\(\int \frac {x^3}{(c+d x)^2 (a+b x^2)^{5/2}} \, dx\) [1289]

Optimal result

Integrand size = 22, antiderivative size = 221 \[ \int \frac {x^3}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {a \left (b c^2-a d^2-2 b c d x\right )}{3 b \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )^{3/2}}-\frac {c \left (3 c \left (b c^2-3 a d^2\right )-4 d \left (2 b c^2-a d^2\right ) x\right )}{3 \left (b c^2+a d^2\right )^3 \sqrt {a+b x^2}}+\frac {c^3 d^2 \sqrt {a+b x^2}}{\left (b c^2+a d^2\right )^3 (c+d x)}+\frac {c^2 d \left (2 b c^2-3 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 10.32 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.03 \[ \int \frac {x^3}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {-a^3 d^4 (c+d x)+3 a^2 b c d^2 \left (4 c^2+c d x-2 d^2 x^2\right )+b^3 c^3 x^2 \left (-3 c^2+5 c d x+11 d^2 x^2\right )+a b^2 c \left (-2 c^4+4 c^3 d x+21 c^2 d^2 x^2+5 c d^3 x^3-4 d^4 x^4\right )}{3 b \left (b c^2+a d^2\right )^3 (c+d x) \left (a+b x^2\right )^{3/2}}+\frac {c^2 d \left (2 b c^2-3 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{7/2}} \] Input:

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

Output:

(-(a^3*d^4*(c + d*x)) + 3*a^2*b*c*d^2*(4*c^2 + c*d*x - 2*d^2*x^2) + b^3*c^ 
3*x^2*(-3*c^2 + 5*c*d*x + 11*d^2*x^2) + a*b^2*c*(-2*c^4 + 4*c^3*d*x + 21*c 
^2*d^2*x^2 + 5*c*d^3*x^3 - 4*d^4*x^4))/(3*b*(b*c^2 + a*d^2)^3*(c + d*x)*(a 
 + b*x^2)^(3/2)) + (c^2*d*(2*b*c^2 - 3*a*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[ 
b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(b*c^2 + a*d^2)^(7/2)
 

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {601, 25, 2178, 27, 679, 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.

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

\(\Big \downarrow \) 601

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2178

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 679

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 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 601

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c 
+ d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* 
(2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] 
 && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
 

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 2178

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po 
lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia 
lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + 
b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x 
)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 
2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1561\) vs. \(2(205)=410\).

Time = 0.39 (sec) , antiderivative size = 1562, normalized size of antiderivative = 7.07

method	result	size
default	\(\text {Expression too large to display}\)	\(1562\)

Input:

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

Output:

-1/3/d^2/b/(b*x^2+a)^(3/2)-2*c/d^3*(1/3*x/a/(b*x^2+a)^(3/2)+2/3/a^2/(b*x^2 
+a)^(1/2)*x)+3/d^4*c^2*(1/3/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d) 
+(a*d^2+b*c^2)/d^2)^(3/2)+b*c*d/(a*d^2+b*c^2)*(2/3*(2*b*(x+c/d)-2*b*c/d)/( 
4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b 
*c^2)/d^2)^(3/2)+16/3*b/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)^2*(2*b*(x+c/ 
d)-2*b*c/d)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))+1/(a*d^ 
2+b*c^2)*d^2*(1/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^ 
2)/d^2)^(1/2)+2*b*c*d/(a*d^2+b*c^2)*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^ 
2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2 
)-1/(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))))-c^3/d^5*(-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)^(3/2)+5*b*c*d/(a*d^2+b*c^2)* 
(1/3/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/ 
2)+b*c*d/(a*d^2+b*c^2)*(2/3*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4 
*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+16/3*b 
/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)^2*(2*b*(x+c/d)-2*b*c/d)/(b*(x+c/d)^ 
2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))+1/(a*d^2+b*c^2)*d^2*(1/(a*d^2+ 
b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+2*b*c*d/( 
a*d^2+b*c^2)*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 879 vs. \(2 (206) = 412\).

Time = 0.38 (sec) , antiderivative size = 1784, normalized size of antiderivative = 8.07 \[ \int \frac {x^3}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/6*(3*(2*a^2*b^2*c^5*d - 3*a^3*b*c^3*d^3 + (2*b^4*c^4*d^2 - 3*a*b^3*c^2 
*d^4)*x^5 + (2*b^4*c^5*d - 3*a*b^3*c^3*d^3)*x^4 + 2*(2*a*b^3*c^4*d^2 - 3*a 
^2*b^2*c^2*d^4)*x^3 + 2*(2*a*b^3*c^5*d - 3*a^2*b^2*c^3*d^3)*x^2 + (2*a^2*b 
^2*c^4*d^2 - 3*a^3*b*c^2*d^4)*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*(2*a*b^3*c^7 - 1 
0*a^2*b^2*c^5*d^2 - 11*a^3*b*c^3*d^4 + a^4*c*d^6 - (11*b^4*c^5*d^2 + 7*a*b 
^3*c^3*d^4 - 4*a^2*b^2*c*d^6)*x^4 - 5*(b^4*c^6*d + 2*a*b^3*c^4*d^3 + a^2*b 
^2*c^2*d^5)*x^3 + 3*(b^4*c^7 - 6*a*b^3*c^5*d^2 - 5*a^2*b^2*c^3*d^4 + 2*a^3 
*b*c*d^6)*x^2 - (4*a*b^3*c^6*d + 7*a^2*b^2*c^4*d^3 + 2*a^3*b*c^2*d^5 - a^4 
*d^7)*x)*sqrt(b*x^2 + a))/(a^2*b^5*c^9 + 4*a^3*b^4*c^7*d^2 + 6*a^4*b^3*c^5 
*d^4 + 4*a^5*b^2*c^3*d^6 + a^6*b*c*d^8 + (b^7*c^8*d + 4*a*b^6*c^6*d^3 + 6* 
a^2*b^5*c^4*d^5 + 4*a^3*b^4*c^2*d^7 + a^4*b^3*d^9)*x^5 + (b^7*c^9 + 4*a*b^ 
6*c^7*d^2 + 6*a^2*b^5*c^5*d^4 + 4*a^3*b^4*c^3*d^6 + a^4*b^3*c*d^8)*x^4 + 2 
*(a*b^6*c^8*d + 4*a^2*b^5*c^6*d^3 + 6*a^3*b^4*c^4*d^5 + 4*a^4*b^3*c^2*d^7 
+ a^5*b^2*d^9)*x^3 + 2*(a*b^6*c^9 + 4*a^2*b^5*c^7*d^2 + 6*a^3*b^4*c^5*d^4 
+ 4*a^4*b^3*c^3*d^6 + a^5*b^2*c*d^8)*x^2 + (a^2*b^5*c^8*d + 4*a^3*b^4*c^6* 
d^3 + 6*a^4*b^3*c^4*d^5 + 4*a^5*b^2*c^2*d^7 + a^6*b*d^9)*x), 1/3*(3*(2*a^2 
*b^2*c^5*d - 3*a^3*b*c^3*d^3 + (2*b^4*c^4*d^2 - 3*a*b^3*c^2*d^4)*x^5 + (2* 
b^4*c^5*d - 3*a*b^3*c^3*d^3)*x^4 + 2*(2*a*b^3*c^4*d^2 - 3*a^2*b^2*c^2*d...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 869 vs. \(2 (206) = 412\).

Time = 0.13 (sec) , antiderivative size = 869, normalized size of antiderivative = 3.93 \[ \int \frac {x^3}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

-5*b^2*c^5*x/(sqrt(b*x^2 + a)*a*b^3*c^6*d + 3*sqrt(b*x^2 + a)*a^2*b^2*c^4* 
d^3 + 3*sqrt(b*x^2 + a)*a^3*b*c^2*d^5 + sqrt(b*x^2 + a)*a^4*d^7) - 5/3*b^2 
*c^5*x/((b*x^2 + a)^(3/2)*a*b^2*c^4*d^3 + 2*(b*x^2 + a)^(3/2)*a^2*b*c^2*d^ 
5 + (b*x^2 + a)^(3/2)*a^3*d^7) - 10/3*b^2*c^5*x/(sqrt(b*x^2 + a)*a^2*b^2*c 
^4*d^3 + 2*sqrt(b*x^2 + a)*a^3*b*c^2*d^5 + sqrt(b*x^2 + a)*a^4*d^7) - 5*b* 
c^4/(sqrt(b*x^2 + a)*b^3*c^6 + 3*sqrt(b*x^2 + a)*a*b^2*c^4*d^2 + 3*sqrt(b* 
x^2 + a)*a^2*b*c^2*d^4 + sqrt(b*x^2 + a)*a^3*d^6) - 5/3*b*c^4/((b*x^2 + a) 
^(3/2)*b^2*c^4*d^2 + 2*(b*x^2 + a)^(3/2)*a*b*c^2*d^4 + (b*x^2 + a)^(3/2)*a 
^2*d^6) + 3*b*c^3*x/(sqrt(b*x^2 + a)*a*b^2*c^4*d + 2*sqrt(b*x^2 + a)*a^2*b 
*c^2*d^3 + sqrt(b*x^2 + a)*a^3*d^5) + 7/3*b*c^3*x/((b*x^2 + a)^(3/2)*a*b*c 
^2*d^3 + (b*x^2 + a)^(3/2)*a^2*d^5) + 14/3*b*c^3*x/(sqrt(b*x^2 + a)*a^2*b* 
c^2*d^3 + sqrt(b*x^2 + a)*a^3*d^5) + c^3/((b*x^2 + a)^(3/2)*b*c^2*d^3*x + 
(b*x^2 + a)^(3/2)*a*d^5*x + (b*x^2 + a)^(3/2)*b*c^3*d^2 + (b*x^2 + a)^(3/2 
)*a*c*d^4) + 3*c^2/(sqrt(b*x^2 + a)*b^2*c^4 + 2*sqrt(b*x^2 + a)*a*b*c^2*d^ 
2 + sqrt(b*x^2 + a)*a^2*d^4) + c^2/((b*x^2 + a)^(3/2)*b*c^2*d^2 + (b*x^2 + 
 a)^(3/2)*a*d^4) - 4/3*c*x/(sqrt(b*x^2 + a)*a^2*d^3) - 2/3*c*x/((b*x^2 + a 
)^(3/2)*a*d^3) - 5*b*c^4*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqr 
t(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(7/2)*d^6) + 3*c^2*arcsinh(b*c*x/(s 
qrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(5 
/2)*d^4) - 1/3/((b*x^2 + a)^(3/2)*b*d^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1448 vs. \(2 (206) = 412\).

Time = 0.31 (sec) , antiderivative size = 1448, normalized size of antiderivative = 6.55 \[ \int \frac {x^3}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/3*((6*b^(3/2)*c^4*d^4*log(abs(-b*c*d + sqrt(b*c^2 + a*d^2)*sqrt(b)*abs( 
d))) - 9*a*sqrt(b)*c^2*d^6*log(abs(-b*c*d + sqrt(b*c^2 + a*d^2)*sqrt(b)*ab 
s(d))) + 11*sqrt(b*c^2 + a*d^2)*b*c^3*d^3*abs(d) - 4*sqrt(b*c^2 + a*d^2)*a 
*c*d^5*abs(d))*sgn(1/(d*x + c))*sgn(d)/(sqrt(b*c^2 + a*d^2)*b^(7/2)*c^6*ab 
s(d) + 3*sqrt(b*c^2 + a*d^2)*a*b^(5/2)*c^4*d^2*abs(d) + 3*sqrt(b*c^2 + a*d 
^2)*a^2*b^(3/2)*c^2*d^4*abs(d) + sqrt(b*c^2 + a*d^2)*a^3*sqrt(b)*d^6*abs(d 
)) - ((11*a^2*b^3*c^3*d^14*sgn(1/(d*x + c))^3*sgn(d)^3 - 4*a^3*b^2*c*d^16* 
sgn(1/(d*x + c))^3*sgn(d)^3)/(a^2*b^4*c^6*d^11*sgn(1/(d*x + c))^4*sgn(d)^4 
 + 3*a^3*b^3*c^4*d^13*sgn(1/(d*x + c))^4*sgn(d)^4 + 3*a^4*b^2*c^2*d^15*sgn 
(1/(d*x + c))^4*sgn(d)^4 + a^5*b*d^17*sgn(1/(d*x + c))^4*sgn(d)^4) - (3*(1 
3*a^2*b^3*c^4*d^15*sgn(1/(d*x + c))^3*sgn(d)^3 - 7*a^3*b^2*c^2*d^17*sgn(1/ 
(d*x + c))^3*sgn(d)^3)/(a^2*b^4*c^6*d^11*sgn(1/(d*x + c))^4*sgn(d)^4 + 3*a 
^3*b^3*c^4*d^13*sgn(1/(d*x + c))^4*sgn(d)^4 + 3*a^4*b^2*c^2*d^15*sgn(1/(d* 
x + c))^4*sgn(d)^4 + a^5*b*d^17*sgn(1/(d*x + c))^4*sgn(d)^4) - (6*(8*a^2*b 
^3*c^5*d^16*sgn(1/(d*x + c))^3*sgn(d)^3 - 3*a^3*b^2*c^3*d^18*sgn(1/(d*x + 
c))^3*sgn(d)^3 - a^4*b*c*d^20*sgn(1/(d*x + c))^3*sgn(d)^3)/(a^2*b^4*c^6*d^ 
11*sgn(1/(d*x + c))^4*sgn(d)^4 + 3*a^3*b^3*c^4*d^13*sgn(1/(d*x + c))^4*sgn 
(d)^4 + 3*a^4*b^2*c^2*d^15*sgn(1/(d*x + c))^4*sgn(d)^4 + a^5*b*d^17*sgn(1/ 
(d*x + c))^4*sgn(d)^4) - ((23*a^2*b^3*c^6*d^17*sgn(1/(d*x + c))^3*sgn(d)^3 
 + 7*a^3*b^2*c^4*d^19*sgn(1/(d*x + c))^3*sgn(d)^3 - 15*a^4*b*c^2*d^21*s...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1880, normalized size of antiderivative = 8.51 \[ \int \frac {x^3}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(9*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**3*b*c**3*d**3 + 9*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**3*b*c**2*d**4*x - 6*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**2 
*c**5*d - 6*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**2*c**4*d**2*x + 18*sqrt(a*d**2 + b*c**2)*log(sqr 
t(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**2*c**3*d**3*x** 
2 + 18*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**2*c**2*d**4*x**3 - 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*b**3*c**5*d*x**2 - 12*s 
qrt(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**3*c**4*d**2*x**3 + 9*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**3*c**3*d**3*x**4 + 9*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** 
3*c**2*d**4*x**5 - 6*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**4*c**5*d*x**4 - 6*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**4*c**4*d**2*x**5 
- 9*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**3*b*c**3*d**3 - 9*sqrt(a*d**2 + 
b*c**2)*log(c + d*x)*a**3*b*c**2*d**4*x + 6*sqrt(a*d**2 + b*c**2)*log(c + 
d*x)*a**2*b**2*c**5*d + 6*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b**2*...