\(\int \frac {x^3}{(c+d x)^{5/2} (a-b x^2)} \, dx\) [598]

Optimal result

Integrand size = 23, antiderivative size = 201 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )} \, dx=-\frac {2 c^3}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {2 c^2 \left (b c^2-3 a d^2\right )}{d^2 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {a \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{b^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}+\frac {a \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{b^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.25 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )} \, dx=\frac {2 b c^4 (2 c+3 d x)-2 a c^2 d^2 (8 c+9 d x)}{3 \left (b c^2 d-a d^3\right )^2 (c+d x)^{3/2}}-\frac {a \sqrt {-b c-\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{b \left (\sqrt {b} c+\sqrt {a} d\right )^3}-\frac {a \sqrt {-b c+\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{b \left (\sqrt {b} c-\sqrt {a} d\right )^3} \] Input:

Integrate[x^3/((c + d*x)^(5/2)*(a - b*x^2)),x]
 

Output:

(2*b*c^4*(2*c + 3*d*x) - 2*a*c^2*d^2*(8*c + 9*d*x))/(3*(b*c^2*d - a*d^3)^2 
*(c + d*x)^(3/2)) - (a*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)])/(b*(Sqrt[b 
]*c + Sqrt[a]*d)^3) - (a*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)])/(b*(Sqrt 
[b]*c - Sqrt[a]*d)^3)
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {561, 25, 27, 1610, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1610

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^3/((c + d*x)^(5/2)*(a - b*x^2)),x]
 

Output:

(-2*((c^3*d^2)/(3*(b*c^2 - a*d^2)*(c + d*x)^(3/2)) - (c^2*d^2*(b*c^2 - 3*a 
*d^2))/((b*c^2 - a*d^2)^2*Sqrt[c + d*x]) - (a*d^4*ArcTanh[(b^(1/4)*Sqrt[c 
+ d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*b^(3/4)*(Sqrt[b]*c - Sqrt[a]*d)^( 
5/2)) - (a*d^4*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d] 
])/(2*b^(3/4)*(Sqrt[b]*c + Sqrt[a]*d)^(5/2))))/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 1610

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^2)^q/(a 
+ b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4 
*a*c, 0] && IntegerQ[q] && IntegerQ[m]
 

rule 2009

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

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.39

method	result	size
derivativedivides	\(-\frac {2 \left (-\frac {a \,d^{2} b \left (\frac {\left (-2 a b c \,d^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (2 a b c \,d^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {c^{3}}{3 \left (a \,d^{2}-b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}+\frac {c^{2} \left (3 a \,d^{2}-b \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {d x +c}}\right )}{d^{2}}\)	\(279\)
default	\(\frac {\frac {2 a \,d^{2} b \left (\frac {\left (-2 a b c \,d^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (2 a b c \,d^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}-\sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {2 c^{2} \left (3 a \,d^{2}-b \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {d x +c}}+\frac {2 c^{3}}{3 \left (a \,d^{2}-b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}}{d^{2}}\)	\(279\)
pseudoelliptic	\(-\frac {2 \left (d^{2} \left (d x +c \right )^{\frac {3}{2}} a \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\frac {\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{2}+a b c \,d^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\left (d^{2} \left (\frac {\left (-a \,d^{2}-b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{2}+a b c \,d^{2}\right ) \left (d x +c \right )^{\frac {3}{2}} a \,\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\frac {8 \sqrt {a b \,d^{2}}\, \left (-\frac {3}{8} b \,c^{2} d x -\frac {1}{4} b \,c^{3}+\frac {9}{8} a x \,d^{3}+a \,d^{2} c \right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, c^{2}}{3}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\right )}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \left (d x +c \right )^{\frac {3}{2}} \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (a \,d^{2}-b \,c^{2}\right )^{2} d^{2}}\)	\(298\)

Input:

int(x^3/(d*x+c)^(5/2)/(-b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5212 vs. \(2 (159) = 318\).

Time = 0.38 (sec) , antiderivative size = 5212, normalized size of antiderivative = 25.93 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(x^3/(d*x+c)^(5/2)/(-b*x^2+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )} \, dx=- \int \frac {x^{3}}{- a c^{2} \sqrt {c + d x} - 2 a c d x \sqrt {c + d x} - a d^{2} x^{2} \sqrt {c + d x} + b c^{2} x^{2} \sqrt {c + d x} + 2 b c d x^{3} \sqrt {c + d x} + b d^{2} x^{4} \sqrt {c + d x}}\, dx \] Input:

integrate(x**3/(d*x+c)**(5/2)/(-b*x**2+a),x)
 

Output:

-Integral(x**3/(-a*c**2*sqrt(c + d*x) - 2*a*c*d*x*sqrt(c + d*x) - a*d**2*x 
**2*sqrt(c + d*x) + b*c**2*x**2*sqrt(c + d*x) + 2*b*c*d*x**3*sqrt(c + d*x) 
 + b*d**2*x**4*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )} \, dx=\int { -\frac {x^{3}}{{\left (b x^{2} - a\right )} {\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^3/(d*x+c)^(5/2)/(-b*x^2+a),x, algorithm="maxima")
 

Output:

-integrate(x^3/((b*x^2 - a)*(d*x + c)^(5/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1269 vs. \(2 (159) = 318\).

Time = 0.28 (sec) , antiderivative size = 1269, normalized size of antiderivative = 6.31 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(x^3/(d*x+c)^(5/2)/(-b*x^2+a),x, algorithm="giac")
 

Output:

((b^2*c^4*d - 2*a*b*c^2*d^3 + a^2*d^5)^2*(sqrt(a*b)*a*b*c^2 + sqrt(a*b)*a^ 
2*d^2)*abs(b) - (a*b^4*c^7 + a^2*b^3*c^5*d^2 - 5*a^3*b^2*c^3*d^4 + 3*a^4*b 
*c*d^6)*abs(b^2*c^4*d - 2*a*b*c^2*d^3 + a^2*d^5)*abs(b) + 2*(sqrt(a*b)*a*b 
^5*c^10*d^2 - 4*sqrt(a*b)*a^2*b^4*c^8*d^4 + 6*sqrt(a*b)*a^3*b^3*c^6*d^6 - 
4*sqrt(a*b)*a^4*b^2*c^4*d^8 + sqrt(a*b)*a^5*b*c^2*d^10)*abs(b))*arctan(sqr 
t(d*x + c)/sqrt(-(b^3*c^5 - 2*a*b^2*c^3*d^2 + a^2*b*c*d^4 + sqrt((b^3*c^5 
- 2*a*b^2*c^3*d^2 + a^2*b*c*d^4)^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)*(b^3*c^4 - 2*a*b^2*c^2*d^2 + a^2*b*d^4)))/(b^3*c^4 - 2* 
a*b^2*c^2*d^2 + a^2*b*d^4)))/((b^6*c^9 - 4*a*b^5*c^7*d^2 + 6*a^2*b^4*c^5*d 
^4 - 4*a^3*b^3*c^3*d^6 + a^4*b^2*c*d^8 - sqrt(a*b)*b^5*c^8*d + 4*sqrt(a*b) 
*a*b^4*c^6*d^3 - 6*sqrt(a*b)*a^2*b^3*c^4*d^5 + 4*sqrt(a*b)*a^3*b^2*c^2*d^7 
 - sqrt(a*b)*a^4*b*d^9)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(b^2*c^4*d - 2*a*b 
*c^2*d^3 + a^2*d^5)) - ((b^2*c^4*d - 2*a*b*c^2*d^3 + a^2*d^5)^2*(sqrt(a*b) 
*a*b*c^2 + sqrt(a*b)*a^2*d^2)*abs(b) + (a*b^4*c^7 + a^2*b^3*c^5*d^2 - 5*a^ 
3*b^2*c^3*d^4 + 3*a^4*b*c*d^6)*abs(b^2*c^4*d - 2*a*b*c^2*d^3 + a^2*d^5)*ab 
s(b) + 2*(sqrt(a*b)*a*b^5*c^10*d^2 - 4*sqrt(a*b)*a^2*b^4*c^8*d^4 + 6*sqrt( 
a*b)*a^3*b^3*c^6*d^6 - 4*sqrt(a*b)*a^4*b^2*c^4*d^8 + sqrt(a*b)*a^5*b*c^2*d 
^10)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^3*c^5 - 2*a*b^2*c^3*d^2 + a^2*b 
*c*d^4 - sqrt((b^3*c^5 - 2*a*b^2*c^3*d^2 + a^2*b*c*d^4)^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)*(b^3*c^4 - 2*a*b^2*c^2*d^2 + ...
 

Mupad [B] (verification not implemented)

Time = 10.37 (sec) , antiderivative size = 7980, normalized size of antiderivative = 39.70 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

int(x^3/((a - b*x^2)*(c + d*x)^(5/2)),x)
 

Output:

atan((((c + d*x)^(1/2)*(16*a^11*b^2*d^18 + 16*a^3*b^10*c^16*d^2 - 320*a^5* 
b^8*c^12*d^6 + 1024*a^6*b^7*c^10*d^8 - 1440*a^7*b^6*c^8*d^10 + 1024*a^8*b^ 
5*c^6*d^12 - 320*a^9*b^4*c^4*d^14) + ((a^2*d^5*(a^5*b^3)^(1/2) + a^2*b^4*c 
^5 + 5*a^4*b^2*c*d^4 + 10*a^3*b^3*c^3*d^2 + 5*b^2*c^4*d*(a^5*b^3)^(1/2) + 
10*a*b*c^2*d^3*(a^5*b^3)^(1/2))/(4*(b^8*c^10 - a^5*b^3*d^10 - 5*a*b^7*c^8* 
d^2 + 10*a^2*b^6*c^6*d^4 - 10*a^3*b^5*c^4*d^6 + 5*a^4*b^4*c^2*d^8)))^(1/2) 
*(96*a^11*b^3*c*d^20 - (c + d*x)^(1/2)*((a^2*d^5*(a^5*b^3)^(1/2) + a^2*b^4 
*c^5 + 5*a^4*b^2*c*d^4 + 10*a^3*b^3*c^3*d^2 + 5*b^2*c^4*d*(a^5*b^3)^(1/2) 
+ 10*a*b*c^2*d^3*(a^5*b^3)^(1/2))/(4*(b^8*c^10 - a^5*b^3*d^10 - 5*a*b^7*c^ 
8*d^2 + 10*a^2*b^6*c^6*d^4 - 10*a^3*b^5*c^4*d^6 + 5*a^4*b^4*c^2*d^8)))^(1/ 
2)*(64*a*b^14*c^21*d^2 + 64*a^11*b^4*c*d^22 - 640*a^2*b^13*c^19*d^4 + 2880 
*a^3*b^12*c^17*d^6 - 7680*a^4*b^11*c^15*d^8 + 13440*a^5*b^10*c^13*d^10 - 1 
6128*a^6*b^9*c^11*d^12 + 13440*a^7*b^8*c^9*d^14 - 7680*a^8*b^7*c^7*d^16 + 
2880*a^9*b^6*c^5*d^18 - 640*a^10*b^5*c^3*d^20) + 32*a^2*b^12*c^19*d^2 - 16 
0*a^3*b^11*c^17*d^4 + 128*a^4*b^10*c^15*d^6 + 896*a^5*b^9*c^13*d^8 - 3136* 
a^6*b^8*c^11*d^10 + 4928*a^7*b^7*c^9*d^12 - 4480*a^8*b^6*c^7*d^14 + 2432*a 
^9*b^5*c^5*d^16 - 736*a^10*b^4*c^3*d^18))*((a^2*d^5*(a^5*b^3)^(1/2) + a^2* 
b^4*c^5 + 5*a^4*b^2*c*d^4 + 10*a^3*b^3*c^3*d^2 + 5*b^2*c^4*d*(a^5*b^3)^(1/ 
2) + 10*a*b*c^2*d^3*(a^5*b^3)^(1/2))/(4*(b^8*c^10 - a^5*b^3*d^10 - 5*a*b^7 
*c^8*d^2 + 10*a^2*b^6*c^6*d^4 - 10*a^3*b^5*c^4*d^6 + 5*a^4*b^4*c^2*d^8)...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1463, normalized size of antiderivative = 7.28 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )} \, dx =\text {Too large to display} \] Input:

int(x^3/(d*x+c)^(5/2)/(-b*x^2+a),x)
 

Output:

( - 6*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*c*d**5 - 6*sqrt(a)*sq 
rt(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*d**6*x - 18*sqrt(a)*sqrt(c + d*x)*sqr 
t(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sq 
rt(a)*d - b*c)))*a*b*c**3*d**3 - 18*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqr 
t(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c 
)))*a*b*c**2*d**4*x - 18*sqrt(b)*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*c* 
*2*d**4 - 18*sqrt(b)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqr 
t(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*c*d**5*x - 6*s 
qrt(b)*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*c**4*d**2 - 6*sqrt(b)*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*c**3*d**3*x - 3*sqrt(a)*sqrt(c + d*x)*sqrt( 
sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sq 
rt(c + d*x))*a**2*c*d**5 - 3*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d 
+ b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a**2* 
d**6*x - 9*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt 
(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a*b*c**3*d**3 - 9*sq...