\(\int \frac {x^4}{(c+d x)^{5/2} (a-b x^2)^2} \, dx\) [679]

Optimal result

Integrand size = 23, antiderivative size = 305 \[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=-\frac {2 c^4}{3 d \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}-\frac {8 a c^3 d}{\left (b c^2-a d^2\right )^3 \sqrt {c+d x}}-\frac {a \sqrt {c+d x} \left (a^2 d^3-b^2 c^3 x+3 a b c d (c-d x)\right )}{2 b \left (b c^2-a d^2\right )^3 \left (a-b x^2\right )}+\frac {\sqrt {a} \left (6 \sqrt {b} c-\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 b^{5/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{7/2}}-\frac {\sqrt {a} \left (6 \sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{4 b^{5/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{7/2}} \] Output:

-2/3*c^4/d/(-a*d^2+b*c^2)^2/(d*x+c)^(3/2)-8*a*c^3*d/(-a*d^2+b*c^2)^3/(d*x+ 
c)^(1/2)-1/2*a*(d*x+c)^(1/2)*(a^2*d^3-b^2*c^3*x+3*a*b*c*d*(-d*x+c))/b/(-a* 
d^2+b*c^2)^3/(-b*x^2+a)+1/4*a^(1/2)*(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))/b^(5/4)/(b^(1/2)*c-a^(1/2)*d) 
^(7/2)-1/4*a^(1/2)*(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))/b^(5/4)/(b^(1/2)*c+a^(1/2)*d)^(7/2)
 

Mathematica [A] (verified)

Time = 1.54 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.20 \[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\frac {\frac {2 \left (4 b^3 c^6 x^2-3 a^3 d^4 (c+d x)^2+a^2 b c d^2 \left (-53 c^3-57 c^2 d x+9 c d^2 x^2+9 d^3 x^3\right )+a b^2 c^3 \left (-4 c^3+3 c^2 d x+50 c d^2 x^2+51 d^3 x^3\right )\right )}{d \left (-b c^2+a d^2\right )^3 (c+d x)^{3/2} \left (-a+b x^2\right )}-\frac {3 \left (6 \sqrt {a} \sqrt {b} c+a d\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right )^3 \sqrt {-b c-\sqrt {a} \sqrt {b} d}}-\frac {3 \left (-6 \sqrt {a} \sqrt {b} c+a d\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right )^3 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{12 b} \] Input:

Integrate[x^4/((c + d*x)^(5/2)*(a - b*x^2)^2),x]
 

Output:

((2*(4*b^3*c^6*x^2 - 3*a^3*d^4*(c + d*x)^2 + a^2*b*c*d^2*(-53*c^3 - 57*c^2 
*d*x + 9*c*d^2*x^2 + 9*d^3*x^3) + a*b^2*c^3*(-4*c^3 + 3*c^2*d*x + 50*c*d^2 
*x^2 + 51*d^3*x^3)))/(d*(-(b*c^2) + a*d^2)^3*(c + d*x)^(3/2)*(-a + b*x^2)) 
 - (3*(6*Sqrt[a]*Sqrt[b]*c + a*d)*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d] 
*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/((Sqrt[b]*c + Sqrt[a]*d)^3*Sqrt[ 
-(b*c) - Sqrt[a]*Sqrt[b]*d]) - (3*(-6*Sqrt[a]*Sqrt[b]*c + a*d)*ArcTan[(Sqr 
t[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/((S 
qrt[b]*c - Sqrt[a]*d)^3*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]))/(12*b)
 

Rubi [A] (verified)

Time = 2.72 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.39, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {561, 27, 1673, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1673

\(\displaystyle \frac {2 \left (\frac {d^4 \int -\frac {2 \left (\frac {4 a b c^4}{d^2}-\frac {8 a b \left (b c^2-2 a d^2\right ) (c+d x) c^3}{d^2 \left (b c^2-a d^2\right )}-\frac {a^2 b \left (b c^2+3 a d^2\right ) (c+d x)^3 c}{\left (b c^2-a d^2\right )^2}+\frac {a \left (4 b^3 c^6-13 a b^2 d^2 c^4+18 a^2 b d^4 c^2-a^3 d^6\right ) (c+d x)^2}{d^2 \left (b c^2-a d^2\right )^2}\right )}{(c+d x)^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (a^2 d^4-b c (c+d x) \left (3 a d^2+b c^2\right )+6 a b c^2 d^2+b^2 c^4\right )}{4 b \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (-\frac {d^4 \int \frac {\frac {4 a b c^4}{d^2}-\frac {8 a b \left (b c^2-2 a d^2\right ) (c+d x) c^3}{d^2 \left (b c^2-a d^2\right )}-\frac {a^2 b \left (b c^2+3 a d^2\right ) (c+d x)^3 c}{\left (b c^2-a d^2\right )^2}+\frac {a \left (4 b^3 c^6-13 a b^2 d^2 c^4+18 a^2 b d^4 c^2-a^3 d^6\right ) (c+d x)^2}{d^2 \left (b c^2-a d^2\right )^2}}{(c+d x)^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (a^2 d^4-b c (c+d x) \left (3 a d^2+b c^2\right )+6 a b c^2 d^2+b^2 c^4\right )}{4 b \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^5}\)

\(\Big \downarrow \) 2195

\(\displaystyle \frac {2 \left (-\frac {d^4 \int \left (-\frac {4 a b c^4}{\left (b c^2-a d^2\right ) (c+d x)^2}-\frac {16 a^2 b d^2 c^3}{\left (b c^2-a d^2\right )^2 (c+d x)}+\frac {a^2 d^2 \left (-23 b^2 c^4-18 a b d^2 c^2+b \left (17 b c^2+3 a d^2\right ) (c+d x) c+a^2 d^4\right )}{\left (b c^2-a d^2\right )^2 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}\right )d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (a^2 d^4-b c (c+d x) \left (3 a d^2+b c^2\right )+6 a b c^2 d^2+b^2 c^4\right )}{4 b \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^5}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {a d^4 \sqrt {c+d x} \left (a^2 d^4-b c (c+d x) \left (3 a d^2+b c^2\right )+6 a b c^2 d^2+b^2 c^4\right )}{4 b \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}-\frac {d^4 \left (-\frac {a^{3/2} d \left (6 \sqrt {b} c-\sqrt {a} d\right ) \left (\sqrt {a} d+\sqrt {b} c\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}+\frac {a^{3/2} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (\sqrt {a} d+6 \sqrt {b} c\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt [4]{b} \left (\sqrt {a} d+\sqrt {b} c\right )^{5/2}}+\frac {16 a^2 b c^3 d^2}{\sqrt {c+d x} \left (b c^2-a d^2\right )^2}+\frac {4 a b c^4}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\right )}{4 a b \left (b c^2-a d^2\right )}\right )}{d^5}\)

Input:

Int[x^4/((c + d*x)^(5/2)*(a - b*x^2)^2),x]
 

Output:

(2*(-1/4*(a*d^4*Sqrt[c + d*x]*(b^2*c^4 + 6*a*b*c^2*d^2 + a^2*d^4 - b*c*(b* 
c^2 + 3*a*d^2)*(c + d*x)))/(b*(b*c^2 - a*d^2)^3*(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*b*c^4)/(3*(b*c^2 - a*d 
^2)*(c + d*x)^(3/2)) + (16*a^2*b*c^3*d^2)/((b*c^2 - a*d^2)^2*Sqrt[c + d*x] 
) - (a^(3/2)*d*(6*Sqrt[b]*c - Sqrt[a]*d)*(Sqrt[b]*c + Sqrt[a]*d)*ArcTanh[( 
b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*b^(1/4)*(Sqrt[b]*c 
 - Sqrt[a]*d)^(5/2)) + (a^(3/2)*d*(Sqrt[b]*c - Sqrt[a]*d)*(6*Sqrt[b]*c + S 
qrt[a]*d)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2 
*b^(1/4)*(Sqrt[b]*c + Sqrt[a]*d)^(5/2))))/(4*a*b*(b*c^2 - a*d^2))))/d^5
 

Defintions of rubi rules used

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]
 

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(\frac {\frac {2 a \,d^{2} \left (\frac {\left (-\frac {3}{4} a \,d^{2} c -\frac {1}{4} b \,c^{3}\right ) \left (d x +c \right )^{\frac {3}{2}}+\frac {\left (a^{2} d^{4}+6 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {d x +c}}{4 b}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {\left (-a^{2} d^{4}+15 b \,c^{2} d^{2} a +6 b^{2} c^{4}+3 \sqrt {a b \,d^{2}}\, a c \,d^{2}+17 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{8 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (a^{2} d^{4}-15 b \,c^{2} d^{2} a -6 b^{2} c^{4}+3 \sqrt {a b \,d^{2}}\, a c \,d^{2}+17 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{8 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{3}}-\frac {2 c^{4}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{2} \left (d x +c \right )^{\frac {3}{2}}}+\frac {8 c^{3} a \,d^{2}}{\left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {d x +c}}}{d}\)	\(394\)
default	\(\frac {\frac {2 a \,d^{2} \left (\frac {\left (-\frac {3}{4} a \,d^{2} c -\frac {1}{4} b \,c^{3}\right ) \left (d x +c \right )^{\frac {3}{2}}+\frac {\left (a^{2} d^{4}+6 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {d x +c}}{4 b}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {\left (-a^{2} d^{4}+15 b \,c^{2} d^{2} a +6 b^{2} c^{4}+3 \sqrt {a b \,d^{2}}\, a c \,d^{2}+17 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{8 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (a^{2} d^{4}-15 b \,c^{2} d^{2} a -6 b^{2} c^{4}+3 \sqrt {a b \,d^{2}}\, a c \,d^{2}+17 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{8 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{3}}-\frac {2 c^{4}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{2} \left (d x +c \right )^{\frac {3}{2}}}+\frac {8 c^{3} a \,d^{2}}{\left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {d x +c}}}{d}\)	\(394\)
pseudoelliptic	\(-\frac {d^{2} \left (\left (-3 a \,d^{2} c -17 b \,c^{3}\right ) \sqrt {a b \,d^{2}}+a^{2} d^{4}-15 b \,c^{2} d^{2} a -6 b^{2} c^{4}\right ) b a \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right ) \left (d x +c \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (d^{2} \left (\left (3 a \,d^{2} c +17 b \,c^{3}\right ) \sqrt {a b \,d^{2}}+a^{2} d^{4}-15 b \,c^{2} d^{2} a -6 b^{2} c^{4}\right ) b a \left (-b \,x^{2}+a \right ) \left (d x +c \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-2 \left (-\frac {4 b^{3} c^{6} x^{2}}{3}+\frac {4 \left (-\frac {51}{4} d^{3} x^{3}-\frac {25}{2} c \,d^{2} x^{2}-\frac {3}{4} c^{2} d x +c^{3}\right ) a \,c^{3} b^{2}}{3}+\frac {53 d^{2} \left (-\frac {9}{53} d^{3} x^{3}-\frac {9}{53} c \,d^{2} x^{2}+\frac {57}{53} c^{2} d x +c^{3}\right ) a^{2} c b}{3}+a^{3} d^{4} \left (d x +c \right )^{2}\right ) \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\right )}{4 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \left (d x +c \right )^{\frac {3}{2}} \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, d \left (-b \,x^{2}+a \right ) \left (a \,d^{2}-b \,c^{2}\right )^{3} b}\)	\(437\)

Input:

int(x^4/(d*x+c)^(5/2)/(-b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

2/d*(a*d^2/(a*d^2-b*c^2)^3*(((-3/4*a*d^2*c-1/4*b*c^3)*(d*x+c)^(3/2)+1/4*(a 
^2*d^4+6*a*b*c^2*d^2+b^2*c^4)/b*(d*x+c)^(1/2))/(-b*(d*x+c)^2+2*b*c*(d*x+c) 
+a*d^2-b*c^2)+1/8*(-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))-1/8*(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)))-1/3*c^4/(a*d^2-b*c^2)^2/(d*x+c)^(3/2) 
+4*c^3*a*d^2/(a*d^2-b*c^2)^3/(d*x+c)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8773 vs. \(2 (245) = 490\).

Time = 6.14 (sec) , antiderivative size = 8773, normalized size of antiderivative = 28.76 \[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^4/(d*x+c)^(5/2)/(-b*x^2+a)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

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

integrate(x**4/(d*x+c)**(5/2)/(-b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^4/(d*x+c)^(5/2)/(-b*x^2+a)^2,x, algorithm="maxima")
 

Output:

integrate(x^4/((b*x^2 - a)^2*(d*x + c)^(5/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2081 vs. \(2 (245) = 490\).

Time = 0.43 (sec) , antiderivative size = 2081, normalized size of antiderivative = 6.82 \[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^4/(d*x+c)^(5/2)/(-b*x^2+a)^2,x, algorithm="giac")
 

Output:

1/12*(3*((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^5 + 3*sqrt(a*b)*a^2*c*d^7)*abs(b) - (23*a*b^6*c^10* 
d^5 - 51*a^2*b^5*c^8*d^7 + 14*a^3*b^4*c^6*d^9 + 34*a^4*b^3*c^4*d^11 - 21*a 
^5*b^2*c^2*d^13 + a^6*b*d^15)*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^5 - 21*sqrt(a*b)*a* 
b^9*c^15*d^7 - sqrt(a*b)*a^2*b^8*c^13*d^9 + 111*sqrt(a*b)*a^3*b^7*c^11*d^1 
1 - 225*sqrt(a*b)*a^4*b^6*c^9*d^13 + 209*sqrt(a*b)*a^5*b^5*c^7*d^15 - 99*s 
qrt(a*b)*a^6*b^4*c^5*d^17 + 21*sqrt(a*b)*a^7*b^3*c^3*d^19 - sqrt(a*b)*a^8* 
b^2*c*d^21)*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)))/((b^9*c^13 - 6*a*b^8*c^11*d^2 + 15*a^2*b^7*c^9*d^4 
- 20*a^3*b^6*c^7*d^6 + 15*a^4*b^5*c^5*d^8 - 6*a^5*b^4*c^3*d^10 + a^6*b^3*c 
*d^12 - sqrt(a*b)*b^8*c^12*d + 6*sqrt(a*b)*a*b^7*c^10*d^3 - 15*sqrt(a*b)*a 
^2*b^6*c^8*d^5 + 20*sqrt(a*b)*a^3*b^5*c^6*d^7 - 15*sqrt(a*b)*a^4*b^4*c^4*d 
^9 + 6*sqrt(a*b)*a^5*b^3*c^2*d^11 - sqrt(a*b)*a^6*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^2*c^2*d^5 - a^3*b 
*d^7)) - 3*((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^...
 

Mupad [B] (verification not implemented)

Time = 12.90 (sec) , antiderivative size = 12755, normalized size of antiderivative = 41.82 \[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int(x^4/((a - b*x^2)^2*(c + d*x)^(5/2)),x)
 

Output:

- atan(-(((-(a^4*d^9*(a^3*b^5)^(1/2) - 36*a*b^7*c^9 + 5*a^5*b^3*c*d^8 - 67 
3*a^2*b^6*c^7*d^2 - 861*a^3*b^5*c^5*d^4 - 35*a^4*b^4*c^3*d^6 + 240*b^4*c^8 
*d*(a^3*b^5)^(1/2) + 1015*a*b^3*c^6*d^3*(a^3*b^5)^(1/2) - 27*a^3*b*c^2*d^7 
*(a^3*b^5)^(1/2) + 371*a^2*b^2*c^4*d^5*(a^3*b^5)^(1/2))/(64*(b^12*c^14 - a 
^7*b^5*d^14 - 7*a*b^11*c^12*d^2 + 21*a^2*b^10*c^10*d^4 - 35*a^3*b^9*c^8*d^ 
6 + 35*a^4*b^8*c^6*d^8 - 21*a^5*b^7*c^4*d^10 + 7*a^6*b^6*c^2*d^12)))^(1/2) 
*(256*a^16*b^7*d^31 + (c + d*x)^(1/2)*(-(a^4*d^9*(a^3*b^5)^(1/2) - 36*a*b^ 
7*c^9 + 5*a^5*b^3*c*d^8 - 673*a^2*b^6*c^7*d^2 - 861*a^3*b^5*c^5*d^4 - 35*a 
^4*b^4*c^3*d^6 + 240*b^4*c^8*d*(a^3*b^5)^(1/2) + 1015*a*b^3*c^6*d^3*(a^3*b 
^5)^(1/2) - 27*a^3*b*c^2*d^7*(a^3*b^5)^(1/2) + 371*a^2*b^2*c^4*d^5*(a^3*b^ 
5)^(1/2))/(64*(b^12*c^14 - a^7*b^5*d^14 - 7*a*b^11*c^12*d^2 + 21*a^2*b^10* 
c^10*d^4 - 35*a^3*b^9*c^8*d^6 + 35*a^4*b^8*c^6*d^8 - 21*a^5*b^7*c^4*d^10 + 
 7*a^6*b^6*c^2*d^12)))^(1/2)*(2048*a*b^24*c^31*d^2 - 2048*a^16*b^9*c*d^32 
- 30720*a^2*b^23*c^29*d^4 + 215040*a^3*b^22*c^27*d^6 - 931840*a^4*b^21*c^2 
5*d^8 + 2795520*a^5*b^20*c^23*d^10 - 6150144*a^6*b^19*c^21*d^12 + 10250240 
*a^7*b^18*c^19*d^14 - 13178880*a^8*b^17*c^17*d^16 + 13178880*a^9*b^16*c^15 
*d^18 - 10250240*a^10*b^15*c^13*d^20 + 6150144*a^11*b^14*c^11*d^22 - 27955 
20*a^12*b^13*c^9*d^24 + 931840*a^13*b^12*c^7*d^26 - 215040*a^14*b^11*c^5*d 
^28 + 30720*a^15*b^10*c^3*d^30) - 5888*a^2*b^21*c^28*d^3 + 66048*a^3*b^20* 
c^26*d^5 - 333056*a^4*b^19*c^24*d^7 + 988160*a^5*b^18*c^22*d^9 - 188390...
 

Reduce [B] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 4619, normalized size of antiderivative = 15.14 \[ \int \frac {x^4}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(x^4/(d*x+c)^(5/2)/(-b*x^2+a)^2,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**3*b*c**2*d**5 + 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**3*b*c*d**6*x + 192*sqrt(a)*sqrt(c + 
 d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(s 
qrt(b)*sqrt(a)*d - b*c)))*a**2*b**2*c**4*d**3 + 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**2*b**2*c**3*d**4*x - 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**2*c**2*d**5*x**2 - 12*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**2*c*d**6*x**3 + 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*b**3*c**6*d + 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*b** 
3*c**5*d**2*x - 192*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*at 
an((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**3*c**4* 
d**3*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**3*c**3*d**4 
*x**3 - 36*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sq...