\right )^{2} \left (x +\frac {c}{d}\right )}+\frac {3 b^{2} c^{7} \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 )}{2 d^{6} \left (a \,d^{2}+b \,c^{2}\right )^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\)	\(716\)
default	\(\frac {\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}}}}{d^{3}}+\frac {6 c^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d^{5} \sqrt {b}}-\frac {3 c \sqrt {b \,x^{2}+a}}{b \,d^{4}}+\frac {10 c^{3} \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 {5 c^{4} \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}}-\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}}}}{2 \left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {3 b c d \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 )}{2 \left (a \,d^{2}+b \,c^{2}\right )}+\frac {b \,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 )}{2 \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{8}}\)	\(883\)

Input:

int(x^5/(d*x+c)^3/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(-d*x+6*c)*(b*x^2+a)^(1/2)/b/d^4-1/2/b^(3/2)/d^3*ln(b^(1/2)*x+(b*x^2+ 
a)^(1/2))*a+6*c^2/d^5*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)+10/d^6*c^3/((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^5*c^4/(a*d^2+b*c^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)-11/2*b/d^6*c^5/(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))+1/2/d^6*c^5/(a*d 
^2+b*c^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)^(1/2)+ 
3/2*b/d^5*c^6/(a*d^2+b*c^2)^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)+3/2*b^2/d^6*c^7/(a*d^2+b*c^2)^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 [F(-1)]

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

integrate(x^5/(d*x+c)^3/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate(x**5/(d*x+c)**3/(b*x**2+a)**(1/2),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.62 \[ \int \frac {x^5}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\frac {3 \, \sqrt {b x^{2} + a} b c^{6}}{2 \, {\left (b^{2} c^{4} d^{5} x + 2 \, a b c^{2} d^{7} x + a^{2} d^{9} x + b^{2} c^{5} d^{4} + 2 \, a b c^{3} d^{6} + a^{2} c d^{8}\right )}} + \frac {\sqrt {b x^{2} + a} c^{5}}{2 \, {\left (b c^{2} d^{6} x^{2} + a d^{8} x^{2} + 2 \, b c^{3} d^{5} x + 2 \, a c d^{7} x + b c^{4} d^{4} + a c^{2} d^{6}\right )}} - \frac {5 \, \sqrt {b x^{2} + a} c^{4}}{b c^{2} d^{5} x + a d^{7} x + b c^{3} d^{4} + a c d^{6}} + \frac {\sqrt {b x^{2} + a} x}{2 \, b d^{3}} + \frac {6 \, c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{5}} - \frac {a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}} d^{3}} - \frac {3 \, b^{2} c^{7} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} d^{10}} + \frac {11 \, b c^{5} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{8}} - \frac {10 \, c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{6}} - \frac {3 \, \sqrt {b x^{2} + a} c}{b d^{4}} \] Input:

integrate(x^5/(d*x+c)^3/(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

3/2*sqrt(b*x^2 + a)*b*c^6/(b^2*c^4*d^5*x + 2*a*b*c^2*d^7*x + a^2*d^9*x + b 
^2*c^5*d^4 + 2*a*b*c^3*d^6 + a^2*c*d^8) + 1/2*sqrt(b*x^2 + a)*c^5/(b*c^2*d 
^6*x^2 + a*d^8*x^2 + 2*b*c^3*d^5*x + 2*a*c*d^7*x + b*c^4*d^4 + a*c^2*d^6) 
- 5*sqrt(b*x^2 + a)*c^4/(b*c^2*d^5*x + a*d^7*x + b*c^3*d^4 + a*c*d^6) + 1/ 
2*sqrt(b*x^2 + a)*x/(b*d^3) + 6*c^2*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^5) - 
 1/2*a*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*d^3) - 3/2*b^2*c^7*arcsinh(b*c*x/(s 
qrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(5 
/2)*d^10) + 11/2*b*c^5*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) - 10*c^3*arcsinh(b*c*x/(sq 
rt(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/(b*d^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (242) = 484\).

Time = 0.16 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.86 \[ \int \frac {x^5}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\frac {1}{2} \, \sqrt {b x^{2} + a} {\left (\frac {x}{b d^{3}} - \frac {6 \, c}{b d^{4}}\right )} - \frac {{\left (12 \, b^{2} c^{7} + 29 \, a b c^{5} d^{2} + 20 \, a^{2} c^{3} d^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b^{2} c^{4} d^{5} + 2 \, a b c^{2} d^{7} + a^{2} d^{9}\right )} \sqrt {-b c^{2} - a d^{2}}} - \frac {8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b^{2} c^{7} d + 11 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a b c^{5} d^{3} + 14 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {5}{2}} c^{8} + 13 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} c^{6} d^{2} - 10 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} \sqrt {b} c^{4} d^{4} - 20 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a b^{2} c^{7} d - 29 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b c^{5} d^{3} + 7 \, a^{2} b^{\frac {3}{2}} c^{6} d^{2} + 10 \, a^{3} \sqrt {b} c^{4} d^{4}}{{\left (b^{2} c^{4} d^{5} + 2 \, a b c^{2} d^{7} + a^{2} d^{9}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{2}} - \frac {{\left (12 \, b c^{2} - a d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {3}{2}} d^{5}} \] Input:

integrate(x^5/(d*x+c)^3/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

1/2*sqrt(b*x^2 + a)*(x/(b*d^3) - 6*c/(b*d^4)) - (12*b^2*c^7 + 29*a*b*c^5*d 
^2 + 20*a^2*c^3*d^4)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c) 
/sqrt(-b*c^2 - a*d^2))/((b^2*c^4*d^5 + 2*a*b*c^2*d^7 + a^2*d^9)*sqrt(-b*c^ 
2 - a*d^2)) - (8*(sqrt(b)*x - sqrt(b*x^2 + a))^3*b^2*c^7*d + 11*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^3*a*b*c^5*d^3 + 14*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^( 
5/2)*c^8 + 13*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2)*c^6*d^2 - 10*(sqrt 
(b)*x - sqrt(b*x^2 + a))^2*a^2*sqrt(b)*c^4*d^4 - 20*(sqrt(b)*x - sqrt(b*x^ 
2 + a))*a*b^2*c^7*d - 29*(sqrt(b)*x - sqrt(b*x^2 + a))*a^2*b*c^5*d^3 + 7*a 
^2*b^(3/2)*c^6*d^2 + 10*a^3*sqrt(b)*c^4*d^4)/((b^2*c^4*d^5 + 2*a*b*c^2*d^7 
 + a^2*d^9)*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*(sqrt(b)*x - sqrt(b*x^2 
 + a))*sqrt(b)*c - a*d)^2) - 1/2*(12*b*c^2 - a*d^2)*log(abs(-sqrt(b)*x + s 
qrt(b*x^2 + a)))/(b^(3/2)*d^5)
 

Mupad [F(-1)]

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

int(x^5/((a + b*x^2)^(1/2)*(c + d*x)^3),x)
 

Output:

int(x^5/((a + b*x^2)^(1/2)*(c + d*x)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 2164, normalized size of antiderivative = 7.99 \[ \int \frac {x^5}{(c+d x)^3 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

int(x^5/(d*x+c)^3/(b*x^2+a)^(1/2),x)
 

Output:

(40*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**4 + 80*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**5*x + 40* 
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**3*d**6*x**2 + 58*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*d**2 + 116*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**6*d**3*x + 58*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**4*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) 
*b**4*c**9 + 48*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**8*d*x + 24*sqrt(a*d**2 + b*c**2)*log( - s 
qrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**4*c**7*d**2*x**2 - 
 40*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b**2*c**5*d**4 - 80*sqrt(a*d** 
2 + b*c**2)*log(c + d*x)*a**2*b**2*c**4*d**5*x - 40*sqrt(a*d**2 + b*c**2)* 
log(c + d*x)*a**2*b**2*c**3*d**6*x**2 - 58*sqrt(a*d**2 + b*c**2)*log(c + d 
*x)*a*b**3*c**7*d**2 - 116*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**3*c**6* 
d**3*x - 58*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**3*c**5*d**4*x**2 - 24* 
sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**4*c**9 - 48*sqrt(a*d**2 + b*c**2)*lo 
g(c + d*x)*b**4*c**8*d*x - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**4*c...