\(\int \frac {x^3 (c+d x)^{3/2}}{(a-b x^2)^2} \, dx\) [647]

Optimal result

Integrand size = 23, antiderivative size = 221 \[ \int \frac {x^3 (c+d x)^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {2 c \sqrt {c+d x}}{b^2}+\frac {2 (c+d x)^{3/2}}{3 b^2}+\frac {a (c+d x)^{3/2}}{2 b^2 \left (a-b x^2\right )}-\frac {\left (4 \sqrt {b} c-7 \sqrt {a} d\right ) \sqrt {\sqrt {b} c-\sqrt {a} d} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 b^{11/4}}-\frac {\sqrt {\sqrt {b} c+\sqrt {a} d} \left (4 \sqrt {b} c+7 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{4 b^{11/4}} \] Output:

2*c*(d*x+c)^(1/2)/b^2+2/3*(d*x+c)^(3/2)/b^2+1/2*a*(d*x+c)^(3/2)/b^2/(-b*x^ 
2+a)-1/4*(4*b^(1/2)*c-7*a^(1/2)*d)*(b^(1/2)*c-a^(1/2)*d)^(1/2)*arctanh(b^( 
1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/b^(11/4)-1/4*(b^(1/2)*c+a^ 
(1/2)*d)^(1/2)*(4*b^(1/2)*c+7*a^(1/2)*d)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^ 
(1/2)*c+a^(1/2)*d)^(1/2))/b^(11/4)
 

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 (c+d x)^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {\frac {2 b \sqrt {c+d x} \left (-4 b x^2 (4 c+d x)+a (19 c+7 d x)\right )}{a-b x^2}+3 \left (4 \sqrt {b} c+7 \sqrt {a} d\right ) \sqrt {-b c-\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )+3 \left (4 \sqrt {b} c-7 \sqrt {a} d\right ) \sqrt {-b c+\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{12 b^3} \] Input:

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

Output:

((2*b*Sqrt[c + d*x]*(-4*b*x^2*(4*c + d*x) + a*(19*c + 7*d*x)))/(a - b*x^2) 
 + 3*(4*Sqrt[b]*c + 7*Sqrt[a]*d)*Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*ArcTan[( 
Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)] + 
 3*(4*Sqrt[b]*c - 7*Sqrt[a]*d)*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*ArcTan[(Sq 
rt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/(1 
2*b^3)
 

Rubi [A] (verified)

Time = 1.81 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.51, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {561, 25, 27, 1672, 27, 2028, 2195, 2009}

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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1672

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2028

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

\(\Big \downarrow \) 2195

\(\displaystyle -\frac {2 \left (\frac {d^4 \int \left (-\frac {4 a c \left (b c^2-a d^2\right )}{b}-\frac {4 a (c+d x) \left (b c^2-a d^2\right )}{b}-\frac {4 a c \left (b c^2-a d^2\right )^2-a \left (b c^2-a d^2\right ) \left (4 b c^2+7 a d^2\right ) (c+d x)}{b d^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}\right )d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {a d^4 (c+d x)^{3/2}}{4 b^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (\frac {d^4 \left (\frac {a \left (-3 \sqrt {a} \sqrt {b} c d-7 a d^2+4 b c^2\right ) \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 b^{7/4}}+\frac {a \left (\sqrt {a} d+\sqrt {b} c\right )^{3/2} \left (3 \sqrt {a} \sqrt {b} c d-7 a d^2+4 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 b^{7/4}}-\frac {4 a (c+d x)^{3/2} \left (b c^2-a d^2\right )}{3 b}-\frac {4 a c \sqrt {c+d x} \left (b c^2-a d^2\right )}{b}\right )}{4 a b \left (b c^2-a d^2\right )}-\frac {a d^4 (c+d x)^{3/2}}{4 b^2 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^4}\)

Input:

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

Output:

(-2*(-1/4*(a*d^4*(c + d*x)^(3/2))/(b^2*(a - (b*c^2)/d^2 + (2*b*c*(c + d*x) 
)/d^2 - (b*(c + d*x)^2)/d^2)) + (d^4*((-4*a*c*(b*c^2 - a*d^2)*Sqrt[c + d*x 
])/b - (4*a*(b*c^2 - a*d^2)*(c + d*x)^(3/2))/(3*b) + (a*(Sqrt[b]*c - Sqrt[ 
a]*d)^(3/2)*(4*b*c^2 - 3*Sqrt[a]*Sqrt[b]*c*d - 7*a*d^2)*ArcTanh[(b^(1/4)*S 
qrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*b^(7/4)) + (a*(Sqrt[b]*c + 
Sqrt[a]*d)^(3/2)*(4*b*c^2 + 3*Sqrt[a]*Sqrt[b]*c*d - 7*a*d^2)*ArcTanh[(b^(1 
/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*b^(7/4))))/(4*a*b*(b*c 
^2 - a*d^2))))/d^4
 

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 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 1672

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 
4)^(p_), x_Symbol] :> With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q 
, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d + e*x^ 
2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a 
*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), 
 x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c*x^4)^(p + 1)* 
Simp[ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^ 
2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + 
 c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x], x]] /; FreeQ[{a, b, c, d, e}, x 
] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && IGtQ[m/2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2028

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), 
x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ 
{a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] &&  !(E 
qQ[p, 1] && EqQ[u, 1])
 

rule 2195

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ 
Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; 
FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
 

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.18

method	result	size
risch	\(\frac {2 \left (d x +4 c \right ) \sqrt {d x +c}}{3 b^{2}}+\frac {-\frac {a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{2 \left (b \left (d x +c \right )^{2}-2 b c \left (d x +c \right )-a \,d^{2}+b \,c^{2}\right )}+\frac {b \left (-\frac {\left (11 a b c \,d^{2}+7 \sqrt {a b \,d^{2}}\, a \,d^{2}+4 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-11 a b c \,d^{2}+7 \sqrt {a b \,d^{2}}\, a \,d^{2}+4 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2}}{b^{2}}\)	\(261\)
derivativedivides	\(\frac {\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3}+2 c \sqrt {d x +c}}{b^{2}}-\frac {2 \left (-\frac {a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{4 \left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )}+\frac {b \left (\frac {\left (11 a b c \,d^{2}-7 \sqrt {a b \,d^{2}}\, a \,d^{2}-4 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-11 a b c \,d^{2}-7 \sqrt {a b \,d^{2}}\, a \,d^{2}-4 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}\right )}{b^{2}}\)	\(268\)
default	\(\frac {\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3}+2 c \sqrt {d x +c}}{b^{2}}-\frac {2 \left (-\frac {a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{4 \left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )}+\frac {b \left (\frac {\left (11 a b c \,d^{2}-7 \sqrt {a b \,d^{2}}\, a \,d^{2}-4 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-11 a b c \,d^{2}-7 \sqrt {a b \,d^{2}}\, a \,d^{2}-4 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}\right )}{b^{2}}\)	\(268\)
pseudoelliptic	\(-\frac {11 \left (\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\frac {\left (-7 a \,d^{2}-4 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{11}+a b c \,d^{2}\right ) \left (-b \,x^{2}+a \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\left (\left (\frac {\left (7 a \,d^{2}+4 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{11}+a b c \,d^{2}\right ) \left (-b \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-\frac {38 \sqrt {d x +c}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \left (-\frac {16 x^{2} \left (\frac {d x}{4}+c \right ) b}{19}+a \left (\frac {7 d x}{19}+c \right )\right )}{33}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\right )}{4 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, b^{2} \left (-b \,x^{2}+a \right )}\)	\(280\)

Input:

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

Output:

2/3*(d*x+4*c)*(d*x+c)^(1/2)/b^2+1/b^2*(-1/2*a*d^2*(d*x+c)^(3/2)/(b*(d*x+c) 
^2-2*b*c*(d*x+c)-a*d^2+b*c^2)+1/2*b*(-1/2*(11*a*b*c*d^2+7*(a*b*d^2)^(1/2)* 
a*d^2+4*(a*b*d^2)^(1/2)*b*c^2)/b/(a*b*d^2)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b) 
^(1/2)*arctanh(b*(d*x+c)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2))+1/2*(-11*a 
*b*c*d^2+7*(a*b*d^2)^(1/2)*a*d^2+4*(a*b*d^2)^(1/2)*b*c^2)/b/(a*b*d^2)^(1/2 
)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((-b*c+(a*b*d^2) 
^(1/2))*b)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1034 vs. \(2 (164) = 328\).

Time = 0.11 (sec) , antiderivative size = 1034, normalized size of antiderivative = 4.68 \[ \int \frac {x^3 (c+d x)^{3/2}}{\left (a-b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/24*(3*(b^3*x^2 - a*b^2)*sqrt((b^5*sqrt((5184*a*b^2*c^4*d^2 + 7056*a^2*b 
*c^2*d^4 + 2401*a^3*d^6)/b^11) + 16*b*c^3 + 105*a*c*d^2)/b^5)*log(-(1152*b 
^2*c^4 - 2744*a*b*c^2*d^2 - 2401*a^2*d^4)*sqrt(d*x + c) + (7*b^8*sqrt((518 
4*a*b^2*c^4*d^2 + 7056*a^2*b*c^2*d^4 + 2401*a^3*d^6)/b^11) - 288*b^4*c^3 - 
 196*a*b^3*c*d^2)*sqrt((b^5*sqrt((5184*a*b^2*c^4*d^2 + 7056*a^2*b*c^2*d^4 
+ 2401*a^3*d^6)/b^11) + 16*b*c^3 + 105*a*c*d^2)/b^5)) - 3*(b^3*x^2 - a*b^2 
)*sqrt((b^5*sqrt((5184*a*b^2*c^4*d^2 + 7056*a^2*b*c^2*d^4 + 2401*a^3*d^6)/ 
b^11) + 16*b*c^3 + 105*a*c*d^2)/b^5)*log(-(1152*b^2*c^4 - 2744*a*b*c^2*d^2 
 - 2401*a^2*d^4)*sqrt(d*x + c) - (7*b^8*sqrt((5184*a*b^2*c^4*d^2 + 7056*a^ 
2*b*c^2*d^4 + 2401*a^3*d^6)/b^11) - 288*b^4*c^3 - 196*a*b^3*c*d^2)*sqrt((b 
^5*sqrt((5184*a*b^2*c^4*d^2 + 7056*a^2*b*c^2*d^4 + 2401*a^3*d^6)/b^11) + 1 
6*b*c^3 + 105*a*c*d^2)/b^5)) - 3*(b^3*x^2 - a*b^2)*sqrt(-(b^5*sqrt((5184*a 
*b^2*c^4*d^2 + 7056*a^2*b*c^2*d^4 + 2401*a^3*d^6)/b^11) - 16*b*c^3 - 105*a 
*c*d^2)/b^5)*log(-(1152*b^2*c^4 - 2744*a*b*c^2*d^2 - 2401*a^2*d^4)*sqrt(d* 
x + c) + (7*b^8*sqrt((5184*a*b^2*c^4*d^2 + 7056*a^2*b*c^2*d^4 + 2401*a^3*d 
^6)/b^11) + 288*b^4*c^3 + 196*a*b^3*c*d^2)*sqrt(-(b^5*sqrt((5184*a*b^2*c^4 
*d^2 + 7056*a^2*b*c^2*d^4 + 2401*a^3*d^6)/b^11) - 16*b*c^3 - 105*a*c*d^2)/ 
b^5)) + 3*(b^3*x^2 - a*b^2)*sqrt(-(b^5*sqrt((5184*a*b^2*c^4*d^2 + 7056*a^2 
*b*c^2*d^4 + 2401*a^3*d^6)/b^11) - 16*b*c^3 - 105*a*c*d^2)/b^5)*log(-(1152 
*b^2*c^4 - 2744*a*b*c^2*d^2 - 2401*a^2*d^4)*sqrt(d*x + c) - (7*b^8*sqrt...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (164) = 328\).

Time = 0.22 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.86 \[ \int \frac {x^3 (c+d x)^{3/2}}{\left (a-b x^2\right )^2} \, dx=-\frac {{\left (d x + c\right )}^{\frac {3}{2}} a d^{2}}{2 \, {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} - a d^{2}\right )} b^{2}} + \frac {{\left (11 \, \sqrt {a b} b c^{2} d^{2} {\left | b \right |} - {\left (4 \, \sqrt {a b} b c^{2} + 7 \, \sqrt {a b} a d^{2}\right )} d^{2} {\left | b \right |} + 4 \, {\left (b^{2} c^{3} - a b c d^{2}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{3} c + \sqrt {b^{6} c^{2} - {\left (b^{3} c^{2} - a b^{2} d^{2}\right )} b^{3}}}{b^{3}}}}\right )}{4 \, {\left (b^{4} c - \sqrt {a b} b^{3} d\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | d \right |}} - \frac {{\left (11 \, \sqrt {a b} b c^{2} d^{2} {\left | b \right |} - {\left (4 \, \sqrt {a b} b c^{2} + 7 \, \sqrt {a b} a d^{2}\right )} d^{2} {\left | b \right |} - 4 \, {\left (b^{2} c^{3} - a b c d^{2}\right )} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{3} c - \sqrt {b^{6} c^{2} - {\left (b^{3} c^{2} - a b^{2} d^{2}\right )} b^{3}}}{b^{3}}}}\right )}{4 \, {\left (b^{4} c + \sqrt {a b} b^{3} d\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | d \right |}} + \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} b^{4} + 3 \, \sqrt {d x + c} b^{4} c\right )}}{3 \, b^{6}} \] Input:

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

Output:

-1/2*(d*x + c)^(3/2)*a*d^2/(((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 - a*d 
^2)*b^2) + 1/4*(11*sqrt(a*b)*b*c^2*d^2*abs(b) - (4*sqrt(a*b)*b*c^2 + 7*sqr 
t(a*b)*a*d^2)*d^2*abs(b) + 4*(b^2*c^3 - a*b*c*d^2)*abs(b)*abs(d))*arctan(s 
qrt(d*x + c)/sqrt(-(b^3*c + sqrt(b^6*c^2 - (b^3*c^2 - a*b^2*d^2)*b^3))/b^3 
))/((b^4*c - sqrt(a*b)*b^3*d)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(d)) - 1/4*( 
11*sqrt(a*b)*b*c^2*d^2*abs(b) - (4*sqrt(a*b)*b*c^2 + 7*sqrt(a*b)*a*d^2)*d^ 
2*abs(b) - 4*(b^2*c^3 - a*b*c*d^2)*abs(b)*abs(d))*arctan(sqrt(d*x + c)/sqr 
t(-(b^3*c - sqrt(b^6*c^2 - (b^3*c^2 - a*b^2*d^2)*b^3))/b^3))/((b^4*c + sqr 
t(a*b)*b^3*d)*sqrt(-b^2*c + sqrt(a*b)*b*d)*abs(d)) + 2/3*((d*x + c)^(3/2)* 
b^4 + 3*sqrt(d*x + c)*b^4*c)/b^6
 

Mupad [B] (verification not implemented)

Time = 8.84 (sec) , antiderivative size = 1596, normalized size of antiderivative = 7.22 \[ \int \frac {x^3 (c+d x)^{3/2}}{\left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((a^3*d^6*(c + d*x)^(1/2)*(c^3/(4*b^4) - (49*a*d^3*(a*b^11)^(1/2))/(64 
*b^11) - (9*c^2*d*(a*b^11)^(1/2))/(8*b^10) + (105*a*c*d^2)/(64*b^5))^(1/2) 
*98i)/((343*a^4*d^8)/(4*b^3) - (126*a^2*c^4*d^4)/b + (161*a^3*c^2*d^6)/(4* 
b^2) - (23*a^2*c^3*d^5*(a*b^11)^(1/2))/b^7 + (72*a*c^5*d^3*(a*b^11)^(1/2)) 
/b^6 - (49*a^3*c*d^7*(a*b^11)^(1/2))/b^8) + (a^2*c^2*d^4*(c + d*x)^(1/2)*( 
c^3/(4*b^4) - (49*a*d^3*(a*b^11)^(1/2))/(64*b^11) - (9*c^2*d*(a*b^11)^(1/2 
))/(8*b^10) + (105*a*c*d^2)/(64*b^5))^(1/2)*144i)/((343*a^4*d^8)/(4*b^4) - 
 (126*a^2*c^4*d^4)/b^2 + (161*a^3*c^2*d^6)/(4*b^3) - (23*a^2*c^3*d^5*(a*b^ 
11)^(1/2))/b^8 + (72*a*c^5*d^3*(a*b^11)^(1/2))/b^7 - (49*a^3*c*d^7*(a*b^11 
)^(1/2))/b^9) + (a*c^3*d^3*(a*b^11)^(1/2)*(c + d*x)^(1/2)*(c^3/(4*b^4) - ( 
49*a*d^3*(a*b^11)^(1/2))/(64*b^11) - (9*c^2*d*(a*b^11)^(1/2))/(8*b^10) + ( 
105*a*c*d^2)/(64*b^5))^(1/2)*144i)/((343*a^4*b*d^8)/4 - 126*a^2*b^3*c^4*d^ 
4 + (161*a^3*b^2*c^2*d^6)/4 - (23*a^2*c^3*d^5*(a*b^11)^(1/2))/b^3 + (72*a* 
c^5*d^3*(a*b^11)^(1/2))/b^2 - (49*a^3*c*d^7*(a*b^11)^(1/2))/b^4) + (a^2*c* 
d^5*(a*b^11)^(1/2)*(c + d*x)^(1/2)*(c^3/(4*b^4) - (49*a*d^3*(a*b^11)^(1/2) 
)/(64*b^11) - (9*c^2*d*(a*b^11)^(1/2))/(8*b^10) + (105*a*c*d^2)/(64*b^5))^ 
(1/2)*98i)/((343*a^4*b^2*d^8)/4 - 126*a^2*b^4*c^4*d^4 + (161*a^3*b^3*c^2*d 
^6)/4 - (23*a^2*c^3*d^5*(a*b^11)^(1/2))/b^2 + (72*a*c^5*d^3*(a*b^11)^(1/2) 
)/b - (49*a^3*c*d^7*(a*b^11)^(1/2))/b^3))*((16*b^7*c^3 - 49*a*d^3*(a*b^11) 
^(1/2) + 105*a*b^6*c*d^2 - 72*b*c^2*d*(a*b^11)^(1/2))/(64*b^11))^(1/2)*...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 587, normalized size of antiderivative = 2.66 \[ \int \frac {x^3 (c+d x)^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {42 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) a d -42 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) b d \,x^{2}-24 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) a c +24 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) b c \,x^{2}+21 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a d -21 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b d \,x^{2}-21 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a d +21 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b d \,x^{2}+12 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a c -12 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b c \,x^{2}-12 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) a c +12 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) b c \,x^{2}+76 \sqrt {d x +c}\, a b c +28 \sqrt {d x +c}\, a b d x -64 \sqrt {d x +c}\, b^{2} c \,x^{2}-16 \sqrt {d x +c}\, b^{2} d \,x^{3}}{24 b^{3} \left (-b \,x^{2}+a \right )} \] Input:

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

Output:

(42*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)* 
sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*d - 42*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - 
b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*b*d*x 
**2 - 24*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqr 
t(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*c + 24*sqrt(b)*sqrt(sqrt(b)*sqrt(a) 
*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))* 
b*c*x**2 + 21*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sq 
rt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a*d - 21*sqrt(a)*sqrt(sqrt(b)*sqrt 
(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x)) 
*b*d*x**2 - 21*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt(b)*sqrt 
(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a*d + 21*sqrt(a)*sqrt(sqrt(b)*sqrt(a 
)*d + b*c)*log(sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*b*d* 
x**2 + 12*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a 
)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a*c - 12*sqrt(b)*sqrt(sqrt(b)*sqrt(a)* 
d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*b*c 
*x**2 - 12*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt(b)*sqrt(a)* 
d + b*c) + sqrt(b)*sqrt(c + d*x))*a*c + 12*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d 
+ b*c)*log(sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*b*c*x**2 
 + 76*sqrt(c + d*x)*a*b*c + 28*sqrt(c + d*x)*a*b*d*x - 64*sqrt(c + d*x)*b* 
*2*c*x**2 - 16*sqrt(c + d*x)*b**2*d*x**3)/(24*b**3*(a - b*x**2))