\(\int \frac {x^3}{(c+d x)^2 \sqrt {a+b x^2}} \, dx\) [1192]

Optimal result

Integrand size = 22, antiderivative size = 160 \[ \int \frac {x^3}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2}}{b d^2}+\frac {c^3 \sqrt {a+b x^2}}{d^2 \left (b c^2+a d^2\right ) (c+d x)}-\frac {2 c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} d^3}-\frac {c^2 \left (2 b c^2+3 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^3 \left (b c^2+a d^2\right )^{3/2}} \] Output:

(b*x^2+a)^(1/2)/b/d^2+c^3*(b*x^2+a)^(1/2)/d^2/(a*d^2+b*c^2)/(d*x+c)-2*c*ar 
ctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)/d^3-c^2*(3*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^3/(a*d^2+b*c^2)^(3/2)
 

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.08 \[ \int \frac {x^3}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (a d^2 (c+d x)+b c^2 (2 c+d x)\right )}{b \left (b c^2+a d^2\right ) (c+d x)}+\frac {2 c^2 \left (2 b c^2+3 a d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+\frac {2 c \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{d^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {603, 25, 2185, 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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(c^3*Sqrt[a + b*x^2])/(d^2*(b*c^2 + a*d^2)*(c + d*x)) + ((a/b + c^2/d^2)*S 
qrt[a + b*x^2] + (c*((-2*(b*c^2 + a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^ 
2]])/(Sqrt[b]*d) - (c*(2*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*Sqrt[b*c^2 + a*d^2])))/d^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 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(385\) vs. \(2(144)=288\).

Time = 0.39 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.41

method	result	size
risch	\(\frac {\sqrt {b \,x^{2}+a}}{b \,d^{2}}+\frac {c^{3} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{d^{3} \left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}+\frac {c^{4} b \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{4} \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {2 c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d^{3} \sqrt {b}}-\frac {3 c^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{4} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\)	\(386\)
default	\(\frac {\sqrt {b \,x^{2}+a}}{b \,d^{2}}-\frac {2 c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d^{3} \sqrt {b}}-\frac {3 c^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{4} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {c^{3} \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{5}}\)	\(390\)

Input:

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

Output:

(b*x^2+a)^(1/2)/b/d^2+c^3/d^3/(a*d^2+b*c^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)+c^4/d^4*b/(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))-2*c/d^3* 
ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)-3/d^4*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. 346 vs. \(2 (145) = 290\).

Time = 3.84 (sec) , antiderivative size = 1449, normalized size of antiderivative = 9.06 \[ \int \frac {x^3}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*(2*(b^2*c^6 + 2*a*b*c^4*d^2 + a^2*c^2*d^4 + (b^2*c^5*d + 2*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) 
+ (2*b^2*c^5 + 3*a*b*c^3*d^2 + (2*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*(2*b^2*c^5*d + 3*a*b*c^3*d^3 + a^2*c*d^5 + (b^2*c^4*d^2 + 2 
*a*b*c^2*d^4 + a^2*d^6)*x)*sqrt(b*x^2 + a))/(b^3*c^5*d^3 + 2*a*b^2*c^3*d^5 
 + a^2*b*c*d^7 + (b^3*c^4*d^4 + 2*a*b^2*c^2*d^6 + a^2*b*d^8)*x), -((2*b^2* 
c^5 + 3*a*b*c^3*d^2 + (2*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)) - (b^2*c^6 + 2*a*b*c^4*d^2 + a^2*c^2*d^4 
+ (b^2*c^5*d + 2*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) - (2*b^2*c^5*d + 3*a*b*c^3*d^3 + a^2*c*d^5 + (b 
^2*c^4*d^2 + 2*a*b*c^2*d^4 + a^2*d^6)*x)*sqrt(b*x^2 + a))/(b^3*c^5*d^3 + 2 
*a*b^2*c^3*d^5 + a^2*b*c*d^7 + (b^3*c^4*d^4 + 2*a*b^2*c^2*d^6 + a^2*b*d^8) 
*x), 1/2*(4*(b^2*c^6 + 2*a*b*c^4*d^2 + a^2*c^2*d^4 + (b^2*c^5*d + 2*a*b*c^ 
3*d^3 + a^2*c*d^5)*x)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (2*b^2 
*c^5 + 3*a*b*c^3*d^2 + (2*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*s 
qrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + ...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.21 \[ \int \frac {x^3}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} c^{3}}{b c^{2} d^{3} x + a d^{5} x + b c^{3} d^{2} + a c d^{4}} - \frac {2 \, c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{3}} - \frac {b c^{4} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{6}} + \frac {3 \, c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{4}} + \frac {\sqrt {b x^{2} + a}}{b d^{2}} \] Input:

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 805, normalized size of antiderivative = 5.03 \[ \int \frac {x^3}{(c+d x)^2 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*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 + 3*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 + 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**2*c**5 + 2*s 
qrt(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 - 3*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**3*d**2 - 
3*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**2*d**3*x - 2*sqrt(a*d**2 + b*c 
**2)*log(c + d*x)*b**2*c**5 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2*c* 
*4*d*x + sqrt(a + b*x**2)*a**2*c*d**5 + sqrt(a + b*x**2)*a**2*d**6*x + 3*s 
qrt(a + b*x**2)*a*b*c**3*d**3 + 2*sqrt(a + b*x**2)*a*b*c**2*d**4*x + 2*sqr 
t(a + b*x**2)*b**2*c**5*d + sqrt(a + b*x**2)*b**2*c**4*d**2*x + sqrt(b)*lo 
g(sqrt(a + b*x**2) - sqrt(b)*x)*a**2*c**2*d**4 + sqrt(b)*log(sqrt(a + b*x* 
*2) - sqrt(b)*x)*a**2*c*d**5*x + 2*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)* 
x)*a*b*c**4*d**2 + 2*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a*b*c**3*d* 
*3*x + sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*b**2*c**6 + sqrt(b)*log(s 
qrt(a + b*x**2) - sqrt(b)*x)*b**2*c**5*d*x - sqrt(b)*log(sqrt(a + b*x**2) 
+ sqrt(b)*x)*a**2*c**2*d**4 - sqrt(b)*log(sqrt(a + b*x**2) + sqrt(b)*x)*a* 
*2*c*d**5*x - 2*sqrt(b)*log(sqrt(a + b*x**2) + sqrt(b)*x)*a*b*c**4*d**2 - 
2*sqrt(b)*log(sqrt(a + b*x**2) + sqrt(b)*x)*a*b*c**3*d**3*x - sqrt(b)*log( 
sqrt(a + b*x**2) + sqrt(b)*x)*b**2*c**6 - sqrt(b)*log(sqrt(a + b*x**2) ...