\(\int \frac {x (a+b x^2)^{5/2}}{(c+d x)^3} \, dx\) [1145]

Optimal result

Integrand size = 20, antiderivative size = 325 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^3} \, dx=-\frac {2 b c \left (5 b c^2+3 a d^2\right ) \sqrt {a+b x^2}}{d^6}+\frac {3 b \left (8 b c^2+3 a d^2\right ) x \sqrt {a+b x^2}}{8 d^5}+\frac {b^2 x^3 \sqrt {a+b x^2}}{4 d^3}+\frac {c \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}{2 d^6 (c+d x)^2}-\frac {\left (b c^2+a d^2\right ) \left (11 b c^2+2 a d^2\right ) \sqrt {a+b x^2}}{2 d^6 (c+d x)}-\frac {b c \left (a+b x^2\right )^{3/2}}{d^4}+\frac {15 \sqrt {b} \left (8 b^2 c^4+8 a b c^2 d^2+a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 d^7}+\frac {15 b c \sqrt {b c^2+a d^2} \left (2 b c^2+a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^7} \] Output:

-2*b*c*(3*a*d^2+5*b*c^2)*(b*x^2+a)^(1/2)/d^6+3/8*b*(3*a*d^2+8*b*c^2)*x*(b* 
x^2+a)^(1/2)/d^5+1/4*b^2*x^3*(b*x^2+a)^(1/2)/d^3+1/2*c*(a*d^2+b*c^2)^2*(b* 
x^2+a)^(1/2)/d^6/(d*x+c)^2-1/2*(a*d^2+b*c^2)*(2*a*d^2+11*b*c^2)*(b*x^2+a)^ 
(1/2)/d^6/(d*x+c)-b*c*(b*x^2+a)^(3/2)/d^4+15/8*b^(1/2)*(a^2*d^4+8*a*b*c^2* 
d^2+8*b^2*c^4)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/d^7+15/2*b*c*(a*d^2+b*c^ 
2)^(1/2)*(a*d^2+2*b*c^2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a 
)^(1/2))/d^7
 

Mathematica [A] (verified)

Time = 1.62 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.82 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^3} \, dx=-\frac {\frac {d \sqrt {a+b x^2} \left (4 a^2 d^4 (c+2 d x)+a b d^2 \left (100 c^3+155 c^2 d x+38 c d^2 x^2-9 d^3 x^3\right )+2 b^2 \left (60 c^5+90 c^4 d x+20 c^3 d^2 x^2-5 c^2 d^3 x^3+2 c d^4 x^4-d^5 x^5\right )\right )}{(c+d x)^2}+120 b c \sqrt {-b c^2-a d^2} \left (2 b c^2+a d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )+15 \sqrt {b} \left (8 b^2 c^4+8 a b c^2 d^2+a^2 d^4\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 d^7} \] Input:

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

Output:

-1/8*((d*Sqrt[a + b*x^2]*(4*a^2*d^4*(c + 2*d*x) + a*b*d^2*(100*c^3 + 155*c 
^2*d*x + 38*c*d^2*x^2 - 9*d^3*x^3) + 2*b^2*(60*c^5 + 90*c^4*d*x + 20*c^3*d 
^2*x^2 - 5*c^2*d^3*x^3 + 2*c*d^4*x^4 - d^5*x^5)))/(c + d*x)^2 + 120*b*c*Sq 
rt[-(b*c^2) - a*d^2]*(2*b*c^2 + a*d^2)*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqrt[ 
a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]] + 15*Sqrt[b]*(8*b^2*c^4 + 8*a*b*c^2*d^ 
2 + a^2*d^4)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/d^7
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.83, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {590, 27, 681, 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.

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

\(\Big \downarrow \) 590

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 681

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 682

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

\(\displaystyle \frac {5 \left (-\frac {3 b \left (\frac {\frac {4 c \left (a d^2+b c^2\right ) \left (a d^2+2 b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (a^2 d^4+8 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (4 c \left (a d^2+2 b c^2\right )-d x \left (a d^2+4 b c^2\right )\right )}{2 d^2}\right )}{d^2}-\frac {\left (a+b x^2\right )^{3/2} \left (a d^2+4 b c^2+b c d x\right )}{d^2 (c+d x)}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{5/2} (3 c+d x)}{4 d^2 (c+d x)^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {5 \left (-\frac {3 b \left (\frac {\frac {4 c \left (a d^2+b c^2\right ) \left (a d^2+2 b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (a^2 d^4+8 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (4 c \left (a d^2+2 b c^2\right )-d x \left (a d^2+4 b c^2\right )\right )}{2 d^2}\right )}{d^2}-\frac {\left (a+b x^2\right )^{3/2} \left (a d^2+4 b c^2+b c d x\right )}{d^2 (c+d x)}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{5/2} (3 c+d x)}{4 d^2 (c+d x)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (-\frac {3 b \left (\frac {\frac {4 c \left (a d^2+b c^2\right ) \left (a d^2+2 b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^2 d^4+8 a b c^2 d^2+8 b^2 c^4\right )}{\sqrt {b} d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (4 c \left (a d^2+2 b c^2\right )-d x \left (a d^2+4 b c^2\right )\right )}{2 d^2}\right )}{d^2}-\frac {\left (a+b x^2\right )^{3/2} \left (a d^2+4 b c^2+b c d x\right )}{d^2 (c+d x)}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{5/2} (3 c+d x)}{4 d^2 (c+d x)^2}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {5 \left (-\frac {3 b \left (\frac {-\frac {4 c \left (a d^2+b c^2\right ) \left (a d^2+2 b c^2\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^2 d^4+8 a b c^2 d^2+8 b^2 c^4\right )}{\sqrt {b} d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (4 c \left (a d^2+2 b c^2\right )-d x \left (a d^2+4 b c^2\right )\right )}{2 d^2}\right )}{d^2}-\frac {\left (a+b x^2\right )^{3/2} \left (a d^2+4 b c^2+b c d x\right )}{d^2 (c+d x)}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{5/2} (3 c+d x)}{4 d^2 (c+d x)^2}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

((3*c + d*x)*(a + b*x^2)^(5/2))/(4*d^2*(c + d*x)^2) + (5*(-(((4*b*c^2 + a* 
d^2 + b*c*d*x)*(a + b*x^2)^(3/2))/(d^2*(c + d*x))) - (3*b*(((4*c*(2*b*c^2 
+ a*d^2) - d*(4*b*c^2 + a*d^2)*x)*Sqrt[a + b*x^2])/(2*d^2) + (-(((8*b^2*c^ 
4 + 8*a*b*c^2*d^2 + a^2*d^4)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b 
]*d)) - (4*c*Sqrt[b*c^2 + a*d^2]*(2*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)))/d^2))/(4*d^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 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 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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. \(1014\) vs. \(2(289)=578\).

Time = 0.46 (sec) , antiderivative size = 1015, normalized size of antiderivative = 3.12

method	result	size
risch	\(\text {Expression too large to display}\)	\(1015\)
default	\(\text {Expression too large to display}\)	\(3555\)

Input:

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

Output:

-1/8*b*(-2*b*d^3*x^3+8*b*c*d^2*x^2-9*a*d^3*x-24*b*c^2*d*x+56*a*c*d^2+80*b* 
c^3)*(b*x^2+a)^(1/2)/d^6+1/8/d^6*(8/d^3*(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)*(-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)))+15*b^(1/2)*(a^ 
2*d^4+8*a*b*c^2*d^2+8*b^2*c^4)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))-8*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^4*(-1/2/(a*d^2+b*c^2)*d^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*c*d/(a* 
d^2+b*c^2)*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d 
^2+b*c^2)/d^2)^(1/2)-b*c*d/(a*d^2+b*c^2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*( 
a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2- 
2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))+1/2*b/(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)))+24*b*c/d^2*(3*a^2*d^4+10*a*b*c^2*d^2+7*b^2*c^4)/((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 = 3.64 (sec) , antiderivative size = 1812, normalized size of antiderivative = 5.58 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/16*(15*(8*b^2*c^6 + 8*a*b*c^4*d^2 + a^2*c^2*d^4 + (8*b^2*c^4*d^2 + 8*a* 
b*c^2*d^4 + a^2*d^6)*x^2 + 2*(8*b^2*c^5*d + 8*a*b*c^3*d^3 + a^2*c*d^5)*x)* 
sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 60*(2*b^2*c^5 + 
a*b*c^3*d^2 + (2*b^2*c^3*d^2 + a*b*c*d^4)*x^2 + 2*(2*b^2*c^4*d + a*b*c^2*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*(2*b^2*d^6*x^5 - 4*b^2*c*d^5*x^4 - 120*b^2 
*c^5*d - 100*a*b*c^3*d^3 - 4*a^2*c*d^5 + (10*b^2*c^2*d^4 + 9*a*b*d^6)*x^3 
- 2*(20*b^2*c^3*d^3 + 19*a*b*c*d^5)*x^2 - (180*b^2*c^4*d^2 + 155*a*b*c^2*d 
^4 + 8*a^2*d^6)*x)*sqrt(b*x^2 + a))/(d^9*x^2 + 2*c*d^8*x + c^2*d^7), 1/16* 
(120*(2*b^2*c^5 + a*b*c^3*d^2 + (2*b^2*c^3*d^2 + a*b*c*d^4)*x^2 + 2*(2*b^2 
*c^4*d + a*b*c^2*d^3)*x)*sqrt(-b*c^2 - a*d^2)*arctan(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 
)) + 15*(8*b^2*c^6 + 8*a*b*c^4*d^2 + a^2*c^2*d^4 + (8*b^2*c^4*d^2 + 8*a*b* 
c^2*d^4 + a^2*d^6)*x^2 + 2*(8*b^2*c^5*d + 8*a*b*c^3*d^3 + a^2*c*d^5)*x)*sq 
rt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(2*b^2*d^6*x^5 - 
 4*b^2*c*d^5*x^4 - 120*b^2*c^5*d - 100*a*b*c^3*d^3 - 4*a^2*c*d^5 + (10*b^2 
*c^2*d^4 + 9*a*b*d^6)*x^3 - 2*(20*b^2*c^3*d^3 + 19*a*b*c*d^5)*x^2 - (180*b 
^2*c^4*d^2 + 155*a*b*c^2*d^4 + 8*a^2*d^6)*x)*sqrt(b*x^2 + a))/(d^9*x^2 + 2 
*c*d^8*x + c^2*d^7), -1/8*(15*(8*b^2*c^6 + 8*a*b*c^4*d^2 + a^2*c^2*d^4 ...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 659 vs. \(2 (290) = 580\).

Time = 0.12 (sec) , antiderivative size = 659, normalized size of antiderivative = 2.03 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-15/4*b^4*c^6*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*c^2*d^7 + a*sqrt(b)*d^9) - 1 
5/4*a*b^3*c^4*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*c^2*d^5 + a*sqrt(b)*d^7) + 1 
5/4*sqrt(b*x^2 + a)*b^3*c^4*x/(b*c^2*d^5 + a*d^7) - 5/2*(b*x^2 + a)^(3/2)* 
b^2*c^3/(b*c^2*d^4 + a*d^6) + 5/2*(b*x^2 + a)^(3/2)*b^2*c^2*x/(b*c^2*d^3 + 
 a*d^5) + 15/4*sqrt(b*x^2 + a)*a*b^2*c^2*x/(b*c^2*d^3 + a*d^5) - 3/2*(b*x^ 
2 + a)^(5/2)*b*c^2/(b*c^2*d^3*x + a*d^5*x + b*c^3*d^2 + a*c*d^4) + 1/2*(b* 
x^2 + a)^(7/2)*c/(b*c^2*d^2*x^2 + a*d^4*x^2 + 2*b*c^3*d*x + 2*a*c*d^3*x + 
b*c^4 + a*c^2*d^2) - 1/2*(b*x^2 + a)^(5/2)*b*c/(b*c^2*d^2 + a*d^4) - (b*x^ 
2 + a)^(5/2)/(d^3*x + c*d^2) + 15/4*sqrt(b*x^2 + a)*b^2*c^2*x/d^5 + 5/4*(b 
*x^2 + a)^(3/2)*b*x/d^3 + 15/8*sqrt(b*x^2 + a)*a*b*x/d^3 + 75/4*b^(5/2)*c^ 
4*arcsinh(b*x/sqrt(a*b))/d^7 + 15*a*b^(3/2)*c^2*arcsinh(b*x/sqrt(a*b))/d^5 
 + 15/8*a^2*sqrt(b)*arcsinh(b*x/sqrt(a*b))/d^3 - 15/2*sqrt(a + b*c^2/d^2)* 
b^2*c^3*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + 
c)))/d^6 - 15/2*(a + b*c^2/d^2)^(3/2)*b*c*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x 
 + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^4 - 15*sqrt(b*x^2 + a)*b^2*c^3/d^ 
6 - 5/2*(b*x^2 + a)^(3/2)*b*c/d^4 - 15/2*sqrt(b*x^2 + a)*a*b*c/d^4
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (290) = 580\).

Time = 0.22 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.95 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^3} \, dx=\frac {1}{8} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, x {\left (\frac {b^{2} x}{d^{3}} - \frac {4 \, b^{2} c}{d^{4}}\right )} + \frac {3 \, {\left (8 \, b^{4} c^{2} d^{20} + 3 \, a b^{3} d^{22}\right )}}{b^{2} d^{25}}\right )} x - \frac {8 \, {\left (10 \, b^{4} c^{3} d^{19} + 7 \, a b^{3} c d^{21}\right )}}{b^{2} d^{25}}\right )} - \frac {15 \, {\left (8 \, b^{\frac {5}{2}} c^{4} + 8 \, a b^{\frac {3}{2}} c^{2} d^{2} + a^{2} \sqrt {b} d^{4}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, d^{7}} - \frac {15 \, {\left (2 \, b^{3} c^{5} + 3 \, a b^{2} c^{3} d^{2} + a^{2} b c d^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{\sqrt {-b c^{2} - a d^{2}} d^{7}} - \frac {12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b^{3} c^{5} d + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a b^{2} c^{3} d^{3} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a^{2} b c d^{5} + 22 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {7}{2}} c^{6} + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {5}{2}} c^{4} d^{2} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {3}{2}} c^{2} d^{4} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} \sqrt {b} d^{6} - 32 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a b^{3} c^{5} d - 37 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b^{2} c^{3} d^{3} - 5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{3} b c d^{5} + 11 \, a^{2} b^{\frac {5}{2}} c^{4} d^{2} + 13 \, a^{3} b^{\frac {3}{2}} c^{2} d^{4} + 2 \, a^{4} \sqrt {b} d^{6}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{2} d^{7}} \] Input:

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

Output:

1/8*sqrt(b*x^2 + a)*((2*x*(b^2*x/d^3 - 4*b^2*c/d^4) + 3*(8*b^4*c^2*d^20 + 
3*a*b^3*d^22)/(b^2*d^25))*x - 8*(10*b^4*c^3*d^19 + 7*a*b^3*c*d^21)/(b^2*d^ 
25)) - 15/8*(8*b^(5/2)*c^4 + 8*a*b^(3/2)*c^2*d^2 + a^2*sqrt(b)*d^4)*log(ab 
s(-sqrt(b)*x + sqrt(b*x^2 + a)))/d^7 - 15*(2*b^3*c^5 + 3*a*b^2*c^3*d^2 + a 
^2*b*c*d^4)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b* 
c^2 - a*d^2))/(sqrt(-b*c^2 - a*d^2)*d^7) - (12*(sqrt(b)*x - sqrt(b*x^2 + a 
))^3*b^3*c^5*d + 15*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a*b^2*c^3*d^3 + 3*(sqr 
t(b)*x - sqrt(b*x^2 + a))^3*a^2*b*c*d^5 + 22*(sqrt(b)*x - sqrt(b*x^2 + a)) 
^2*b^(7/2)*c^6 + 15*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(5/2)*c^4*d^2 - 9* 
(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(3/2)*c^2*d^4 - 2*(sqrt(b)*x - sqrt( 
b*x^2 + a))^2*a^3*sqrt(b)*d^6 - 32*(sqrt(b)*x - sqrt(b*x^2 + a))*a*b^3*c^5 
*d - 37*(sqrt(b)*x - sqrt(b*x^2 + a))*a^2*b^2*c^3*d^3 - 5*(sqrt(b)*x - sqr 
t(b*x^2 + a))*a^3*b*c*d^5 + 11*a^2*b^(5/2)*c^4*d^2 + 13*a^3*b^(3/2)*c^2*d^ 
4 + 2*a^4*sqrt(b)*d^6)/(((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*d^7)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 1245, normalized size of antiderivative = 3.83 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

int(x*(b*x^2+a)^(5/2)/(d*x+c)^3,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*b*c**3*d**2 + 240*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 + 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*b*c*d**4*x**2 + 240*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 + 480*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 + 24 
0*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**3*d**2*x**2 - 120*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a* 
b*c**3*d**2 - 240*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**2*d**3*x - 120 
*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c*d**4*x**2 - 240*sqrt(a*d**2 + b* 
c**2)*log(c + d*x)*b**2*c**5 - 480*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2 
*c**4*d*x - 240*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2*c**3*d**2*x**2 - 8 
*sqrt(a + b*x**2)*a**2*c*d**5 - 16*sqrt(a + b*x**2)*a**2*d**6*x - 200*sqrt 
(a + b*x**2)*a*b*c**3*d**3 - 310*sqrt(a + b*x**2)*a*b*c**2*d**4*x - 76*sqr 
t(a + b*x**2)*a*b*c*d**5*x**2 + 18*sqrt(a + b*x**2)*a*b*d**6*x**3 - 240*sq 
rt(a + b*x**2)*b**2*c**5*d - 360*sqrt(a + b*x**2)*b**2*c**4*d**2*x - 80*sq 
rt(a + b*x**2)*b**2*c**3*d**3*x**2 + 20*sqrt(a + b*x**2)*b**2*c**2*d**4*x* 
*3 - 8*sqrt(a + b*x**2)*b**2*c*d**5*x**4 + 4*sqrt(a + b*x**2)*b**2*d**6*x* 
*5 - 15*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a**2*c**2*d**4 - 30*s...