\(\int \frac {(a+b x^2)^{5/2}}{x (c+d x)} \, dx\) [1131]

Optimal result

Integrand size = 22, antiderivative size = 253 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x (c+d x)} \, dx=-\frac {b c \left (b c^2+2 a d^2\right ) \sqrt {a+b x^2}}{d^4}+\frac {b \left (4 b c^2+9 a d^2\right ) x \sqrt {a+b x^2}}{8 d^3}+\frac {b^2 x^3 \sqrt {a+b x^2}}{4 d}-\frac {b c \left (a+b x^2\right )^{3/2}}{3 d^2}+\frac {\sqrt {b} \left (8 b^2 c^4+20 a b c^2 d^2+15 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 d^5}+\frac {\left (b c^2+a d^2\right )^{5/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c d^5}-\frac {a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c} \] Output:

-b*c*(2*a*d^2+b*c^2)*(b*x^2+a)^(1/2)/d^4+1/8*b*(9*a*d^2+4*b*c^2)*x*(b*x^2+ 
a)^(1/2)/d^3+1/4*b^2*x^3*(b*x^2+a)^(1/2)/d-1/3*b*c*(b*x^2+a)^(3/2)/d^2+1/8 
*b^(1/2)*(15*a^2*d^4+20*a*b*c^2*d^2+8*b^2*c^4)*arctanh(b^(1/2)*x/(b*x^2+a) 
^(1/2))/d^5+(a*d^2+b*c^2)^(5/2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/( 
b*x^2+a)^(1/2))/c/d^5-a^(5/2)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/c
 

Mathematica [A] (verified)

Time = 1.54 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x (c+d x)} \, dx=\frac {b c d \sqrt {a+b x^2} \left (a d^2 (-56 c+27 d x)+b \left (-24 c^3+12 c^2 d x-8 c d^2 x^2+6 d^3 x^3\right )\right )-48 \left (-b c^2-a d^2\right )^{5/2} \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )+6 \sqrt {b} c \left (8 b^2 c^4+20 a b c^2 d^2+15 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )-24 a^{5/2} d^5 \log (x)+24 a^{5/2} d^5 \log \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{24 c d^5} \] Input:

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

Output:

(b*c*d*Sqrt[a + b*x^2]*(a*d^2*(-56*c + 27*d*x) + b*(-24*c^3 + 12*c^2*d*x - 
 8*c*d^2*x^2 + 6*d^3*x^3)) - 48*(-(b*c^2) - a*d^2)^(5/2)*ArcTan[(Sqrt[-(b* 
c^2) - a*d^2]*x)/(Sqrt[a]*(c + d*x) - c*Sqrt[a + b*x^2])] + 6*Sqrt[b]*c*(8 
*b^2*c^4 + 20*a*b*c^2*d^2 + 15*a^2*d^4)*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sq 
rt[a + b*x^2])] - 24*a^(5/2)*d^5*Log[x] + 24*a^(5/2)*d^5*Log[-Sqrt[a] + Sq 
rt[a + b*x^2]])/(24*c*d^5)
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.16, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {606, 243, 60, 60, 73, 221, 682, 27, 682, 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.

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

\(\Big \downarrow \) 606

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 682

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 682

\(\displaystyle \frac {a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )}{2 c}-\frac {\frac {\frac {\int \frac {b \left (a d \left (4 b^2 c^4+9 a b d^2 c^2+8 a^2 d^4\right )-b c \left (8 b^2 c^4+20 a b d^2 c^2+15 a^2 d^4\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^2}+\frac {\sqrt {a+b x^2} \left (8 \left (a d^2+b c^2\right )^2-b c d x \left (7 a d^2+4 b c^2\right )\right )}{2 d^2}}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (4 \left (a d^2+b c^2\right )-3 b c d x\right )}{12 d^2}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )}{2 c}-\frac {\frac {\frac {\int \frac {a d \left (4 b^2 c^4+9 a b d^2 c^2+8 a^2 d^4\right )-b c \left (8 b^2 c^4+20 a b d^2 c^2+15 a^2 d^4\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{2 d^2}+\frac {\sqrt {a+b x^2} \left (8 \left (a d^2+b c^2\right )^2-b c d x \left (7 a d^2+4 b c^2\right )\right )}{2 d^2}}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (4 \left (a d^2+b c^2\right )-3 b c d x\right )}{12 d^2}}{c}\)

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {a \left (a \left (2 \sqrt {a+b x^2}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b x^2\right )^{3/2}\right )}{2 c}-\frac {\frac {\frac {-\frac {\sqrt {b} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (15 a^2 d^4+20 a b c^2 d^2+8 b^2 c^4\right )}{d}-\frac {8 \left (a d^2+b c^2\right )^{5/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (8 \left (a d^2+b c^2\right )^2-b c d x \left (7 a d^2+4 b c^2\right )\right )}{2 d^2}}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (4 \left (a d^2+b c^2\right )-3 b c d x\right )}{12 d^2}}{c}\)

Input:

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

Output:

-((((4*(b*c^2 + a*d^2) - 3*b*c*d*x)*(a + b*x^2)^(3/2))/(12*d^2) + (((8*(b* 
c^2 + a*d^2)^2 - b*c*d*(4*b*c^2 + 7*a*d^2)*x)*Sqrt[a + b*x^2])/(2*d^2) + ( 
-((Sqrt[b]*c*(8*b^2*c^4 + 20*a*b*c^2*d^2 + 15*a^2*d^4)*ArcTanh[(Sqrt[b]*x) 
/Sqrt[a + b*x^2]])/d) - (8*(b*c^2 + a*d^2)^(5/2)*ArcTanh[(a*d - b*c*x)/(Sq 
rt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/d)/(2*d^2))/(4*d^2))/c) + (a*((2*(a + 
 b*x^2)^(3/2))/3 + a*(2*Sqrt[a + b*x^2] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*x^2 
]/Sqrt[a]])))/(2*c)
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, 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 221

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

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 606

Int[(((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] : 
> Simp[a/c   Int[(c + d*x)^(n + 1)*((a + b*x^2)^(p - 1)/x), x], x] - Simp[1 
/c   Int[(c + d*x)^n*(a*d - b*c*x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, 
 b, c, d}, x] && GtQ[p, 0] && ILtQ[n, 0]
 

rule 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(903\) vs. \(2(219)=438\).

Time = 0.37 (sec) , antiderivative size = 904, normalized size of antiderivative = 3.57

method	result	size
default	\(\frac {\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{c}-\frac {\frac {\left (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 )^{\frac {5}{2}}}{5}-\frac {b c \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right ) \left (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 )^{\frac {3}{2}}}{8 b}+\frac {3 \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b 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}}}}{4 b}+\frac {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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 )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{d}+\frac {\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {\left (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 )^{\frac {3}{2}}}{3}-\frac {b c \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b 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}}}}{4 b}+\frac {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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 )}{8 b^{\frac {3}{2}}}\right )}{d}+\frac {\left (a \,d^{2}+b \,c^{2}\right ) \left (\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 {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}-\frac {\left (a \,d^{2}+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} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{2}}\right )}{d^{2}}}{c}\)	\(904\)

Input:

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

Output:

1/c*(1/5*(b*x^2+a)^(5/2)+a*(1/3*(b*x^2+a)^(3/2)+a*((b*x^2+a)^(1/2)-a^(1/2) 
*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x))))-1/c*(1/5*(b*(x+c/d)^2-2*b*c/d*(x 
+c/d)+(a*d^2+b*c^2)/d^2)^(5/2)-b*c/d*(1/8*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/ 
d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+3/16*(4*b*(a*d^2+b*c^2)/d^2- 
4*b^2*c^2/d^2)/b*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d) 
+(a*d^2+b*c^2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2 
)*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2) 
/d^2)^(1/2))))+(a*d^2+b*c^2)/d^2*(1/3*(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*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d* 
(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2) 
/b^(3/2)*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2 
+b*c^2)/d^2)^(1/2)))+(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)-b^(1/2)*c/d*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2 
-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))-(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)))))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x (c+d x)} \, dx=\frac {{\left (\frac {12 \, \sqrt {b x^{2} + a} b^{2} c^{3} x}{d^{4}} + \frac {6 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b c x}{d^{2}} + \frac {21 \, \sqrt {b x^{2} + a} a b c x}{d^{2}} + \frac {24 \, b^{\frac {5}{2}} c^{5} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{6}} + \frac {60 \, a b^{\frac {3}{2}} c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{4}} + \frac {45 \, a^{2} \sqrt {b} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{2}} - \frac {24 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {2 \, b c x}{\sqrt {a b} {\left | 2 \, d x + 2 \, c \right |}} - \frac {2 \, a d}{\sqrt {a b} {\left | 2 \, d x + 2 \, c \right |}}\right )}{d} - \frac {24 \, a^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{d} - \frac {24 \, \sqrt {b x^{2} + a} b^{2} c^{4}}{d^{5}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b c^{2}}{d^{3}} - \frac {48 \, \sqrt {b x^{2} + a} a b c^{2}}{d^{3}}\right )} d}{24 \, c} \] Input:

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

Output:

1/24*(12*sqrt(b*x^2 + a)*b^2*c^3*x/d^4 + 6*(b*x^2 + a)^(3/2)*b*c*x/d^2 + 2 
1*sqrt(b*x^2 + a)*a*b*c*x/d^2 + 24*b^(5/2)*c^5*arcsinh(b*x/sqrt(a*b))/d^6 
+ 60*a*b^(3/2)*c^3*arcsinh(b*x/sqrt(a*b))/d^4 + 45*a^2*sqrt(b)*c*arcsinh(b 
*x/sqrt(a*b))/d^2 - 24*(a + b*c^2/d^2)^(5/2)*arcsinh(2*b*c*x/(sqrt(a*b)*ab 
s(2*d*x + 2*c)) - 2*a*d/(sqrt(a*b)*abs(2*d*x + 2*c)))/d - 24*a^(5/2)*arcsi 
nh(a/(sqrt(a*b)*abs(x)))/d - 24*sqrt(b*x^2 + a)*b^2*c^4/d^5 - 8*(b*x^2 + a 
)^(3/2)*b*c^2/d^3 - 48*sqrt(b*x^2 + a)*a*b*c^2/d^3)*d/c
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x (c+d x)} \, dx=\frac {48 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (-\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a^{2} d^{4}+96 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (-\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a b \,c^{2} d^{2}+48 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (-\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) b^{2} c^{4}-48 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a^{2} d^{4}-96 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a b \,c^{2} d^{2}-48 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) b^{2} c^{4}-112 \sqrt {b \,x^{2}+a}\, a b \,c^{2} d^{3}+54 \sqrt {b \,x^{2}+a}\, a b c \,d^{4} x -48 \sqrt {b \,x^{2}+a}\, b^{2} c^{4} d +24 \sqrt {b \,x^{2}+a}\, b^{2} c^{3} d^{2} x -16 \sqrt {b \,x^{2}+a}\, b^{2} c^{2} d^{3} x^{2}+12 \sqrt {b \,x^{2}+a}\, b^{2} c \,d^{4} x^{3}+24 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {a}\right ) a^{2} d^{5}-24 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {a}\right ) a^{2} d^{5}-45 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {b}\, x \right ) a^{2} c \,d^{4}-60 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {b}\, x \right ) a b \,c^{3} d^{2}-24 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {b}\, x \right ) b^{2} c^{5}+45 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {b}\, x \right ) a^{2} c \,d^{4}+60 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {b}\, x \right ) a b \,c^{3} d^{2}+24 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {b}\, x \right ) b^{2} c^{5}}{48 c \,d^{5}} \] Input:

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

Output:

(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)*a**2*d**4 + 96*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*c**2*d**2 + 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**2*c**4 
 - 48*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*d**4 - 96*sqrt(a*d**2 + b*c* 
*2)*log(c + d*x)*a*b*c**2*d**2 - 48*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b** 
2*c**4 - 112*sqrt(a + b*x**2)*a*b*c**2*d**3 + 54*sqrt(a + b*x**2)*a*b*c*d* 
*4*x - 48*sqrt(a + b*x**2)*b**2*c**4*d + 24*sqrt(a + b*x**2)*b**2*c**3*d** 
2*x - 16*sqrt(a + b*x**2)*b**2*c**2*d**3*x**2 + 12*sqrt(a + b*x**2)*b**2*c 
*d**4*x**3 + 24*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a**2*d**5 - 24*sqr 
t(a)*log(sqrt(a + b*x**2) + sqrt(a))*a**2*d**5 - 45*sqrt(b)*log(sqrt(a + b 
*x**2) - sqrt(b)*x)*a**2*c*d**4 - 60*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b 
)*x)*a*b*c**3*d**2 - 24*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*b**2*c** 
5 + 45*sqrt(b)*log(sqrt(a + b*x**2) + sqrt(b)*x)*a**2*c*d**4 + 60*sqrt(b)* 
log(sqrt(a + b*x**2) + sqrt(b)*x)*a*b*c**3*d**2 + 24*sqrt(b)*log(sqrt(a + 
b*x**2) + sqrt(b)*x)*b**2*c**5)/(48*c*d**5)