\(\int \frac {x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx\) [1201]

Optimal result

Integrand size = 22, antiderivative size = 210 \[ \int \frac {x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\frac {c^3 \sqrt {a+b x^2}}{2 d^2 \left (b c^2+a d^2\right ) (c+d x)^2}-\frac {3 c^2 \left (b c^2+2 a d^2\right ) \sqrt {a+b x^2}}{2 d^2 \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} d^3}+\frac {c \left (2 b^2 c^4+5 a b c^2 d^2+6 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^3 \left (b c^2+a d^2\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.20 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=-\frac {\frac {c^2 d \sqrt {a+b x^2} \left (b c^2 (2 c+3 d x)+a d^2 (5 c+6 d x)\right )}{\left (b c^2+a d^2\right )^2 (c+d x)^2}-\frac {2 c \left (2 b^2 c^4+5 a b c^2 d^2+6 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}}+\frac {2 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{2 d^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.21, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {603, 25, 2182, 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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

\(\displaystyle \frac {-\frac {\frac {c \left (6 a^2 d^4+5 a b c^2 d^2+2 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {2 \left (a d^2+b c^2\right )^2 \int \frac {1}{\sqrt {b x^2+a}}dx}{d}}{d^2 \left (a d^2+b c^2\right )}-\frac {3 c^2 \sqrt {a+b x^2} \left (2 a+\frac {b c^2}{d^2}\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^3 \sqrt {a+b x^2}}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {-\frac {\frac {c \left (6 a^2 d^4+5 a b c^2 d^2+2 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {2 \left (a d^2+b c^2\right )^2 \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}}{d^2 \left (a d^2+b c^2\right )}-\frac {3 c^2 \sqrt {a+b x^2} \left (2 a+\frac {b c^2}{d^2}\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^3 \sqrt {a+b x^2}}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

\(\displaystyle \frac {-\frac {-\frac {c \left (6 a^2 d^4+5 a b c^2 d^2+2 b^2 c^4\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )^2}{\sqrt {b} d}}{d^2 \left (a d^2+b c^2\right )}-\frac {3 c^2 \sqrt {a+b x^2} \left (2 a+\frac {b c^2}{d^2}\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^3 \sqrt {a+b x^2}}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

(c^3*Sqrt[a + b*x^2])/(2*d^2*(b*c^2 + a*d^2)*(c + d*x)^2) + ((-3*c^2*(2*a 
+ (b*c^2)/d^2)*Sqrt[a + b*x^2])/((b*c^2 + a*d^2)*(c + d*x)) - ((-2*(b*c^2 
+ a*d^2)^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d) - (c*(2*b^2*c 
^4 + 5*a*b*c^2*d^2 + 6*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^2*(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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(816\) vs. \(2(188)=376\).

Time = 0.42 (sec) , antiderivative size = 817, normalized size of antiderivative = 3.89

method	result	size
default	\(\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d^{3} \sqrt {b}}+\frac {3 c \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^{4} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {3 c^{2} \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^{5}}-\frac {c^{3} \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^{6}}\)	\(817\)

Input:

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

Output:

1/d^3*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)+3*c/d^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))+3/d^5*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)))-c^3/d^6*(-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)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (189) = 378\).

Time = 52.82 (sec) , antiderivative size = 2397, normalized size of antiderivative = 11.41 \[ \int \frac {x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(2*(b^3*c^8 + 3*a*b^2*c^6*d^2 + 3*a^2*b*c^4*d^4 + a^3*c^2*d^6 + (b^3* 
c^6*d^2 + 3*a*b^2*c^4*d^4 + 3*a^2*b*c^2*d^6 + a^3*d^8)*x^2 + 2*(b^3*c^7*d 
+ 3*a*b^2*c^5*d^3 + 3*a^2*b*c^3*d^5 + a^3*c*d^7)*x)*sqrt(b)*log(-2*b*x^2 - 
 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + (2*b^3*c^7 + 5*a*b^2*c^5*d^2 + 6*a^2*b 
*c^3*d^4 + (2*b^3*c^5*d^2 + 5*a*b^2*c^3*d^4 + 6*a^2*b*c*d^6)*x^2 + 2*(2*b^ 
3*c^6*d + 5*a*b^2*c^4*d^3 + 6*a^2*b*c^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*( 
2*b^3*c^7*d + 7*a*b^2*c^5*d^3 + 5*a^2*b*c^3*d^5 + 3*(b^3*c^6*d^2 + 3*a*b^2 
*c^4*d^4 + 2*a^2*b*c^2*d^6)*x)*sqrt(b*x^2 + a))/(b^4*c^8*d^3 + 3*a*b^3*c^6 
*d^5 + 3*a^2*b^2*c^4*d^7 + a^3*b*c^2*d^9 + (b^4*c^6*d^5 + 3*a*b^3*c^4*d^7 
+ 3*a^2*b^2*c^2*d^9 + a^3*b*d^11)*x^2 + 2*(b^4*c^7*d^4 + 3*a*b^3*c^5*d^6 + 
 3*a^2*b^2*c^3*d^8 + a^3*b*c*d^10)*x), 1/2*((2*b^3*c^7 + 5*a*b^2*c^5*d^2 + 
 6*a^2*b*c^3*d^4 + (2*b^3*c^5*d^2 + 5*a*b^2*c^3*d^4 + 6*a^2*b*c*d^6)*x^2 + 
 2*(2*b^3*c^6*d + 5*a*b^2*c^4*d^3 + 6*a^2*b*c^2*d^5)*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)) + (b^3*c^8 + 3*a*b^2*c^6*d^2 + 3*a^2*b*c 
^4*d^4 + a^3*c^2*d^6 + (b^3*c^6*d^2 + 3*a*b^2*c^4*d^4 + 3*a^2*b*c^2*d^6 + 
a^3*d^8)*x^2 + 2*(b^3*c^7*d + 3*a*b^2*c^5*d^3 + 3*a^2*b*c^3*d^5 + a^3*c*d^ 
7)*x)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - (2*b^3*...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (189) = 378\).

Time = 0.06 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.80 \[ \int \frac {x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\frac {3 \, \sqrt {b x^{2} + a} b c^{4}}{2 \, {\left (b^{2} c^{4} d^{3} x + 2 \, a b c^{2} d^{5} x + a^{2} d^{7} x + b^{2} c^{5} d^{2} + 2 \, a b c^{3} d^{4} + a^{2} c d^{6}\right )}} + \frac {\sqrt {b x^{2} + a} c^{3}}{2 \, {\left (b c^{2} d^{4} x^{2} + a d^{6} x^{2} + 2 \, b c^{3} d^{3} x + 2 \, a c d^{5} x + b c^{4} d^{2} + a c^{2} d^{4}\right )}} - \frac {3 \, \sqrt {b x^{2} + a} c^{2}}{b c^{2} d^{3} x + a d^{5} x + b c^{3} d^{2} + a c d^{4}} + \frac {\operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{3}} - \frac {3 \, b^{2} c^{5} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} d^{8}} + \frac {7 \, b c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{6}} - \frac {3 \, c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{4}} \] Input:

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

Output:

3/2*sqrt(b*x^2 + a)*b*c^4/(b^2*c^4*d^3*x + 2*a*b*c^2*d^5*x + a^2*d^7*x + b 
^2*c^5*d^2 + 2*a*b*c^3*d^4 + a^2*c*d^6) + 1/2*sqrt(b*x^2 + a)*c^3/(b*c^2*d 
^4*x^2 + a*d^6*x^2 + 2*b*c^3*d^3*x + 2*a*c*d^5*x + b*c^4*d^2 + a*c^2*d^4) 
- 3*sqrt(b*x^2 + a)*c^2/(b*c^2*d^3*x + a*d^5*x + b*c^3*d^2 + a*c*d^4) + ar 
csinh(b*x/sqrt(a*b))/(sqrt(b)*d^3) - 3/2*b^2*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)^(5/2)*d^8) 
+ 7/2*b*c^3*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^6) - 3*c*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^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (189) = 378\).

Time = 0.15 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.20 \[ \int \frac {x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=-\frac {{\left (2 \, b^{2} c^{5} + 5 \, a b c^{3} d^{2} + 6 \, a^{2} c d^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b^{2} c^{4} d^{3} + 2 \, a b c^{2} d^{5} + a^{2} d^{7}\right )} \sqrt {-b c^{2} - a d^{2}}} - \frac {4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b^{2} c^{5} d + 7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a b c^{3} d^{3} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {5}{2}} c^{6} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} c^{4} d^{2} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} \sqrt {b} c^{2} d^{4} - 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a b^{2} c^{5} d - 17 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b c^{3} d^{3} + 3 \, a^{2} b^{\frac {3}{2}} c^{4} d^{2} + 6 \, a^{3} \sqrt {b} c^{2} d^{4}}{{\left (b^{2} c^{4} d^{3} + 2 \, a b c^{2} d^{5} + a^{2} d^{7}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{2}} - \frac {\log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{\sqrt {b} d^{3}} \] Input:

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

Output:

-(2*b^2*c^5 + 5*a*b*c^3*d^2 + 6*a^2*c*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^3 + 2*a*b*c^2*d^5 
 + a^2*d^7)*sqrt(-b*c^2 - a*d^2)) - (4*(sqrt(b)*x - sqrt(b*x^2 + a))^3*b^2 
*c^5*d + 7*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a*b*c^3*d^3 + 6*(sqrt(b)*x - sq 
rt(b*x^2 + a))^2*b^(5/2)*c^6 + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2) 
*c^4*d^2 - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*sqrt(b)*c^2*d^4 - 8*(sqrt 
(b)*x - sqrt(b*x^2 + a))*a*b^2*c^5*d - 17*(sqrt(b)*x - sqrt(b*x^2 + a))*a^ 
2*b*c^3*d^3 + 3*a^2*b^(3/2)*c^4*d^2 + 6*a^3*sqrt(b)*c^2*d^4)/((b^2*c^4*d^3 
 + 2*a*b*c^2*d^5 + a^2*d^7)*((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) - log(abs(-sqrt(b)*x + sqrt(b*x 
^2 + a)))/(sqrt(b)*d^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1729, normalized size of antiderivative = 8.23 \[ \int \frac {x^3}{(c+d x)^3 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(6*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*c**3*d**4 + 12*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*c**2*d**5*x + 6*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*c*d**6*x**2 + 5*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**2*c**5*d**2 + 10*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**2*c** 
4*d**3*x + 5*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**2*c**3*d**4*x**2 + 2*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**3*c**7 + 4*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)*b**3*c**6*d*x + 2*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**3*c**5*d**2*x**2 - 6*sqrt(a*d**2 + b*c 
**2)*log(c + d*x)*a**2*b*c**3*d**4 - 12*sqrt(a*d**2 + b*c**2)*log(c + d*x) 
*a**2*b*c**2*d**5*x - 6*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c*d**6*x 
**2 - 5*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**5*d**2 - 10*sqrt(a*d* 
*2 + b*c**2)*log(c + d*x)*a*b**2*c**4*d**3*x - 5*sqrt(a*d**2 + b*c**2)*log 
(c + d*x)*a*b**2*c**3*d**4*x**2 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b** 
3*c**7 - 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**3*c**6*d*x - 2*sqrt(a*d** 
2 + b*c**2)*log(c + d*x)*b**3*c**5*d**2*x**2 - 5*sqrt(a + b*x**2)*a**2*...