\(\int \frac {x (a+b x^2)^{3/2}}{(c+d x)^3} \, dx\) [1097]

Optimal result

Integrand size = 20, antiderivative size = 231 \[ \int \frac {x \left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=-\frac {3 b c \sqrt {a+b x^2}}{d^4}+\frac {b x \sqrt {a+b x^2}}{2 d^3}+\frac {c \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}{2 d^4 (c+d x)^2}-\frac {\left (7 b c^2+2 a d^2\right ) \sqrt {a+b x^2}}{2 d^4 (c+d x)}+\frac {3 \sqrt {b} \left (4 b c^2+a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 d^5}+\frac {3 b c \left (4 b c^2+3 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^5 \sqrt {b c^2+a d^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.83 \[ \int \frac {x \left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=-\frac {\frac {d \sqrt {a+b x^2} \left (a d^2 (c+2 d x)+b \left (12 c^3+18 c^2 d x+4 c d^2 x^2-d^3 x^3\right )\right )}{(c+d x)^2}-\frac {6 b c \left (4 b c^2+3 a d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+3 \sqrt {b} \left (4 b c^2+a d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 d^5} \] Input:

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

Output:

-1/2*((d*Sqrt[a + b*x^2]*(a*d^2*(c + 2*d*x) + b*(12*c^3 + 18*c^2*d*x + 4*c 
*d^2*x^2 - d^3*x^3)))/(c + d*x)^2 - (6*b*c*(4*b*c^2 + 3*a*d^2)*ArcTan[(Sqr 
t[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/Sqrt[-(b*c^2) 
 - a*d^2] + 3*Sqrt[b]*(4*b*c^2 + a*d^2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2] 
])/d^5
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.87, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {590, 27, 681, 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.

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

\(\Big \downarrow \) 590

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 681

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

((2*c + d*x)*(a + b*x^2)^(3/2))/(2*d^2*(c + d*x)^2) + (3*(-(((4*b*c^2 + a* 
d^2 + 2*b*c*d*x)*Sqrt[a + b*x^2])/(d^2*(c + d*x))) - (b*(-(((4*b*c^2 + a*d 
^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - (c*(4*b*c^2 + 3*a 
*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(d*Sqr 
t[b*c^2 + a*d^2])))/d^2))/(2*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 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. \(918\) vs. \(2(201)=402\).

Time = 0.44 (sec) , antiderivative size = 919, normalized size of antiderivative = 3.98

method	result	size
risch	\(-\frac {\left (-d x +6 c \right ) \sqrt {b \,x^{2}+a}\, b}{2 d^{4}}+\frac {\frac {2 \left (a^{2} d^{4}+6 b \,c^{2} d^{2} a +5 b^{2} c^{4}\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}}+\frac {3 \sqrt {b}\, \left (a \,d^{2}+4 b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d}-\frac {2 c \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\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}}}}{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^{4}}+\frac {4 b c \left (3 a \,d^{2}+5 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}}}}}{2 d^{4}}\)	\(919\)
default	\(\text {Expression too large to display}\)	\(2345\)

Input:

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

Output:

-1/2*(-d*x+6*c)*(b*x^2+a)^(1/2)*b/d^4+1/2/d^4*(2/d^3*(a^2*d^4+6*a*b*c^2*d^ 
2+5*b^2*c^4)*(-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)))+3*b^(1/2)*(a*d^2+4*b 
*c^2)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))-2*c*(a^2*d^4+2*a*b*c^2*d^2+b^2*c^4)/ 
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)))+4*b*c/d^2*(3*a*d^2+5*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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (202) = 404\).

Time = 1.52 (sec) , antiderivative size = 1837, normalized size of antiderivative = 7.95 \[ \int \frac {x \left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(3*(4*b^2*c^6 + 5*a*b*c^4*d^2 + a^2*c^2*d^4 + (4*b^2*c^4*d^2 + 5*a*b* 
c^2*d^4 + a^2*d^6)*x^2 + 2*(4*b^2*c^5*d + 5*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) + 3*(4*b^2*c^5 + 3*a 
*b*c^3*d^2 + (4*b^2*c^3*d^2 + 3*a*b*c*d^4)*x^2 + 2*(4*b^2*c^4*d + 3*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*(12*b^2*c^5*d + 13*a*b*c^3*d^3 + a^2*c* 
d^5 - (b^2*c^2*d^4 + a*b*d^6)*x^3 + 4*(b^2*c^3*d^3 + a*b*c*d^5)*x^2 + 2*(9 
*b^2*c^4*d^2 + 10*a*b*c^2*d^4 + a^2*d^6)*x)*sqrt(b*x^2 + a))/(b*c^4*d^5 + 
a*c^2*d^7 + (b*c^2*d^7 + a*d^9)*x^2 + 2*(b*c^3*d^6 + a*c*d^8)*x), 1/4*(6*( 
4*b^2*c^5 + 3*a*b*c^3*d^2 + (4*b^2*c^3*d^2 + 3*a*b*c*d^4)*x^2 + 2*(4*b^2*c 
^4*d + 3*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 
)) + 3*(4*b^2*c^6 + 5*a*b*c^4*d^2 + a^2*c^2*d^4 + (4*b^2*c^4*d^2 + 5*a*b*c 
^2*d^4 + a^2*d^6)*x^2 + 2*(4*b^2*c^5*d + 5*a*b*c^3*d^3 + a^2*c*d^5)*x)*sqr 
t(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(12*b^2*c^5*d + 1 
3*a*b*c^3*d^3 + a^2*c*d^5 - (b^2*c^2*d^4 + a*b*d^6)*x^3 + 4*(b^2*c^3*d^3 + 
 a*b*c*d^5)*x^2 + 2*(9*b^2*c^4*d^2 + 10*a*b*c^2*d^4 + a^2*d^6)*x)*sqrt(b*x 
^2 + a))/(b*c^4*d^5 + a*c^2*d^7 + (b*c^2*d^7 + a*d^9)*x^2 + 2*(b*c^3*d^6 + 
 a*c*d^8)*x), -1/4*(6*(4*b^2*c^6 + 5*a*b*c^4*d^2 + a^2*c^2*d^4 + (4*b^2...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (202) = 404\).

Time = 0.08 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.80 \[ \int \frac {x \left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=-\frac {3 \, \sqrt {b x^{2} + a} b^{2} c^{3}}{2 \, {\left (b c^{2} d^{4} + a d^{6}\right )}} + \frac {3 \, \sqrt {b x^{2} + a} b^{2} c^{2} x}{2 \, {\left (b c^{2} d^{3} + a d^{5}\right )}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b c^{2}}{2 \, {\left (b c^{2} d^{3} x + a d^{5} x + b c^{3} d^{2} + a c d^{4}\right )}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c}{2 \, {\left (b c^{2} d^{2} x^{2} + a d^{4} x^{2} + 2 \, b c^{3} d x + 2 \, a c d^{3} x + b c^{4} + a c^{2} d^{2}\right )}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b c}{2 \, {\left (b c^{2} d^{2} + a d^{4}\right )}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{d^{3} x + c d^{2}} + \frac {3 \, \sqrt {b x^{2} + a} b x}{2 \, d^{3}} + \frac {6 \, b^{\frac {3}{2}} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{5}} + \frac {3 \, a \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, d^{3}} - \frac {3 \, b^{2} c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} d^{6}} - \frac {9 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} b c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, d^{4}} - \frac {9 \, \sqrt {b x^{2} + a} b c}{2 \, d^{4}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (202) = 404\).

Time = 0.15 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.84 \[ \int \frac {x \left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=\frac {1}{2} \, \sqrt {b x^{2} + a} {\left (\frac {b x}{d^{3}} - \frac {6 \, b c}{d^{4}}\right )} - \frac {3 \, {\left (4 \, b^{\frac {3}{2}} c^{2} + a \sqrt {b} d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, d^{5}} - \frac {3 \, {\left (4 \, b^{2} c^{3} + 3 \, a b c d^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{\sqrt {-b c^{2} - a d^{2}} d^{5}} - \frac {8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b^{2} c^{3} d + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a b c d^{3} + 14 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {5}{2}} c^{4} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} c^{2} d^{2} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} \sqrt {b} d^{4} - 20 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a b^{2} c^{3} d - 5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b c d^{3} + 7 \, a^{2} b^{\frac {3}{2}} c^{2} d^{2} + 2 \, a^{3} \sqrt {b} d^{4}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{2} d^{5}} \] Input:

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

Output:

1/2*sqrt(b*x^2 + a)*(b*x/d^3 - 6*b*c/d^4) - 3/2*(4*b^(3/2)*c^2 + a*sqrt(b) 
*d^2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/d^5 - 3*(4*b^2*c^3 + 3*a*b*c* 
d^2)*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^5) - (8*(sqrt(b)*x - sqrt(b*x^2 + a))^3*b^2 
*c^3*d + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a*b*c*d^3 + 14*(sqrt(b)*x - sqr 
t(b*x^2 + a))^2*b^(5/2)*c^4 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2)* 
c^2*d^2 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*sqrt(b)*d^4 - 20*(sqrt(b)* 
x - sqrt(b*x^2 + a))*a*b^2*c^3*d - 5*(sqrt(b)*x - sqrt(b*x^2 + a))*a^2*b*c 
*d^3 + 7*a^2*b^(3/2)*c^2*d^2 + 2*a^3*sqrt(b)*d^4)/(((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^5)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1239, normalized size of antiderivative = 5.36 \[ \int \frac {x \left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(18*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 + 36*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 + 18*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 + 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**2*c**5 + 48*sqrt(a*d**2 + b*c**2)*log( - sqr 
t(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**2*c**4*d*x + 24*sqrt 
(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b* 
c*x)*b**2*c**3*d**2*x**2 - 18*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**3* 
d**2 - 36*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**2*d**3*x - 18*sqrt(a*d 
**2 + b*c**2)*log(c + d*x)*a*b*c*d**4*x**2 - 24*sqrt(a*d**2 + b*c**2)*log( 
c + d*x)*b**2*c**5 - 48*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2*c**4*d*x - 
 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2*c**3*d**2*x**2 - 2*sqrt(a + b* 
x**2)*a**2*c*d**5 - 4*sqrt(a + b*x**2)*a**2*d**6*x - 26*sqrt(a + b*x**2)*a 
*b*c**3*d**3 - 40*sqrt(a + b*x**2)*a*b*c**2*d**4*x - 8*sqrt(a + b*x**2)*a* 
b*c*d**5*x**2 + 2*sqrt(a + b*x**2)*a*b*d**6*x**3 - 24*sqrt(a + b*x**2)*b** 
2*c**5*d - 36*sqrt(a + b*x**2)*b**2*c**4*d**2*x - 8*sqrt(a + b*x**2)*b**2* 
c**3*d**3*x**2 + 2*sqrt(a + b*x**2)*b**2*c**2*d**4*x**3 - 3*sqrt(b)*log(sq 
rt(a + b*x**2) - sqrt(b)*x)*a**2*c**2*d**4 - 6*sqrt(b)*log(sqrt(a + b*x**2 
) - sqrt(b)*x)*a**2*c*d**5*x - 3*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)...