\(\int \frac {x (a+b x^2)^{5/2}}{(c+d x)^2} \, dx\) [1144]

Optimal result

Integrand size = 20, antiderivative size = 310 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^2} \, dx=\frac {\left (b c^2+a d^2\right ) \left (5 b c^2+a d^2\right ) \sqrt {a+b x^2}}{d^6}-\frac {b c \left (8 b c^2+9 a d^2\right ) x \sqrt {a+b x^2}}{4 d^5}-\frac {b^2 c x^3 \sqrt {a+b x^2}}{2 d^3}+\frac {c \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}{d^6 (c+d x)}+\frac {\left (3 b c^2+a d^2\right ) \left (a+b x^2\right )^{3/2}}{3 d^4}+\frac {\left (a+b x^2\right )^{5/2}}{5 d^2}-\frac {\sqrt {b} c \left (24 b^2 c^4+40 a b c^2 d^2+15 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{4 d^7}-\frac {\left (b c^2+a d^2\right )^{3/2} \left (6 b c^2+a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^7} \] Output:

(a*d^2+b*c^2)*(a*d^2+5*b*c^2)*(b*x^2+a)^(1/2)/d^6-1/4*b*c*(9*a*d^2+8*b*c^2 
)*x*(b*x^2+a)^(1/2)/d^5-1/2*b^2*c*x^3*(b*x^2+a)^(1/2)/d^3+c*(a*d^2+b*c^2)^ 
2*(b*x^2+a)^(1/2)/d^6/(d*x+c)+1/3*(a*d^2+3*b*c^2)*(b*x^2+a)^(3/2)/d^4+1/5* 
(b*x^2+a)^(5/2)/d^2-1/4*b^(1/2)*c*(15*a^2*d^4+40*a*b*c^2*d^2+24*b^2*c^4)*a 
rctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/d^7-(a*d^2+b*c^2)^(3/2)*(a*d^2+6*b*c^2)* 
arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^7
 

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.92 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^2} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (4 a^2 d^4 (38 c+23 d x)+a b d^2 \left (540 c^3+285 c^2 d x-91 c d^2 x^2+44 d^3 x^3\right )+6 b^2 \left (60 c^5+30 c^4 d x-10 c^3 d^2 x^2+5 c^2 d^3 x^3-3 c d^4 x^4+2 d^5 x^5\right )\right )}{c+d x}+120 \sqrt {-b c^2-a d^2} \left (6 b^2 c^4+7 a b c^2 d^2+a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )+15 \sqrt {b} c \left (24 b^2 c^4+40 a b c^2 d^2+15 a^2 d^4\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{60 d^7} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(4*a^2*d^4*(38*c + 23*d*x) + a*b*d^2*(540*c^3 + 285*c^ 
2*d*x - 91*c*d^2*x^2 + 44*d^3*x^3) + 6*b^2*(60*c^5 + 30*c^4*d*x - 10*c^3*d 
^2*x^2 + 5*c^2*d^3*x^3 - 3*c*d^4*x^4 + 2*d^5*x^5)))/(c + d*x) + 120*Sqrt[- 
(b*c^2) - a*d^2]*(6*b^2*c^4 + 7*a*b*c^2*d^2 + a^2*d^4)*ArcTan[(Sqrt[b]*(c 
+ d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]] + 15*Sqrt[b]*c*(24*b^2 
*c^4 + 40*a*b*c^2*d^2 + 15*a^2*d^4)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/( 
60*d^7)
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.91, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {590, 25, 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.

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

\(\Big \downarrow \) 590

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 682

\(\displaystyle \frac {\frac {\int \frac {2 b \left (a d \left (3 b c^2+2 a d^2\right )-b c \left (12 b c^2+11 a d^2\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{4 b d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (2 \left (a d^2+6 b c^2\right )-9 b c d x\right )}{6 d^2}}{d^2}+\frac {\left (a+b x^2\right )^{5/2} (6 c+d x)}{5 d^2 (c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (a d \left (3 b c^2+2 a d^2\right )-b c \left (12 b c^2+11 a d^2\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (2 \left (a d^2+6 b c^2\right )-9 b c d x\right )}{6 d^2}}{d^2}+\frac {\left (a+b x^2\right )^{5/2} (6 c+d x)}{5 d^2 (c+d x)}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {\frac {\frac {\int \frac {b \left (a d \left (12 b^2 c^4+17 a b d^2 c^2+4 a^2 d^4\right )-b c \left (24 b^2 c^4+40 a b d^2 c^2+15 a^2 d^4\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^2}+\frac {\sqrt {a+b x^2} \left (4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )-b c d x \left (11 a d^2+12 b c^2\right )\right )}{2 d^2}}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (2 \left (a d^2+6 b c^2\right )-9 b c d x\right )}{6 d^2}}{d^2}+\frac {\left (a+b x^2\right )^{5/2} (6 c+d x)}{5 d^2 (c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {a d \left (12 b^2 c^4+17 a b d^2 c^2+4 a^2 d^4\right )-b c \left (24 b^2 c^4+40 a b d^2 c^2+15 a^2 d^4\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{2 d^2}+\frac {\sqrt {a+b x^2} \left (4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )-b c d x \left (11 a d^2+12 b c^2\right )\right )}{2 d^2}}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (2 \left (a d^2+6 b c^2\right )-9 b c d x\right )}{6 d^2}}{d^2}+\frac {\left (a+b x^2\right )^{5/2} (6 c+d x)}{5 d^2 (c+d x)}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {\frac {\frac {\frac {4 \left (a d^2+b c^2\right )^2 \left (a d^2+6 b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {b c \left (15 a^2 d^4+40 a b c^2 d^2+24 b^2 c^4\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )-b c d x \left (11 a d^2+12 b c^2\right )\right )}{2 d^2}}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (2 \left (a d^2+6 b c^2\right )-9 b c d x\right )}{6 d^2}}{d^2}+\frac {\left (a+b x^2\right )^{5/2} (6 c+d x)}{5 d^2 (c+d x)}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {\frac {4 \left (a d^2+b c^2\right )^2 \left (a d^2+6 b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {b c \left (15 a^2 d^4+40 a b c^2 d^2+24 b^2 c^4\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )-b c d x \left (11 a d^2+12 b c^2\right )\right )}{2 d^2}}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (2 \left (a d^2+6 b c^2\right )-9 b c d x\right )}{6 d^2}}{d^2}+\frac {\left (a+b x^2\right )^{5/2} (6 c+d x)}{5 d^2 (c+d x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {4 \left (a d^2+b c^2\right )^2 \left (a d^2+6 b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\sqrt {b} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (15 a^2 d^4+40 a b c^2 d^2+24 b^2 c^4\right )}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )-b c d x \left (11 a d^2+12 b c^2\right )\right )}{2 d^2}}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (2 \left (a d^2+6 b c^2\right )-9 b c d x\right )}{6 d^2}}{d^2}+\frac {\left (a+b x^2\right )^{5/2} (6 c+d x)}{5 d^2 (c+d x)}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {\frac {-\frac {4 \left (a d^2+6 b c^2\right ) \left (a d^2+b c^2\right )^2 \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}-\frac {\sqrt {b} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (15 a^2 d^4+40 a b c^2 d^2+24 b^2 c^4\right )}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )-b c d x \left (11 a d^2+12 b c^2\right )\right )}{2 d^2}}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (2 \left (a d^2+6 b c^2\right )-9 b c d x\right )}{6 d^2}}{d^2}+\frac {\left (a+b x^2\right )^{5/2} (6 c+d x)}{5 d^2 (c+d x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {-\frac {\sqrt {b} c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (15 a^2 d^4+40 a b c^2 d^2+24 b^2 c^4\right )}{d}-\frac {4 \left (a d^2+6 b c^2\right ) \left (a d^2+b c^2\right )^{3/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )-b c d x \left (11 a d^2+12 b c^2\right )\right )}{2 d^2}}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (2 \left (a d^2+6 b c^2\right )-9 b c d x\right )}{6 d^2}}{d^2}+\frac {\left (a+b x^2\right )^{5/2} (6 c+d x)}{5 d^2 (c+d x)}\)

Input:

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

Output:

((6*c + d*x)*(a + b*x^2)^(5/2))/(5*d^2*(c + d*x)) + (((2*(6*b*c^2 + a*d^2) 
 - 9*b*c*d*x)*(a + b*x^2)^(3/2))/(6*d^2) + (((4*(b*c^2 + a*d^2)*(6*b*c^2 + 
 a*d^2) - b*c*d*(12*b*c^2 + 11*a*d^2)*x)*Sqrt[a + b*x^2])/(2*d^2) + (-((Sq 
rt[b]*c*(24*b^2*c^4 + 40*a*b*c^2*d^2 + 15*a^2*d^4)*ArcTanh[(Sqrt[b]*x)/Sqr 
t[a + b*x^2]])/d) - (4*(b*c^2 + a*d^2)^(3/2)*(6*b*c^2 + a*d^2)*ArcTanh[(a* 
d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/d)/(2*d^2))/(2*d^2))/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 590

Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^p*((c*(2*p + 1) - d*(n + 1)*x)/(d^2*( 
n + 1)*(n + 2*p + 2))), x] + Simp[2*(p/(d^2*(n + 1)*(n + 2*p + 2)))   Int[( 
c + d*x)^(n + 1)*(a + b*x^2)^(p - 1)*(a*d*(n + 1) + b*c*(2*p + 1)*x), x], x 
] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && LtQ[n, -1] &&  !ILtQ[n + 2*p + 
1, 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. \(586\) vs. \(2(276)=552\).

Time = 0.40 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.89

method	result	size
risch	\(\frac {\left (12 b^{2} d^{4} x^{4}-30 b^{2} c \,d^{3} x^{3}+44 a b \,d^{4} x^{2}+60 d^{2} c^{2} x^{2} b^{2}-135 a b c \,d^{3} x -120 b^{2} c^{3} d x +92 a^{2} d^{4}+420 b \,c^{2} d^{2} a +300 b^{2} c^{4}\right ) \sqrt {b \,x^{2}+a}}{60 d^{6}}-\frac {\frac {4 \left (a^{3} d^{6}+9 a^{2} b \,c^{2} d^{4}+15 a \,b^{2} c^{4} d^{2}+7 b^{3} c^{6}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {\sqrt {b}\, c \left (15 a^{2} d^{4}+40 b \,c^{2} d^{2} a +24 b^{2} c^{4}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d}+\frac {4 c \left (a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}}{4 d^{6}}\)	\(587\)
default	\(\text {Expression too large to display}\)	\(2143\)

Input:

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

Output:

1/60*(12*b^2*d^4*x^4-30*b^2*c*d^3*x^3+44*a*b*d^4*x^2+60*b^2*c^2*d^2*x^2-13 
5*a*b*c*d^3*x-120*b^2*c^3*d*x+92*a^2*d^4+420*a*b*c^2*d^2+300*b^2*c^4)*(b*x 
^2+a)^(1/2)/d^6-1/4/d^6*(4/d^2*(a^3*d^6+9*a^2*b*c^2*d^4+15*a*b^2*c^4*d^2+7 
*b^3*c^6)/((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))+b^(1/2)*c*(15*a^2*d^4+40*a*b*c^2*d^2+24*b^2*c^4)/d*ln( 
b^(1/2)*x+(b*x^2+a)^(1/2))+4*c*(a^3*d^6+3*a^2*b*c^2*d^4+3*a*b^2*c^4*d^2+b^ 
3*c^6)/d^3*(-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))))
 

Fricas [A] (verification not implemented)

Time = 19.65 (sec) , antiderivative size = 1612, normalized size of antiderivative = 5.20 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/120*(15*(24*b^2*c^6 + 40*a*b*c^4*d^2 + 15*a^2*c^2*d^4 + (24*b^2*c^5*d + 
 40*a*b*c^3*d^3 + 15*a^2*c*d^5)*x)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a 
)*sqrt(b)*x - a) + 60*(6*b^2*c^5 + 7*a*b*c^3*d^2 + a^2*c*d^4 + (6*b^2*c^4* 
d + 7*a*b*c^2*d^3 + a^2*d^5)*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^2*d^6*x^5 - 
 18*b^2*c*d^5*x^4 + 360*b^2*c^5*d + 540*a*b*c^3*d^3 + 152*a^2*c*d^5 + 2*(1 
5*b^2*c^2*d^4 + 22*a*b*d^6)*x^3 - (60*b^2*c^3*d^3 + 91*a*b*c*d^5)*x^2 + (1 
80*b^2*c^4*d^2 + 285*a*b*c^2*d^4 + 92*a^2*d^6)*x)*sqrt(b*x^2 + a))/(d^8*x 
+ c*d^7), 1/60*(15*(24*b^2*c^6 + 40*a*b*c^4*d^2 + 15*a^2*c^2*d^4 + (24*b^2 
*c^5*d + 40*a*b*c^3*d^3 + 15*a^2*c*d^5)*x)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt 
(b*x^2 + a)) + 30*(6*b^2*c^5 + 7*a*b*c^3*d^2 + a^2*c*d^4 + (6*b^2*c^4*d + 
7*a*b*c^2*d^3 + a^2*d^5)*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)) + (12*b^2*d^6*x^5 - 18*b^ 
2*c*d^5*x^4 + 360*b^2*c^5*d + 540*a*b*c^3*d^3 + 152*a^2*c*d^5 + 2*(15*b^2* 
c^2*d^4 + 22*a*b*d^6)*x^3 - (60*b^2*c^3*d^3 + 91*a*b*c*d^5)*x^2 + (180*b^2 
*c^4*d^2 + 285*a*b*c^2*d^4 + 92*a^2*d^6)*x)*sqrt(b*x^2 + a))/(d^8*x + c*d^ 
7), -1/120*(120*(6*b^2*c^5 + 7*a*b*c^3*d^2 + a^2*c*d^4 + (6*b^2*c^4*d + 7* 
a*b*c^2*d^3 + a^2*d^5)*x)*sqrt(-b*c^2 - a*d^2)*arctan(sqrt(-b*c^2 - a*d...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.14 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c}{d^{3} x + c d^{2}} - \frac {3 \, \sqrt {b x^{2} + a} b^{2} c^{3} x}{d^{5}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b c x}{2 \, d^{3}} - \frac {11 \, \sqrt {b x^{2} + a} a b c x}{4 \, d^{3}} - \frac {6 \, b^{\frac {5}{2}} c^{5} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{7}} - \frac {10 \, a b^{\frac {3}{2}} c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{5}} - \frac {15 \, a^{2} \sqrt {b} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{4 \, d^{3}} + \frac {5 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} b c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{4}} + \frac {{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{2}} + \frac {6 \, \sqrt {b x^{2} + a} b^{2} c^{4}}{d^{6}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b c^{2}}{d^{4}} + \frac {7 \, \sqrt {b x^{2} + a} a b c^{2}}{d^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{5 \, d^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a}{3 \, d^{2}} + \frac {\sqrt {b x^{2} + a} a^{2}}{d^{2}} \] Input:

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

Output:

(b*x^2 + a)^(5/2)*c/(d^3*x + c*d^2) - 3*sqrt(b*x^2 + a)*b^2*c^3*x/d^5 - 3/ 
2*(b*x^2 + a)^(3/2)*b*c*x/d^3 - 11/4*sqrt(b*x^2 + a)*a*b*c*x/d^3 - 6*b^(5/ 
2)*c^5*arcsinh(b*x/sqrt(a*b))/d^7 - 10*a*b^(3/2)*c^3*arcsinh(b*x/sqrt(a*b) 
)/d^5 - 15/4*a^2*sqrt(b)*c*arcsinh(b*x/sqrt(a*b))/d^3 + 5*(a + b*c^2/d^2)^ 
(3/2)*b*c^2*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d* 
x + c)))/d^4 + (a + b*c^2/d^2)^(5/2)*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c) 
) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^2 + 6*sqrt(b*x^2 + a)*b^2*c^4/d^6 + 2* 
(b*x^2 + a)^(3/2)*b*c^2/d^4 + 7*sqrt(b*x^2 + a)*a*b*c^2/d^4 + 1/5*(b*x^2 + 
 a)^(5/2)/d^2 + 1/3*(b*x^2 + a)^(3/2)*a/d^2 + sqrt(b*x^2 + a)*a^2/d^2
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 1039, normalized size of antiderivative = 3.35 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(120*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a* 
d + b*c*x)*a**2*c*d**4 + 120*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)*a**2*d**5*x + 840*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**3*d**2 + 
 840*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**3*x + 720*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**5 + 720*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*d*x 
 - 120*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*c*d**4 - 120*sqrt(a*d**2 + 
b*c**2)*log(c + d*x)*a**2*d**5*x - 840*sqrt(a*d**2 + b*c**2)*log(c + d*x)* 
a*b*c**3*d**2 - 840*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**2*d**3*x - 7 
20*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2*c**5 - 720*sqrt(a*d**2 + b*c**2 
)*log(c + d*x)*b**2*c**4*d*x + 304*sqrt(a + b*x**2)*a**2*c*d**5 + 184*sqrt 
(a + b*x**2)*a**2*d**6*x + 1080*sqrt(a + b*x**2)*a*b*c**3*d**3 + 570*sqrt( 
a + b*x**2)*a*b*c**2*d**4*x - 182*sqrt(a + b*x**2)*a*b*c*d**5*x**2 + 88*sq 
rt(a + b*x**2)*a*b*d**6*x**3 + 720*sqrt(a + b*x**2)*b**2*c**5*d + 360*sqrt 
(a + b*x**2)*b**2*c**4*d**2*x - 120*sqrt(a + b*x**2)*b**2*c**3*d**3*x**2 + 
 60*sqrt(a + b*x**2)*b**2*c**2*d**4*x**3 - 36*sqrt(a + b*x**2)*b**2*c*d**5 
*x**4 + 24*sqrt(a + b*x**2)*b**2*d**6*x**5 + 225*sqrt(b)*log(sqrt(a + b*x* 
*2) - sqrt(b)*x)*a**2*c**2*d**4 + 225*sqrt(b)*log(sqrt(a + b*x**2) - sq...