\(\int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^2} \, dx\) [1006]

Optimal result

Integrand size = 22, antiderivative size = 199 \[ \int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^2} \, dx=\frac {3 c^2 \sqrt {a+b x^2}}{d^4}-\frac {c x \sqrt {a+b x^2}}{d^3}+\frac {c^3 \sqrt {a+b x^2}}{d^4 (c+d x)}+\frac {\left (a+b x^2\right )^{3/2}}{3 b d^2}-\frac {c \left (4 b c^2+a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} d^5}-\frac {c^2 \left (4 b c^2+3 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^5 \sqrt {b c^2+a d^2}} \] Output:

3*c^2*(b*x^2+a)^(1/2)/d^4-c*x*(b*x^2+a)^(1/2)/d^3+c^3*(b*x^2+a)^(1/2)/d^4/ 
(d*x+c)+1/3*(b*x^2+a)^(3/2)/b/d^2-c*(a*d^2+4*b*c^2)*arctanh(b^(1/2)*x/(b*x 
^2+a)^(1/2))/b^(1/2)/d^5-c^2*(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 = 0.83 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^2} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (a d^2 (c+d x)+b \left (12 c^3+6 c^2 d x-2 c d^2 x^2+d^3 x^3\right )\right )}{b (c+d x)}-\frac {6 c^2 \left (4 b c^2+3 a d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+\frac {3 c \left (4 b c^2+a d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{3 d^5} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.32, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {603, 25, 2185, 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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 682

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(c^3*(a + b*x^2)^(3/2))/(d^2*(b*c^2 + a*d^2)*(c + d*x)) + (((a/b + c^2/d^2 
)*(a + b*x^2)^(3/2))/3 + (c*(((c*(4*b*c^2 + 3*a*d^2) - d*(2*b*c^2 + a*d^2) 
*x)*Sqrt[a + b*x^2])/d^2 + ((b*c^2 + a*d^2)*(-(((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)*ArcTan 
h[(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))/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 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]
 

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. \(453\) vs. \(2(177)=354\).

Time = 0.42 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.28

method	result	size
risch	\(\frac {\left (b \,x^{2} d^{2}-3 b c d x +a \,d^{2}+9 b \,c^{2}\right ) \sqrt {b \,x^{2}+a}}{3 b \,d^{4}}-\frac {c \left (\frac {\left (a \,d^{2}+4 b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {c^{2} \left (a \,d^{2}+b \,c^{2}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}+\frac {c \left (3 a \,d^{2}+5 b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{4}}\)	\(454\)
default	\(\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 b \,d^{2}}-\frac {2 c \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{d^{3}}+\frac {3 c^{2} \left (\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}}}-\frac {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}}}\right )}{d}-\frac {\left (a \,d^{2}+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}}}}\right )}{d^{4}}-\frac {c^{3} \left (-\frac {d^{2} \left (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}}\right )^{\frac {3}{2}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \left (\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}}}-\frac {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}}}\right )}{d}-\frac {\left (a \,d^{2}+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}}}}\right )}{a \,d^{2}+b \,c^{2}}+\frac {2 b \,d^{2} \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right ) \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}}}}{4 b}+\frac {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}}}\right )}{8 b^{\frac {3}{2}}}\right )}{a \,d^{2}+b \,c^{2}}\right )}{d^{5}}\)	\(863\)

Input:

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

Output:

1/3*(b*d^2*x^2-3*b*c*d*x+a*d^2+9*b*c^2)*(b*x^2+a)^(1/2)/b/d^4-c/d^4*((a*d^ 
2+4*b*c^2)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)+c^2*(a*d^2+b*c^2)/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)))+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. 366 vs. \(2 (178) = 356\).

Time = 1.53 (sec) , antiderivative size = 1529, normalized size of antiderivative = 7.68 \[ \int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

integrate(x**3*(b*x**2+a)**(1/2)/(d*x+c)**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 = 228, normalized size of antiderivative = 1.15 \[ \int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^2} \, dx=\frac {\sqrt {b x^{2} + a} c^{3}}{d^{5} x + c d^{4}} - \frac {\sqrt {b x^{2} + a} c x}{d^{3}} - \frac {4 \, \sqrt {b} c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{5}} - \frac {a c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{3}} + \frac {b c^{4} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{6}} + \frac {3 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{4}} + \frac {3 \, \sqrt {b x^{2} + a} c^{2}}{d^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{3 \, b d^{2}} \] Input:

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^2} \, dx=\int \frac {x^3\,\sqrt {b\,x^2+a}}{{\left (c+d\,x\right )}^2} \,d x \] 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 = 0.22 (sec) , antiderivative size = 865, normalized size of antiderivative = 4.35 \[ \int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

int(x^3*(b*x^2+a)^(1/2)/(d*x+c)^2,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 + 18*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*b*c**2*d**3*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**5 + 
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**4*d*x - 18*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**3*d* 
*2 - 18*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**2*d**3*x - 24*sqrt(a*d** 
2 + b*c**2)*log(c + d*x)*b**2*c**5 - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x) 
*b**2*c**4*d*x + 2*sqrt(a + b*x**2)*a**2*c*d**5 + 2*sqrt(a + b*x**2)*a**2* 
d**6*x + 26*sqrt(a + b*x**2)*a*b*c**3*d**3 + 14*sqrt(a + b*x**2)*a*b*c**2* 
d**4*x - 4*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 + 12*sqrt(a + b*x**2)*b**2*c**4*d** 
2*x - 4*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(sqrt(a + b*x**2) - sqrt(b)*x)*a**2*c**2*d**4 + 
 3*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a**2*c*d**5*x + 15*sqrt(b)*lo 
g(sqrt(a + b*x**2) - sqrt(b)*x)*a*b*c**4*d**2 + 15*sqrt(b)*log(sqrt(a + b* 
x**2) - sqrt(b)*x)*a*b*c**3*d**3*x + 12*sqrt(b)*log(sqrt(a + b*x**2) - sqr 
t(b)*x)*b**2*c**6 + 12*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*b**2*c**5 
*d*x - 3*sqrt(b)*log(sqrt(a + b*x**2) + sqrt(b)*x)*a**2*c**2*d**4 - 3*sqrt 
(b)*log(sqrt(a + b*x**2) + sqrt(b)*x)*a**2*c*d**5*x - 15*sqrt(b)*log(sq...