\(\int \frac {x^3}{(c+d x)^{3/2} (a-b x^2)^3} \, dx\) [727]

Optimal result

Integrand size = 23, antiderivative size = 341 \[ \int \frac {x^3}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx=-\frac {2 c^3 d^2}{\left (b c^2-a d^2\right )^3 \sqrt {c+d x}}+\frac {\sqrt {c+d x} \left (a^2 d^4-b^2 c^3 (8 c-19 d x)-3 a b c d^2 (7 c-3 d x)\right )}{16 b \left (b c^2-a d^2\right )^3 \left (a-b x^2\right )}+\frac {a \sqrt {c+d x} \left (a d^2+b c (c-2 d x)\right )}{4 b \left (b c^2-a d^2\right )^2 \left (a-b x^2\right )^2}+\frac {3 d \left (6 \sqrt {b} c-\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 \sqrt {a} b^{5/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{7/2}}-\frac {3 d \left (6 \sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 \sqrt {a} b^{5/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{7/2}} \] Output:

-2*c^3*d^2/(-a*d^2+b*c^2)^3/(d*x+c)^(1/2)+1/16*(d*x+c)^(1/2)*(a^2*d^4-b^2* 
c^3*(-19*d*x+8*c)-3*a*b*c*d^2*(-3*d*x+7*c))/b/(-a*d^2+b*c^2)^3/(-b*x^2+a)+ 
1/4*a*(d*x+c)^(1/2)*(a*d^2+b*c*(-2*d*x+c))/b/(-a*d^2+b*c^2)^2/(-b*x^2+a)^2 
+3/32*d*(6*b^(1/2)*c-a^(1/2)*d)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a 
^(1/2)*d)^(1/2))/a^(1/2)/b^(5/4)/(b^(1/2)*c-a^(1/2)*d)^(7/2)-3/32*d*(6*b^( 
1/2)*c+a^(1/2)*d)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2 
))/a^(1/2)/b^(5/4)/(b^(1/2)*c+a^(1/2)*d)^(7/2)
 

Mathematica [A] (verified)

Time = 3.44 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.17 \[ \int \frac {x^3}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx=\frac {d \left (\frac {2 \left (3 a^3 d^4 (c+d x)+b^3 c^3 x^2 \left (-8 c^2+11 c d x+51 d^2 x^2\right )+a^2 b d^2 \left (53 c^3+4 c^2 d x-16 c d^2 x^2+d^3 x^3\right )+a b^2 c \left (4 c^4-7 c^3 d x-96 c^2 d^2 x^2-12 c d^3 x^3+9 d^4 x^4\right )\right )}{d \left (-b c^2+a d^2\right )^3 \sqrt {c+d x} \left (a-b x^2\right )^2}-\frac {3 \left (6 \sqrt {b} c+\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \left (\sqrt {b} c+\sqrt {a} d\right )^3 \sqrt {-b c-\sqrt {a} \sqrt {b} d}}+\frac {3 \left (6 \sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {a} \left (\sqrt {b} c-\sqrt {a} d\right )^3 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}\right )}{32 b} \] Input:

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

Output:

(d*((2*(3*a^3*d^4*(c + d*x) + b^3*c^3*x^2*(-8*c^2 + 11*c*d*x + 51*d^2*x^2) 
 + a^2*b*d^2*(53*c^3 + 4*c^2*d*x - 16*c*d^2*x^2 + d^3*x^3) + a*b^2*c*(4*c^ 
4 - 7*c^3*d*x - 96*c^2*d^2*x^2 - 12*c*d^3*x^3 + 9*d^4*x^4)))/(d*(-(b*c^2) 
+ a*d^2)^3*Sqrt[c + d*x]*(a - b*x^2)^2) - (3*(6*Sqrt[b]*c + Sqrt[a]*d)*Arc 
Tan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]* 
d)])/(Sqrt[a]*(Sqrt[b]*c + Sqrt[a]*d)^3*Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]) 
+ (3*(6*Sqrt[b]*c - Sqrt[a]*d)*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sq 
rt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/(Sqrt[a]*(Sqrt[b]*c - Sqrt[a]*d)^3* 
Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d])))/(32*b)
 

Rubi [A] (verified)

Time = 3.33 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.70, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {561, 25, 27, 1673, 27, 2198, 27, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1673

\(\displaystyle -\frac {2 \left (\frac {d^4 \int -\frac {2 \left (\frac {8 a b c^3}{d^2}-\frac {10 a^2 b (c+d x)^2 c}{b c^2-a d^2}-\frac {a \left (8 b^2 c^4-21 a b d^2 c^2+a^2 d^4\right ) (c+d x)}{d^2 \left (b c^2-a d^2\right )}\right )}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (\frac {a d^2+3 b c^2}{b c^2-a d^2}-\frac {2 b c (c+d x)}{b c^2-a d^2}\right )}{8 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {d^4 \int \frac {\frac {8 a b c^3}{d^2}-\frac {10 a^2 b (c+d x)^2 c}{b c^2-a d^2}-\frac {a \left (8 b^2 c^4-21 a b d^2 c^2+a^2 d^4\right ) (c+d x)}{d^2 \left (b c^2-a d^2\right )}}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (\frac {a d^2+3 b c^2}{b c^2-a d^2}-\frac {2 b c (c+d x)}{b c^2-a d^2}\right )}{8 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 2198

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (\frac {d^4 \int -\frac {2 \left (\frac {32 a^2 b^2 c^3}{d^4}-\frac {a^2 b^2 \left (19 b c^2+9 a d^2\right ) (c+d x)^2 c}{d^4 \left (b c^2-a d^2\right )}+\frac {a^2 b \left (5 b^2 c^4+54 a b d^2 c^2-3 a^2 d^4\right ) (c+d x)}{d^4 \left (b c^2-a d^2\right )}\right )}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {a \sqrt {c+d x} \left (\frac {-a^2 d^4+30 a b c^2 d^2+27 b^2 c^4}{b c^2-a d^2}-\frac {b c (c+d x) \left (9 a d^2+19 b c^2\right )}{b c^2-a d^2}\right )}{4 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (\frac {a d^2+3 b c^2}{b c^2-a d^2}-\frac {2 b c (c+d x)}{b c^2-a d^2}\right )}{8 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (-\frac {d^4 \int \frac {\frac {32 a^2 b^2 c^3}{d^4}-\frac {a^2 b^2 \left (19 b c^2+9 a d^2\right ) (c+d x)^2 c}{d^4 \left (b c^2-a d^2\right )}+\frac {a^2 b \left (5 b^2 c^4+54 a b d^2 c^2-3 a^2 d^4\right ) (c+d x)}{d^4 \left (b c^2-a d^2\right )}}{(c+d x) \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {a \sqrt {c+d x} \left (\frac {-a^2 d^4+30 a b c^2 d^2+27 b^2 c^4}{b c^2-a d^2}-\frac {b c (c+d x) \left (9 a d^2+19 b c^2\right )}{b c^2-a d^2}\right )}{4 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (\frac {a d^2+3 b c^2}{b c^2-a d^2}-\frac {2 b c (c+d x)}{b c^2-a d^2}\right )}{8 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 2195

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (-\frac {d^4 \int \left (\frac {32 a^2 b^2 c^3}{d^2 \left (a d^2-b c^2\right ) (c+d x)}+\frac {3 a^2 b \left (-23 b^2 c^4-18 a b d^2 c^2+b \left (17 b c^2+3 a d^2\right ) (c+d x) c+a^2 d^4\right )}{d^2 \left (b c^2-a d^2\right ) \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}\right )d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {a \sqrt {c+d x} \left (\frac {-a^2 d^4+30 a b c^2 d^2+27 b^2 c^4}{b c^2-a d^2}-\frac {b c (c+d x) \left (9 a d^2+19 b c^2\right )}{b c^2-a d^2}\right )}{4 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (\frac {a d^2+3 b c^2}{b c^2-a d^2}-\frac {2 b c (c+d x)}{b c^2-a d^2}\right )}{8 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (-\frac {d^4 \left (-\frac {a \sqrt {c+d x} \left (\frac {-a^2 d^4+30 a b c^2 d^2+27 b^2 c^4}{b c^2-a d^2}-\frac {b c (c+d x) \left (9 a d^2+19 b c^2\right )}{b c^2-a d^2}\right )}{4 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {d^4 \left (-\frac {3 a^{3/2} b^{3/4} \left (6 \sqrt {b} c-\sqrt {a} d\right ) \left (\sqrt {a} d+\sqrt {b} c\right )^2 \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 d^3 \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}+\frac {3 a^{3/2} b^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right )^2 \left (\sqrt {a} d+6 \sqrt {b} c\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 d^3 \left (\sqrt {a} d+\sqrt {b} c\right )^{3/2}}+\frac {32 a^2 b^2 c^3}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\right )}{4 a b \left (b c^2-a d^2\right )}\right )}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (\frac {a d^2+3 b c^2}{b c^2-a d^2}-\frac {2 b c (c+d x)}{b c^2-a d^2}\right )}{8 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

Input:

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

Output:

(-2*(-1/8*(a*d^4*Sqrt[c + d*x]*((3*b*c^2 + a*d^2)/(b*c^2 - a*d^2) - (2*b*c 
*(c + d*x))/(b*c^2 - a*d^2)))/(b*(b*c^2 - a*d^2)*(a - (b*c^2)/d^2 + (2*b*c 
*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)^2) - (d^4*(-1/4*(a*Sqrt[c + d*x]*(( 
27*b^2*c^4 + 30*a*b*c^2*d^2 - a^2*d^4)/(b*c^2 - a*d^2) - (b*c*(19*b*c^2 + 
9*a*d^2)*(c + d*x))/(b*c^2 - a*d^2)))/((b*c^2 - a*d^2)*(a - (b*c^2)/d^2 + 
(2*b*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)) - (d^4*((32*a^2*b^2*c^3)/(d^ 
2*(b*c^2 - a*d^2)*Sqrt[c + d*x]) - (3*a^(3/2)*b^(3/4)*(6*Sqrt[b]*c - Sqrt[ 
a]*d)*(Sqrt[b]*c + Sqrt[a]*d)^2*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[ 
b]*c - Sqrt[a]*d]])/(2*d^3*(Sqrt[b]*c - Sqrt[a]*d)^(3/2)) + (3*a^(3/2)*b^( 
3/4)*(Sqrt[b]*c - Sqrt[a]*d)^2*(6*Sqrt[b]*c + Sqrt[a]*d)*ArcTanh[(b^(1/4)* 
Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*d^3*(Sqrt[b]*c + Sqrt[a]*d 
)^(3/2))))/(4*a*b*(b*c^2 - a*d^2))))/(8*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 1673

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[x^m*(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])/x^m + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5 
) - a*b*g)/x^m + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x], x], x]] /; Fr 
eeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] 
&& ILtQ[m/2, 0]
 

rule 2009

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

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]
 

rule 2198

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x], d = Coeff[Pol 
ynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Polynomial 
Remainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4) 
^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 
 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[x^m*(a + b*x^2 + 
 c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*Qx)/x^m + (b^2*d*(2* 
p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - 
m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x 
^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]
 

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.32

method	result	size
default	\(2 d^{2} \left (\frac {c^{3}}{\left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {d x +c}}+\frac {\frac {\left (\frac {9}{32} a b c \,d^{2}+\frac {19}{32} c^{3} b^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}+\left (\frac {1}{32} a^{2} d^{4}-\frac {3}{2} b \,c^{2} d^{2} a -\frac {65}{32} b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {5}{2}}-\frac {c \left (19 a^{2} d^{4}-58 b \,c^{2} d^{2} a -73 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32}+\frac {3 \left (a^{3} d^{6}+13 a^{2} b \,c^{2} d^{4}-5 a \,b^{2} c^{4} d^{2}-9 b^{3} c^{6}\right ) \sqrt {d x +c}}{32 b}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {3 \left (-a^{2} d^{4}+15 b \,c^{2} d^{2} a +6 b^{2} c^{4}+3 \sqrt {a b \,d^{2}}\, a c \,d^{2}+17 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {3 \left (a^{2} d^{4}-15 b \,c^{2} d^{2} a -6 b^{2} c^{4}+3 \sqrt {a b \,d^{2}}\, a c \,d^{2}+17 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{\left (a \,d^{2}-b \,c^{2}\right )^{3}}\right )\)	\(449\)
derivativedivides	\(-2 d^{2} \left (-\frac {c^{3}}{\left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {d x +c}}-\frac {\frac {\left (\frac {9}{32} a b c \,d^{2}+\frac {19}{32} c^{3} b^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}+\left (\frac {1}{32} a^{2} d^{4}-\frac {3}{2} b \,c^{2} d^{2} a -\frac {65}{32} b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {5}{2}}-\frac {c \left (19 a^{2} d^{4}-58 b \,c^{2} d^{2} a -73 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32}+\frac {3 \left (a^{3} d^{6}+13 a^{2} b \,c^{2} d^{4}-5 a \,b^{2} c^{4} d^{2}-9 b^{3} c^{6}\right ) \sqrt {d x +c}}{32 b}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {3 \left (-a^{2} d^{4}+15 b \,c^{2} d^{2} a +6 b^{2} c^{4}+3 \sqrt {a b \,d^{2}}\, a c \,d^{2}+17 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {3 \left (a^{2} d^{4}-15 b \,c^{2} d^{2} a -6 b^{2} c^{4}+3 \sqrt {a b \,d^{2}}\, a c \,d^{2}+17 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{\left (a \,d^{2}-b \,c^{2}\right )^{3}}\right )\)	\(451\)
pseudoelliptic	\(-\frac {3 \left (d^{2} b \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {d x +c}\, \left (\left (-3 a \,d^{2} c -17 b \,c^{3}\right ) \sqrt {a b \,d^{2}}+a^{2} d^{4}-15 b \,c^{2} d^{2} a -6 b^{2} c^{4}\right ) \left (-b \,x^{2}+a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\left (d^{2} \left (\left (3 a \,d^{2} c +17 b \,c^{3}\right ) \sqrt {a b \,d^{2}}+a^{2} d^{4}-15 b \,c^{2} d^{2} a -6 b^{2} c^{4}\right ) b \sqrt {d x +c}\, \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-2 \left (\left (-\frac {8}{3} c^{5} x^{2}+\frac {11}{3} c^{4} d \,x^{3}+17 c^{3} d^{2} x^{4}\right ) b^{3}+\frac {4 \left (\frac {9}{4} d^{4} x^{4}-3 c \,d^{3} x^{3}-24 d^{2} c^{2} x^{2}-\frac {7}{4} c^{3} d x +c^{4}\right ) a c \,b^{2}}{3}+\frac {53 \left (\frac {1}{53} d^{3} x^{3}-\frac {16}{53} c \,d^{2} x^{2}+\frac {4}{53} c^{2} d x +c^{3}\right ) d^{2} a^{2} b}{3}+a^{3} d^{4} \left (d x +c \right )\right ) \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\right )}{32 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \sqrt {d x +c}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (a \,d^{2}-b \,c^{2}\right )^{3} \left (-b \,x^{2}+a \right )^{2} b}\)	\(464\)

Input:

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

Output:

2*d^2*(c^3/(a*d^2-b*c^2)^3/(d*x+c)^(1/2)+1/(a*d^2-b*c^2)^3*(((9/32*a*b*c*d 
^2+19/32*c^3*b^2)*(d*x+c)^(7/2)+(1/32*a^2*d^4-3/2*b*c^2*d^2*a-65/32*b^2*c^ 
4)*(d*x+c)^(5/2)-1/32*c*(19*a^2*d^4-58*a*b*c^2*d^2-73*b^2*c^4)*(d*x+c)^(3/ 
2)+3/32*(a^3*d^6+13*a^2*b*c^2*d^4-5*a*b^2*c^4*d^2-9*b^3*c^6)/b*(d*x+c)^(1/ 
2))/(-b*(d*x+c)^2+2*b*c*(d*x+c)+a*d^2-b*c^2)^2+3/64*(-a^2*d^4+15*b*c^2*d^2 
*a+6*b^2*c^4+3*(a*b*d^2)^(1/2)*a*c*d^2+17*(a*b*d^2)^(1/2)*b*c^3)/(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))-3/64*(a^2*d^4-15*b*c^2*d^2*a-6*b^2*c^4+3*(a*b*d^2)^ 
(1/2)*a*c*d^2+17*(a*b*d^2)^(1/2)*b*c^3)/(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))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8991 vs. \(2 (283) = 566\).

Time = 6.42 (sec) , antiderivative size = 8991, normalized size of antiderivative = 26.37 \[ \int \frac {x^3}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2186 vs. \(2 (283) = 566\).

Time = 0.47 (sec) , antiderivative size = 2186, normalized size of antiderivative = 6.41 \[ \int \frac {x^3}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-2*c^3*d^2/((b^3*c^6 - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 - a^3*d^6)*sqrt(d 
*x + c)) + 3/32*((b^4*c^6*d - 3*a*b^3*c^4*d^3 + 3*a^2*b^2*c^2*d^5 - a^3*b* 
d^7)^2*(17*sqrt(a*b)*a*b*c^3*d^2 + 3*sqrt(a*b)*a^2*c*d^4)*abs(b) - (23*a*b 
^6*c^10*d^2 - 51*a^2*b^5*c^8*d^4 + 14*a^3*b^4*c^6*d^6 + 34*a^4*b^3*c^4*d^8 
 - 21*a^5*b^2*c^2*d^10 + a^6*b*d^12)*abs(b^4*c^6*d - 3*a*b^3*c^4*d^3 + 3*a 
^2*b^2*c^2*d^5 - a^3*b*d^7)*abs(b) + (6*sqrt(a*b)*b^10*c^17*d^2 - 21*sqrt( 
a*b)*a*b^9*c^15*d^4 - sqrt(a*b)*a^2*b^8*c^13*d^6 + 111*sqrt(a*b)*a^3*b^7*c 
^11*d^8 - 225*sqrt(a*b)*a^4*b^6*c^9*d^10 + 209*sqrt(a*b)*a^5*b^5*c^7*d^12 
- 99*sqrt(a*b)*a^6*b^4*c^5*d^14 + 21*sqrt(a*b)*a^7*b^3*c^3*d^16 - sqrt(a*b 
)*a^8*b^2*c*d^18)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^5*c^7 - 3*a*b^4*c^ 
5*d^2 + 3*a^2*b^3*c^3*d^4 - a^3*b^2*c*d^6 + sqrt((b^5*c^7 - 3*a*b^4*c^5*d^ 
2 + 3*a^2*b^3*c^3*d^4 - a^3*b^2*c*d^6)^2 - (b^5*c^8 - 4*a*b^4*c^6*d^2 + 6* 
a^2*b^3*c^4*d^4 - 4*a^3*b^2*c^2*d^6 + a^4*b*d^8)*(b^5*c^6 - 3*a*b^4*c^4*d^ 
2 + 3*a^2*b^3*c^2*d^4 - a^3*b^2*d^6)))/(b^5*c^6 - 3*a*b^4*c^4*d^2 + 3*a^2* 
b^3*c^2*d^4 - a^3*b^2*d^6)))/((a*b^9*c^13 - 6*a^2*b^8*c^11*d^2 + 15*a^3*b^ 
7*c^9*d^4 - 20*a^4*b^6*c^7*d^6 + 15*a^5*b^5*c^5*d^8 - 6*a^6*b^4*c^3*d^10 + 
 a^7*b^3*c*d^12 - sqrt(a*b)*a*b^8*c^12*d + 6*sqrt(a*b)*a^2*b^7*c^10*d^3 - 
15*sqrt(a*b)*a^3*b^6*c^8*d^5 + 20*sqrt(a*b)*a^4*b^5*c^6*d^7 - 15*sqrt(a*b) 
*a^5*b^4*c^4*d^9 + 6*sqrt(a*b)*a^6*b^3*c^2*d^11 - sqrt(a*b)*a^7*b^2*d^13)* 
sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(b^4*c^6*d - 3*a*b^3*c^4*d^3 + 3*a^2*b^...
 

Mupad [B] (verification not implemented)

Time = 14.24 (sec) , antiderivative size = 13107, normalized size of antiderivative = 38.44 \[ \int \frac {x^3}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((((-(9*(a^4*d^11*(a^3*b^5)^(1/2) - 36*a*b^7*c^9*d^2 + 5*a^5*b^3*c*d^1 
0 - 673*a^2*b^6*c^7*d^4 - 861*a^3*b^5*c^5*d^6 - 35*a^4*b^4*c^3*d^8 + 240*b 
^4*c^8*d^3*(a^3*b^5)^(1/2) + 1015*a*b^3*c^6*d^5*(a^3*b^5)^(1/2) - 27*a^3*b 
*c^2*d^9*(a^3*b^5)^(1/2) + 371*a^2*b^2*c^4*d^7*(a^3*b^5)^(1/2)))/(4096*(a^ 
2*b^12*c^14 - a^9*b^5*d^14 - 7*a^3*b^11*c^12*d^2 + 21*a^4*b^10*c^10*d^4 - 
35*a^5*b^9*c^8*d^6 + 35*a^6*b^8*c^6*d^8 - 21*a^7*b^7*c^4*d^10 + 7*a^8*b^6* 
c^2*d^12)))^(1/2)*(3145728*a^15*b^7*d^32 + (c + d*x)^(1/2)*(-(9*(a^4*d^11* 
(a^3*b^5)^(1/2) - 36*a*b^7*c^9*d^2 + 5*a^5*b^3*c*d^10 - 673*a^2*b^6*c^7*d^ 
4 - 861*a^3*b^5*c^5*d^6 - 35*a^4*b^4*c^3*d^8 + 240*b^4*c^8*d^3*(a^3*b^5)^( 
1/2) + 1015*a*b^3*c^6*d^5*(a^3*b^5)^(1/2) - 27*a^3*b*c^2*d^9*(a^3*b^5)^(1/ 
2) + 371*a^2*b^2*c^4*d^7*(a^3*b^5)^(1/2)))/(4096*(a^2*b^12*c^14 - a^9*b^5* 
d^14 - 7*a^3*b^11*c^12*d^2 + 21*a^4*b^10*c^10*d^4 - 35*a^5*b^9*c^8*d^6 + 3 
5*a^6*b^8*c^6*d^8 - 21*a^7*b^7*c^4*d^10 + 7*a^8*b^6*c^2*d^12)))^(1/2)*(671 
08864*a*b^24*c^31*d^2 - 67108864*a^16*b^9*c*d^32 - 1006632960*a^2*b^23*c^2 
9*d^4 + 7046430720*a^3*b^22*c^27*d^6 - 30534533120*a^4*b^21*c^25*d^8 + 916 
03599360*a^5*b^20*c^23*d^10 - 201527918592*a^6*b^19*c^21*d^12 + 3358798643 
20*a^7*b^18*c^19*d^14 - 431845539840*a^8*b^17*c^17*d^16 + 431845539840*a^9 
*b^16*c^15*d^18 - 335879864320*a^10*b^15*c^13*d^20 + 201527918592*a^11*b^1 
4*c^11*d^22 - 91603599360*a^12*b^13*c^9*d^24 + 30534533120*a^13*b^12*c^7*d 
^26 - 7046430720*a^14*b^11*c^5*d^28 + 1006632960*a^15*b^10*c^3*d^30) - ...
 

Reduce [B] (verification not implemented)

Time = 3.15 (sec) , antiderivative size = 3600, normalized size of antiderivative = 10.56 \[ \int \frac {x^3}{(c+d x)^{3/2} \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(12*sqrt(a)*sqrt(c + d*x)*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**4*b*c*d**5 + 192*sqrt(a)* 
sqrt(c + d*x)*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**3*b**2*c**3*d**3 - 24*sqrt(a)*sqrt(c 
+ d*x)*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**3*b**2*c*d**5*x**2 + 36*sqrt(a)*sqrt(c + d*x 
)*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**2*b**3*c**5*d - 384*sqrt(a)*sqrt(c + d*x)*sqrt(sq 
rt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a 
)*d - b*c)))*a**2*b**3*c**3*d**3*x**2 + 12*sqrt(a)*sqrt(c + d*x)*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**2*b**3*c*d**5*x**4 - 72*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*s 
qrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b 
*c)))*a*b**4*c**5*d*x**2 + 192*sqrt(a)*sqrt(c + d*x)*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 
*b**4*c**3*d**3*x**4 + 36*sqrt(a)*sqrt(c + d*x)*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**5*c 
**5*d*x**4 - 6*sqrt(b)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((s 
qrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**5*d**6 + 108*s 
qrt(b)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*...