\(\int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^3} \, dx\) [1016]

Optimal result

Integrand size = 22, antiderivative size = 249 \[ \int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^3} \, dx=-\frac {3 c \sqrt {a+b x^2}}{d^4}+\frac {x \sqrt {a+b x^2}}{2 d^3}+\frac {c^3 \sqrt {a+b x^2}}{2 d^4 (c+d x)^2}-\frac {c^2 \left (7 b c^2+6 a d^2\right ) \sqrt {a+b x^2}}{2 d^4 \left (b c^2+a d^2\right ) (c+d x)}+\frac {\left (12 b c^2+a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b} d^5}+\frac {c \left (12 b^2 c^4+19 a b c^2 d^2+6 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^5 \left (b c^2+a d^2\right )^{3/2}} \] Output:

-3*c*(b*x^2+a)^(1/2)/d^4+1/2*x*(b*x^2+a)^(1/2)/d^3+1/2*c^3*(b*x^2+a)^(1/2) 
/d^4/(d*x+c)^2-1/2*c^2*(6*a*d^2+7*b*c^2)*(b*x^2+a)^(1/2)/d^4/(a*d^2+b*c^2) 
/(d*x+c)+1/2*(a*d^2+12*b*c^2)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)/d 
^5+1/2*c*(6*a^2*d^4+19*a*b*c^2*d^2+12*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^5/(a*d^2+b*c^2)^(3/2)
 

Mathematica [A] (verified)

Time = 1.70 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (b c^2 \left (-12 c^3-18 c^2 d x-4 c d^2 x^2+d^3 x^3\right )+a d^2 \left (-11 c^3-17 c^2 d x-4 c d^2 x^2+d^3 x^3\right )\right )}{\left (b c^2+a d^2\right ) (c+d x)^2}-\frac {2 c \left (12 b^2 c^4+19 a b c^2 d^2+6 a^2 d^4\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}-\frac {\left (12 b c^2+a d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{2 d^5} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(b*c^2*(-12*c^3 - 18*c^2*d*x - 4*c*d^2*x^2 + d^3*x^3) 
+ a*d^2*(-11*c^3 - 17*c^2*d*x - 4*c*d^2*x^2 + d^3*x^3)))/((b*c^2 + a*d^2)* 
(c + d*x)^2) - (2*c*(12*b^2*c^4 + 19*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)^(3/2) - ((12*b*c^2 + a*d^2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/Sqrt 
[b])/(2*d^5)
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.42, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {603, 25, 2182, 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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2182

\(\displaystyle \frac {-\frac {\int \frac {\left (a c \left (\frac {3 b c^2}{d}+4 a d\right )-2 \left (\frac {6 b^2 c^4}{d^2}+8 a b c^2+a^2 d^2\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{3/2} \left (6 a+\frac {5 b c^2}{d^2}\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^3 \left (a+b x^2\right )^{3/2}}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {-\frac {\frac {\int \frac {2 b \left (b c^2+a d^2\right ) \left (a c d \left (6 b c^2+5 a d^2\right )-\left (b c^2+a d^2\right ) \left (12 b c^2+a d^2\right ) x\right )}{d^2 (c+d x) \sqrt {b x^2+a}}dx}{2 b d^2}+\frac {\sqrt {a+b x^2} \left (c \left (6 a^2 d^2+19 a b c^2+\frac {12 b^2 c^4}{d^2}\right )-d x \left (a^2 d^2+8 a b c^2+\frac {6 b^2 c^4}{d^2}\right )\right )}{d^2}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{3/2} \left (6 a+\frac {5 b c^2}{d^2}\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^3 \left (a+b x^2\right )^{3/2}}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\frac {\left (a d^2+b c^2\right ) \int \frac {a c d \left (6 b c^2+5 a d^2\right )-\left (b c^2+a d^2\right ) \left (12 b c^2+a d^2\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{d^4}+\frac {\sqrt {a+b x^2} \left (c \left (6 a^2 d^2+19 a b c^2+\frac {12 b^2 c^4}{d^2}\right )-d x \left (a^2 d^2+8 a b c^2+\frac {6 b^2 c^4}{d^2}\right )\right )}{d^2}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{3/2} \left (6 a+\frac {5 b c^2}{d^2}\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^3 \left (a+b x^2\right )^{3/2}}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {-\frac {\frac {\left (a d^2+b c^2\right ) \left (\frac {c \left (6 a^2 d^4+19 a b c^2 d^2+12 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (a d^2+b c^2\right ) \left (a d^2+12 b c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}\right )}{d^4}+\frac {\sqrt {a+b x^2} \left (c \left (6 a^2 d^2+19 a b c^2+\frac {12 b^2 c^4}{d^2}\right )-d x \left (a^2 d^2+8 a b c^2+\frac {6 b^2 c^4}{d^2}\right )\right )}{d^2}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{3/2} \left (6 a+\frac {5 b c^2}{d^2}\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^3 \left (a+b x^2\right )^{3/2}}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {-\frac {\frac {\left (a d^2+b c^2\right ) \left (\frac {c \left (6 a^2 d^4+19 a b c^2 d^2+12 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (a d^2+b c^2\right ) \left (a d^2+12 b c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{d^4}+\frac {\sqrt {a+b x^2} \left (c \left (6 a^2 d^2+19 a b c^2+\frac {12 b^2 c^4}{d^2}\right )-d x \left (a^2 d^2+8 a b c^2+\frac {6 b^2 c^4}{d^2}\right )\right )}{d^2}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{3/2} \left (6 a+\frac {5 b c^2}{d^2}\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^3 \left (a+b x^2\right )^{3/2}}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\frac {\left (a d^2+b c^2\right ) \left (\frac {c \left (6 a^2 d^4+19 a b c^2 d^2+12 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right ) \left (a d^2+12 b c^2\right )}{\sqrt {b} d}\right )}{d^4}+\frac {\sqrt {a+b x^2} \left (c \left (6 a^2 d^2+19 a b c^2+\frac {12 b^2 c^4}{d^2}\right )-d x \left (a^2 d^2+8 a b c^2+\frac {6 b^2 c^4}{d^2}\right )\right )}{d^2}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{3/2} \left (6 a+\frac {5 b c^2}{d^2}\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^3 \left (a+b x^2\right )^{3/2}}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {-\frac {\frac {\left (a d^2+b c^2\right ) \left (-\frac {c \left (6 a^2 d^4+19 a b c^2 d^2+12 b^2 c^4\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right ) \left (a d^2+12 b c^2\right )}{\sqrt {b} d}\right )}{d^4}+\frac {\sqrt {a+b x^2} \left (c \left (6 a^2 d^2+19 a b c^2+\frac {12 b^2 c^4}{d^2}\right )-d x \left (a^2 d^2+8 a b c^2+\frac {6 b^2 c^4}{d^2}\right )\right )}{d^2}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{3/2} \left (6 a+\frac {5 b c^2}{d^2}\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^3 \left (a+b x^2\right )^{3/2}}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\frac {\left (a d^2+b c^2\right ) \left (-\frac {c \left (6 a^2 d^4+19 a b c^2 d^2+12 b^2 c^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d \sqrt {a d^2+b c^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right ) \left (a d^2+12 b c^2\right )}{\sqrt {b} d}\right )}{d^4}+\frac {\sqrt {a+b x^2} \left (c \left (6 a^2 d^2+19 a b c^2+\frac {12 b^2 c^4}{d^2}\right )-d x \left (a^2 d^2+8 a b c^2+\frac {6 b^2 c^4}{d^2}\right )\right )}{d^2}}{a d^2+b c^2}-\frac {c^2 \left (a+b x^2\right )^{3/2} \left (6 a+\frac {5 b c^2}{d^2}\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {c^3 \left (a+b x^2\right )^{3/2}}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\)

Input:

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

Output:

(c^3*(a + b*x^2)^(3/2))/(2*d^2*(b*c^2 + a*d^2)*(c + d*x)^2) + (-((c^2*(6*a 
 + (5*b*c^2)/d^2)*(a + b*x^2)^(3/2))/((b*c^2 + a*d^2)*(c + d*x))) - (((c*( 
19*a*b*c^2 + (12*b^2*c^4)/d^2 + 6*a^2*d^2) - d*(8*a*b*c^2 + (6*b^2*c^4)/d^ 
2 + a^2*d^2)*x)*Sqrt[a + b*x^2])/d^2 + ((b*c^2 + a*d^2)*(-(((b*c^2 + a*d^2 
)*(12*b*c^2 + a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - 
(c*(12*b^2*c^4 + 19*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^4)/(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 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 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. \(893\) vs. \(2(219)=438\).

Time = 0.43 (sec) , antiderivative size = 894, normalized size of antiderivative = 3.59

method	result	size
risch	\(-\frac {\left (-d x +6 c \right ) \sqrt {b \,x^{2}+a}}{2 d^{4}}+\frac {\frac {\left (a \,d^{2}+12 b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {2 c^{2} \left (3 a \,d^{2}+5 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 {2 c^{3} \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}}}}{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 {2 c \left (3 a \,d^{2}+10 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}}\)	\(894\)
default	\(\text {Expression too large to display}\)	\(1745\)

Input:

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

Output:

-1/2*(-d*x+6*c)*(b*x^2+a)^(1/2)/d^4+1/2/d^4*((a*d^2+12*b*c^2)/d*ln(b^(1/2) 
*x+(b*x^2+a)^(1/2))/b^(1/2)+2*c^2/d^3*(3*a*d^2+5*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)))-2*c^3*(a*d^2+b*c^2)/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)))+2*c/d^2*(3*a*d^2+10*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. 617 vs. \(2 (220) = 440\).

Time = 34.25 (sec) , antiderivative size = 2533, normalized size of antiderivative = 10.17 \[ \int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*((12*b^3*c^8 + 25*a*b^2*c^6*d^2 + 14*a^2*b*c^4*d^4 + a^3*c^2*d^6 + (1 
2*b^3*c^6*d^2 + 25*a*b^2*c^4*d^4 + 14*a^2*b*c^2*d^6 + a^3*d^8)*x^2 + 2*(12 
*b^3*c^7*d + 25*a*b^2*c^5*d^3 + 14*a^2*b*c^3*d^5 + a^3*c*d^7)*x)*sqrt(b)*l 
og(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + (12*b^3*c^7 + 19*a*b^2*c^ 
5*d^2 + 6*a^2*b*c^3*d^4 + (12*b^3*c^5*d^2 + 19*a*b^2*c^3*d^4 + 6*a^2*b*c*d 
^6)*x^2 + 2*(12*b^3*c^6*d + 19*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*(12*b^3*c^7*d + 23*a*b^2*c^5*d^3 + 11*a^2*b*c^3*d^5 - (b 
^3*c^4*d^4 + 2*a*b^2*c^2*d^6 + a^2*b*d^8)*x^3 + 4*(b^3*c^5*d^3 + 2*a*b^2*c 
^3*d^5 + a^2*b*c*d^7)*x^2 + (18*b^3*c^6*d^2 + 35*a*b^2*c^4*d^4 + 17*a^2*b* 
c^2*d^6)*x)*sqrt(b*x^2 + a))/(b^3*c^6*d^5 + 2*a*b^2*c^4*d^7 + a^2*b*c^2*d^ 
9 + (b^3*c^4*d^7 + 2*a*b^2*c^2*d^9 + a^2*b*d^11)*x^2 + 2*(b^3*c^5*d^6 + 2* 
a*b^2*c^3*d^8 + a^2*b*c*d^10)*x), 1/4*(2*(12*b^3*c^7 + 19*a*b^2*c^5*d^2 + 
6*a^2*b*c^3*d^4 + (12*b^3*c^5*d^2 + 19*a*b^2*c^3*d^4 + 6*a^2*b*c*d^6)*x^2 
+ 2*(12*b^3*c^6*d + 19*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)) + (12*b^3*c^8 + 25*a*b^2*c^6*d^2 + 14 
*a^2*b*c^4*d^4 + a^3*c^2*d^6 + (12*b^3*c^6*d^2 + 25*a*b^2*c^4*d^4 + 14*a^2 
*b*c^2*d^6 + a^3*d^8)*x^2 + 2*(12*b^3*c^7*d + 25*a*b^2*c^5*d^3 + 14*a^2...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.66 \[ \int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\frac {\sqrt {b x^{2} + a} b c^{4}}{2 \, {\left (b c^{2} d^{5} x + a d^{7} x + b c^{3} d^{4} + a c d^{6}\right )}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} c^{3}}{2 \, {\left (b c^{2} d^{4} x^{2} + a d^{6} x^{2} + 2 \, b c^{3} d^{3} x + 2 \, a c d^{5} x + b c^{4} d^{2} + a c^{2} d^{4}\right )}} - \frac {\sqrt {b x^{2} + a} b c^{3}}{2 \, {\left (b c^{2} d^{4} + a d^{6}\right )}} - \frac {3 \, \sqrt {b x^{2} + a} c^{2}}{d^{5} x + c d^{4}} + \frac {\sqrt {b x^{2} + a} x}{2 \, d^{3}} + \frac {6 \, \sqrt {b} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{5}} + \frac {a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b} d^{3}} + \frac {b^{2} c^{5} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{8}} - \frac {7 \, b c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} d^{6}} - \frac {3 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{4}} - \frac {3 \, \sqrt {b x^{2} + a} c}{d^{4}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (220) = 440\).

Time = 0.16 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.87 \[ \int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\frac {1}{2} \, \sqrt {b x^{2} + a} {\left (\frac {x}{d^{3}} - \frac {6 \, c}{d^{4}}\right )} + \frac {{\left (12 \, b^{2} c^{5} + 19 \, a b c^{3} d^{2} + 6 \, a^{2} c d^{4}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b c^{2} d^{5} + a d^{7}\right )} \sqrt {-b c^{2} - a d^{2}}} - \frac {8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b^{2} c^{5} d + 7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a b c^{3} d^{3} + 14 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {5}{2}} c^{6} + 5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} c^{4} d^{2} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} \sqrt {b} c^{2} d^{4} - 20 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a b^{2} c^{5} d - 17 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b c^{3} d^{3} + 7 \, a^{2} b^{\frac {3}{2}} c^{4} d^{2} + 6 \, a^{3} \sqrt {b} c^{2} d^{4}}{{\left (b c^{2} d^{5} + a d^{7}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{2}} - \frac {{\left (12 \, b c^{2} + a d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, \sqrt {b} d^{5}} \] Input:

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

Output:

1/2*sqrt(b*x^2 + a)*(x/d^3 - 6*c/d^4) + (12*b^2*c^5 + 19*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*c^2*d^5 + a*d^7)*sqrt(-b*c^2 - a*d^2)) - (8*(sqrt(b)*x - sq 
rt(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 
 + 14*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(5/2)*c^6 + 5*(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 - 20*(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 + 7*a^2*b^(3/2)*c^4*d^2 + 6*a^3*sqrt(b)* 
c^2*d^4)/((b*c^2*d^5 + a*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) - 1/2*(12*b*c^2 + a*d^2)*log( 
abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/(sqrt(b)*d^5)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\int \frac {x^3\,\sqrt {b\,x^2+a}}{{\left (c+d\,x\right )}^3} \,d x \] 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 = 1816, normalized size of antiderivative = 7.29 \[ \int \frac {x^3 \sqrt {a+b x^2}}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(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**3*d**4 + 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)*a**2*b*c**2*d**5*x + 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*d**6*x**2 + 38*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*s 
qrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**5*d**2 + 76*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 + 38*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 + 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**3*c**7 + 
48*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**6*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**3*c**5*d**2*x**2 - 12*sqrt(a*d* 
*2 + b*c**2)*log(c + d*x)*a**2*b*c**3*d**4 - 24*sqrt(a*d**2 + b*c**2)*log( 
c + d*x)*a**2*b*c**2*d**5*x - 12*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b 
*c*d**6*x**2 - 38*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**5*d**2 - 76 
*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**4*d**3*x - 38*sqrt(a*d**2 + 
b*c**2)*log(c + d*x)*a*b**2*c**3*d**4*x**2 - 24*sqrt(a*d**2 + b*c**2)*log( 
c + d*x)*b**3*c**7 - 48*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**3*c**6*d*x - 
 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**3*c**5*d**2*x**2 - 22*sqrt(a ...