\(\int \frac {x^3}{(c+d x)^3 (a+b x^2)^{5/2}} \, dx\) [1300]

Optimal result

Integrand size = 22, antiderivative size = 330 \[ \int \frac {x^3}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {a \left (c \left (b c^2-3 a d^2\right )-d \left (3 b c^2-a d^2\right ) x\right )}{3 \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^{3/2}}-\frac {3 c \left (b^2 c^4-8 a b c^2 d^2+3 a^2 d^4\right )-2 d \left (6 b^2 c^4-11 a b c^2 d^2+a^2 d^4\right ) x}{3 \left (b c^2+a d^2\right )^4 \sqrt {a+b x^2}}+\frac {c^3 d^2 \sqrt {a+b x^2}}{2 \left (b c^2+a d^2\right )^3 (c+d x)^2}+\frac {c^2 d^2 \left (5 b c^2-6 a d^2\right ) \sqrt {a+b x^2}}{2 \left (b c^2+a d^2\right )^4 (c+d x)}+\frac {c d \left (6 b^2 c^4-23 a b c^2 d^2+6 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 \left (b c^2+a d^2\right )^{9/2}} \] Output:

1/3*a*(c*(-3*a*d^2+b*c^2)-d*(-a*d^2+3*b*c^2)*x)/(a*d^2+b*c^2)^3/(b*x^2+a)^ 
(3/2)-1/3*(3*c*(3*a^2*d^4-8*a*b*c^2*d^2+b^2*c^4)-2*d*(a^2*d^4-11*a*b*c^2*d 
^2+6*b^2*c^4)*x)/(a*d^2+b*c^2)^4/(b*x^2+a)^(1/2)+1/2*c^3*d^2*(b*x^2+a)^(1/ 
2)/(a*d^2+b*c^2)^3/(d*x+c)^2+1/2*c^2*d^2*(-6*a*d^2+5*b*c^2)*(b*x^2+a)^(1/2 
)/(a*d^2+b*c^2)^4/(d*x+c)+1/2*c*d*(6*a^2*d^4-23*a*b*c^2*d^2+6*b^2*c^4)*arc 
tanh((-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)^(9/2)
 

Mathematica [A] (verified)

Time = 10.93 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.99 \[ \int \frac {x^3}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {1}{6} \left (\frac {\sqrt {a+b x^2} \left (\frac {3 c^3 d^2 \left (b c^2+a d^2\right )}{(c+d x)^2}+\frac {3 c^2 d^2 \left (5 b c^2-6 a d^2\right )}{c+d x}+\frac {2 a \left (b c^2+a d^2\right ) \left (b c^2 (c-3 d x)+a d^2 (-3 c+d x)\right )}{\left (a+b x^2\right )^2}+\frac {4 a b c^2 d^2 (12 c-11 d x)-6 b^2 c^4 (c-4 d x)+2 a^2 d^4 (-9 c+2 d x)}{a+b x^2}\right )}{\left (b c^2+a d^2\right )^4}-\frac {3 c d \left (6 b^2 c^4-23 a b c^2 d^2+6 a^2 d^4\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{9/2}}+\frac {3 c d \left (6 b^2 c^4-23 a b c^2 d^2+6 a^2 d^4\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{9/2}}\right ) \] Input:

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

Output:

((Sqrt[a + b*x^2]*((3*c^3*d^2*(b*c^2 + a*d^2))/(c + d*x)^2 + (3*c^2*d^2*(5 
*b*c^2 - 6*a*d^2))/(c + d*x) + (2*a*(b*c^2 + a*d^2)*(b*c^2*(c - 3*d*x) + a 
*d^2*(-3*c + d*x)))/(a + b*x^2)^2 + (4*a*b*c^2*d^2*(12*c - 11*d*x) - 6*b^2 
*c^4*(c - 4*d*x) + 2*a^2*d^4*(-9*c + 2*d*x))/(a + b*x^2)))/(b*c^2 + a*d^2) 
^4 - (3*c*d*(6*b^2*c^4 - 23*a*b*c^2*d^2 + 6*a^2*d^4)*Log[c + d*x])/(b*c^2 
+ a*d^2)^(9/2) + (3*c*d*(6*b^2*c^4 - 23*a*b*c^2*d^2 + 6*a^2*d^4)*Log[a*d - 
 b*c*x + Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2]])/(b*c^2 + a*d^2)^(9/2))/6
 

Rubi [A] (verified)

Time = 2.76 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {601, 25, 2178, 27, 2182, 25, 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.

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

\(\Big \downarrow \) 601

\(\displaystyle \frac {a \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}-\frac {\int -\frac {-\frac {2 a^2 \left (3 b c^2-a d^2\right ) x^3 d^4}{\left (b c^2+a d^2\right )^3}-\frac {3 a^2 c \left (5 b c^2+a d^2\right ) x^2 d^3}{\left (b c^2+a d^2\right )^3}+\frac {a^2 c^3 \left (3 b c^2-a d^2\right ) d}{\left (b c^2+a d^2\right )^3}+\frac {3 a c^2 \left (b c^2-a d^2\right ) x}{\left (b c^2+a d^2\right )^2}}{(c+d x)^3 \left (b x^2+a\right )^{3/2}}dx}{3 a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {-\frac {2 a^2 \left (3 b c^2-a d^2\right ) x^3 d^4}{\left (b c^2+a d^2\right )^3}-\frac {3 a^2 c \left (5 b c^2+a d^2\right ) x^2 d^3}{\left (b c^2+a d^2\right )^3}+\frac {a^2 c^3 \left (3 b c^2-a d^2\right ) d}{\left (b c^2+a d^2\right )^3}+\frac {3 a c^2 \left (b c^2-a d^2\right ) x}{\left (b c^2+a d^2\right )^2}}{(c+d x)^3 \left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {a \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2178

\(\displaystyle \frac {-\frac {\int \frac {3 \left (\frac {a^2 b d \left (3 b^2 c^4-8 a b d^2 c^2+a^2 d^4\right ) c^3}{\left (b c^2+a d^2\right )^4}+\frac {3 a^2 b d^2 \left (b^2 c^4-6 a b d^2 c^2+a^2 d^4\right ) x c^2}{\left (b c^2+a d^2\right )^4}+\frac {a^2 b d^3 \left (b^2 c^4-8 a b d^2 c^2+3 a^2 d^4\right ) x^2 c}{\left (b c^2+a d^2\right )^4}\right )}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a b}-\frac {a \left (3 c \left (3 a^2 d^4-8 a b c^2 d^2+b^2 c^4\right )-2 d x \left (a^2 d^4-11 a b c^2 d^2+6 b^2 c^4\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}+\frac {a \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \int \frac {\frac {a^2 b d \left (3 b^2 c^4-8 a b d^2 c^2+a^2 d^4\right ) c^3}{\left (b c^2+a d^2\right )^4}+\frac {3 a^2 b d^2 \left (b^2 c^4-6 a b d^2 c^2+a^2 d^4\right ) x c^2}{\left (b c^2+a d^2\right )^4}+\frac {a^2 b d^3 \left (b^2 c^4-8 a b d^2 c^2+3 a^2 d^4\right ) x^2 c}{\left (b c^2+a d^2\right )^4}}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a b}-\frac {a \left (3 c \left (3 a^2 d^4-8 a b c^2 d^2+b^2 c^4\right )-2 d x \left (a^2 d^4-11 a b c^2 d^2+6 b^2 c^4\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}+\frac {a \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {-\frac {3 \left (-\frac {\int -\frac {a^2 b c d \left (6 b \left (b c^2-3 a d^2\right ) c^3+d \left (b^2 c^4-17 a b d^2 c^2+6 a^2 d^4\right ) x\right )}{\left (b c^2+a d^2\right )^3 (c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {a^2 b c^3 d^2 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a \left (3 c \left (3 a^2 d^4-8 a b c^2 d^2+b^2 c^4\right )-2 d x \left (a^2 d^4-11 a b c^2 d^2+6 b^2 c^4\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}+\frac {a \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {3 \left (\frac {\int \frac {a^2 b c d \left (6 b \left (b c^2-3 a d^2\right ) c^3+d \left (b^2 c^4-17 a b d^2 c^2+6 a^2 d^4\right ) x\right )}{\left (b c^2+a d^2\right )^3 (c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {a^2 b c^3 d^2 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a \left (3 c \left (3 a^2 d^4-8 a b c^2 d^2+b^2 c^4\right )-2 d x \left (a^2 d^4-11 a b c^2 d^2+6 b^2 c^4\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}+\frac {a \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \left (\frac {a^2 b c d \int \frac {6 b \left (b c^2-3 a d^2\right ) c^3+d \left (b^2 c^4-17 a b d^2 c^2+6 a^2 d^4\right ) x}{(c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )^4}-\frac {a^2 b c^3 d^2 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a \left (3 c \left (3 a^2 d^4-8 a b c^2 d^2+b^2 c^4\right )-2 d x \left (a^2 d^4-11 a b c^2 d^2+6 b^2 c^4\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}+\frac {a \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {-\frac {3 \left (\frac {a^2 b c d \left (\left (6 a^2 d^4-23 a b c^2 d^2+6 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx-\frac {c d \sqrt {a+b x^2} \left (5 b c^2-6 a d^2\right )}{c+d x}\right )}{2 \left (a d^2+b c^2\right )^4}-\frac {a^2 b c^3 d^2 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a \left (3 c \left (3 a^2 d^4-8 a b c^2 d^2+b^2 c^4\right )-2 d x \left (a^2 d^4-11 a b c^2 d^2+6 b^2 c^4\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}+\frac {a \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {-\frac {3 \left (\frac {a^2 b c d \left (-\left (6 a^2 d^4-23 a b c^2 d^2+6 b^2 c^4\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}-\frac {c d \sqrt {a+b x^2} \left (5 b c^2-6 a d^2\right )}{c+d x}\right )}{2 \left (a d^2+b c^2\right )^4}-\frac {a^2 b c^3 d^2 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a \left (3 c \left (3 a^2 d^4-8 a b c^2 d^2+b^2 c^4\right )-2 d x \left (a^2 d^4-11 a b c^2 d^2+6 b^2 c^4\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}+\frac {a \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {3 \left (\frac {a^2 b c d \left (-\frac {\left (6 a^2 d^4-23 a b c^2 d^2+6 b^2 c^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a d^2+b c^2}}-\frac {c d \sqrt {a+b x^2} \left (5 b c^2-6 a d^2\right )}{c+d x}\right )}{2 \left (a d^2+b c^2\right )^4}-\frac {a^2 b c^3 d^2 \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}-\frac {a \left (3 c \left (3 a^2 d^4-8 a b c^2 d^2+b^2 c^4\right )-2 d x \left (a^2 d^4-11 a b c^2 d^2+6 b^2 c^4\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}}{3 a}+\frac {a \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}\)

Input:

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

Output:

(a*(c*(b*c^2 - 3*a*d^2) - d*(3*b*c^2 - a*d^2)*x))/(3*(b*c^2 + a*d^2)^3*(a 
+ b*x^2)^(3/2)) + (-((a*(3*c*(b^2*c^4 - 8*a*b*c^2*d^2 + 3*a^2*d^4) - 2*d*( 
6*b^2*c^4 - 11*a*b*c^2*d^2 + a^2*d^4)*x))/((b*c^2 + a*d^2)^4*Sqrt[a + b*x^ 
2])) - (3*(-1/2*(a^2*b*c^3*d^2*Sqrt[a + b*x^2])/((b*c^2 + a*d^2)^3*(c + d* 
x)^2) + (a^2*b*c*d*(-((c*d*(5*b*c^2 - 6*a*d^2)*Sqrt[a + b*x^2])/(c + d*x)) 
 - ((6*b^2*c^4 - 23*a*b*c^2*d^2 + 6*a^2*d^4)*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]))/(2*(b*c^2 + a*d^2)^ 
4)))/(a*b))/(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]
 

rule 2182

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3146\) vs. \(2(306)=612\).

Time = 0.42 (sec) , antiderivative size = 3147, normalized size of antiderivative = 9.54

method	result	size
default	\(\text {Expression too large to display}\)	\(3147\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1298 vs. \(2 (307) = 614\).

Time = 1.29 (sec) , antiderivative size = 2622, normalized size of antiderivative = 7.95 \[ \int \frac {x^3}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/12*(3*(6*a^2*b^2*c^7*d - 23*a^3*b*c^5*d^3 + 6*a^4*c^3*d^5 + (6*b^4*c^5* 
d^3 - 23*a*b^3*c^3*d^5 + 6*a^2*b^2*c*d^7)*x^6 + 2*(6*b^4*c^6*d^2 - 23*a*b^ 
3*c^4*d^4 + 6*a^2*b^2*c^2*d^6)*x^5 + (6*b^4*c^7*d - 11*a*b^3*c^5*d^3 - 40* 
a^2*b^2*c^3*d^5 + 12*a^3*b*c*d^7)*x^4 + 4*(6*a*b^3*c^6*d^2 - 23*a^2*b^2*c^ 
4*d^4 + 6*a^3*b*c^2*d^6)*x^3 + (12*a*b^3*c^7*d - 40*a^2*b^2*c^5*d^3 - 11*a 
^3*b*c^3*d^5 + 6*a^4*c*d^7)*x^2 + 2*(6*a^2*b^2*c^6*d^2 - 23*a^3*b*c^4*d^4 
+ 6*a^4*c^2*d^6)*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)*sqr 
t(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) - 2*(4*a*b^3*c^9 - 58*a^2*b^2*c^7 
*d^2 - 23*a^3*b*c^5*d^4 + 39*a^4*c^3*d^6 - (39*b^4*c^6*d^3 - 23*a*b^3*c^4* 
d^5 - 58*a^2*b^2*c^2*d^7 + 4*a^3*b*d^9)*x^5 - 5*(12*b^4*c^7*d^2 + a*b^3*c^ 
5*d^4 - 13*a^2*b^2*c^3*d^6 - 2*a^3*b*c*d^8)*x^4 - 2*(6*b^4*c^8*d + 56*a*b^ 
3*c^6*d^3 - 8*a^2*b^2*c^4*d^5 - 55*a^3*b*c^2*d^7 + 3*a^4*d^9)*x^3 + 2*(3*b 
^4*c^9 - 55*a*b^3*c^7*d^2 - 8*a^2*b^2*c^5*d^4 + 56*a^3*b*c^3*d^6 + 6*a^4*c 
*d^8)*x^2 - 5*(2*a*b^3*c^8*d + 13*a^2*b^2*c^6*d^3 - a^3*b*c^4*d^5 - 12*a^4 
*c^2*d^7)*x)*sqrt(b*x^2 + a))/(a^2*b^5*c^12 + 5*a^3*b^4*c^10*d^2 + 10*a^4* 
b^3*c^8*d^4 + 10*a^5*b^2*c^6*d^6 + 5*a^6*b*c^4*d^8 + a^7*c^2*d^10 + (b^7*c 
^10*d^2 + 5*a*b^6*c^8*d^4 + 10*a^2*b^5*c^6*d^6 + 10*a^3*b^4*c^4*d^8 + 5*a^ 
4*b^3*c^2*d^10 + a^5*b^2*d^12)*x^6 + 2*(b^7*c^11*d + 5*a*b^6*c^9*d^3 + 10* 
a^2*b^5*c^7*d^5 + 10*a^3*b^4*c^5*d^7 + 5*a^4*b^3*c^3*d^9 + a^5*b^2*c*d^...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1636 vs. \(2 (307) = 614\).

Time = 0.17 (sec) , antiderivative size = 1636, normalized size of antiderivative = 4.96 \[ \int \frac {x^3}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-35/2*b^3*c^6*x/(sqrt(b*x^2 + a)*a*b^4*c^8*d + 4*sqrt(b*x^2 + a)*a^2*b^3*c 
^6*d^3 + 6*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^5 + 4*sqrt(b*x^2 + a)*a^4*b*c^2*d 
^7 + sqrt(b*x^2 + a)*a^5*d^9) - 35/6*b^3*c^6*x/((b*x^2 + a)^(3/2)*a*b^3*c^ 
6*d^3 + 3*(b*x^2 + a)^(3/2)*a^2*b^2*c^4*d^5 + 3*(b*x^2 + a)^(3/2)*a^3*b*c^ 
2*d^7 + (b*x^2 + a)^(3/2)*a^4*d^9) - 35/3*b^3*c^6*x/(sqrt(b*x^2 + a)*a^2*b 
^3*c^6*d^3 + 3*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^5 + 3*sqrt(b*x^2 + a)*a^4*b*c 
^2*d^7 + sqrt(b*x^2 + a)*a^5*d^9) - 35/2*b^2*c^5/(sqrt(b*x^2 + a)*b^4*c^8 
+ 4*sqrt(b*x^2 + a)*a*b^3*c^6*d^2 + 6*sqrt(b*x^2 + a)*a^2*b^2*c^4*d^4 + 4* 
sqrt(b*x^2 + a)*a^3*b*c^2*d^6 + sqrt(b*x^2 + a)*a^4*d^8) - 35/6*b^2*c^5/(( 
b*x^2 + a)^(3/2)*b^3*c^6*d^2 + 3*(b*x^2 + a)^(3/2)*a*b^2*c^4*d^4 + 3*(b*x^ 
2 + a)^(3/2)*a^2*b*c^2*d^6 + (b*x^2 + a)^(3/2)*a^3*d^8) + 35/2*b^2*c^4*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) + 21/2*b^2*c^4*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) + 21*b^2*c^4*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) + 7/2*b*c^4/((b*x^2 + 
 a)^(3/2)*b^2*c^4*d^3*x + 2*(b*x^2 + a)^(3/2)*a*b*c^2*d^5*x + (b*x^2 + a)^ 
(3/2)*a^2*d^7*x + (b*x^2 + a)^(3/2)*b^2*c^5*d^2 + 2*(b*x^2 + a)^(3/2)*a*b* 
c^3*d^4 + (b*x^2 + a)^(3/2)*a^2*c*d^6) + 35/2*b*c^3/(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 ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2241 vs. \(2 (307) = 614\).

Time = 0.28 (sec) , antiderivative size = 2241, normalized size of antiderivative = 6.79 \[ \int \frac {x^3}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

(6*b^2*c^5*d - 23*a*b*c^3*d^3 + 6*a^2*c*d^5)*arctan(((sqrt(b)*x - sqrt(b*x 
^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b^4*c^8 + 4*a*b^3*c^6*d^2 
+ 6*a^2*b^2*c^4*d^4 + 4*a^3*b*c^2*d^6 + a^4*d^8)*sqrt(-b*c^2 - a*d^2)) + 1 
/3*(((2*(6*a*b^16*c^28*d + 61*a^2*b^15*c^26*d^3 + 265*a^3*b^14*c^24*d^5 + 
606*a^4*b^13*c^22*d^7 + 616*a^5*b^12*c^20*d^9 - 473*a^6*b^11*c^18*d^11 - 2 
673*a^7*b^10*c^16*d^13 - 4620*a^8*b^9*c^14*d^15 - 4818*a^9*b^8*c^12*d^17 - 
 3333*a^10*b^7*c^10*d^19 - 1529*a^11*b^6*c^8*d^21 - 434*a^12*b^5*c^6*d^23 
- 60*a^13*b^4*c^4*d^25 + a^14*b^3*c^2*d^27 + a^15*b^2*d^29)*x/(a*b^17*c^32 
 + 16*a^2*b^16*c^30*d^2 + 120*a^3*b^15*c^28*d^4 + 560*a^4*b^14*c^26*d^6 + 
1820*a^5*b^13*c^24*d^8 + 4368*a^6*b^12*c^22*d^10 + 8008*a^7*b^11*c^20*d^12 
 + 11440*a^8*b^10*c^18*d^14 + 12870*a^9*b^9*c^16*d^16 + 11440*a^10*b^8*c^1 
4*d^18 + 8008*a^11*b^7*c^12*d^20 + 4368*a^12*b^6*c^10*d^22 + 1820*a^13*b^5 
*c^8*d^24 + 560*a^14*b^4*c^6*d^26 + 120*a^15*b^3*c^4*d^28 + 16*a^16*b^2*c^ 
2*d^30 + a^17*b*d^32) - 3*(a*b^16*c^29 + 4*a^2*b^15*c^27*d^2 - 27*a^3*b^14 
*c^25*d^4 - 272*a^4*b^13*c^23*d^6 - 1067*a^5*b^12*c^21*d^8 - 2508*a^6*b^11 
*c^19*d^10 - 3927*a^7*b^10*c^17*d^12 - 4224*a^8*b^9*c^15*d^14 - 3069*a^9*b 
^8*c^13*d^16 - 1364*a^10*b^7*c^11*d^18 - 209*a^11*b^6*c^9*d^20 + 144*a^12* 
b^5*c^7*d^22 + 103*a^13*b^4*c^5*d^24 + 28*a^14*b^3*c^3*d^26 + 3*a^15*b^2*c 
*d^28)/(a*b^17*c^32 + 16*a^2*b^16*c^30*d^2 + 120*a^3*b^15*c^28*d^4 + 560*a 
^4*b^14*c^26*d^6 + 1820*a^5*b^13*c^24*d^8 + 4368*a^6*b^12*c^22*d^10 + 8...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 3295, normalized size of antiderivative = 9.98 \[ \int \frac {x^3}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(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**4*c**3*d**5 + 36*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**4*c**2*d**6*x + 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 
**4*c*d**7*x**2 - 69*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**5*d**3 - 138*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**4* 
d**4*x - 33*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**5*x**2 + 72*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**2*d**6* 
x**3 + 36*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*d**7*x**4 + 18*sqrt(a*d**2 + b*c**2)*log( - sq 
rt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**2*c**7*d + 36* 
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**6*d**2*x - 120*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**3*x**2 - 2 
76*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**4*x**3 - 120*sqrt(a*d**2 + b*c**2)*log( - sq 
rt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**2*c**3*d**5*x* 
*4 + 36*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c...