\(\int \frac {x^4}{(c+d x)^3 (a+b x^2)^{5/2}} \, dx\) [1299]

Optimal result

Integrand size = 22, antiderivative size = 319 \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {a \left (a d \left (3 b c^2-a d^2\right )+b c \left (b c^2-3 a d^2\right ) x\right )}{3 b \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^{3/2}}-\frac {2 c \left (9 a c d \left (b c^2-a d^2\right )+\left (2 b^2 c^4-13 a b c^2 d^2+3 a^2 d^4\right ) x\right )}{3 \left (b c^2+a d^2\right )^4 \sqrt {a+b x^2}}-\frac {c^4 d \sqrt {a+b x^2}}{2 \left (b c^2+a d^2\right )^3 (c+d x)^2}-\frac {c^3 d \left (3 b c^2-8 a d^2\right ) \sqrt {a+b x^2}}{2 \left (b c^2+a d^2\right )^4 (c+d x)}-\frac {c^2 \left (2 b^2 c^4-21 a b c^2 d^2+12 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 \left (b c^2+a d^2\right )^{9/2}} \] Output:

1/3*a*(a*d*(-a*d^2+3*b*c^2)+b*c*(-3*a*d^2+b*c^2)*x)/b/(a*d^2+b*c^2)^3/(b*x 
^2+a)^(3/2)-2/3*c*(9*a*c*d*(-a*d^2+b*c^2)+(3*a^2*d^4-13*a*b*c^2*d^2+2*b^2* 
c^4)*x)/(a*d^2+b*c^2)^4/(b*x^2+a)^(1/2)-1/2*c^4*d*(b*x^2+a)^(1/2)/(a*d^2+b 
*c^2)^3/(d*x+c)^2-1/2*c^3*d*(-8*a*d^2+3*b*c^2)*(b*x^2+a)^(1/2)/(a*d^2+b*c^ 
2)^4/(d*x+c)-1/2*c^2*(12*a^2*d^4-21*a*b*c^2*d^2+2*b^2*c^4)*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)^(9/2)
 

Mathematica [A] (verified)

Time = 10.80 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.02 \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {1}{6} \left (\frac {\sqrt {a+b x^2} \left (-\frac {3 c^4 d \left (b c^2+a d^2\right )}{(c+d x)^2}+\frac {3 \left (-3 b c^5 d+8 a c^3 d^3\right )}{c+d x}+\frac {2 a \left (b c^2+a d^2\right ) \left (-a^2 d^3+b^2 c^3 x+3 a b c d (c-d x)\right )}{b \left (a+b x^2\right )^2}+\frac {4 \left (-2 b^2 c^5 x+3 a^2 c d^3 (3 c-d x)+a b c^3 d (-9 c+13 d x)\right )}{a+b x^2}\right )}{\left (b c^2+a d^2\right )^4}+\frac {3 c^2 \left (2 b^2 c^4-21 a b c^2 d^2+12 a^2 d^4\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{9/2}}-\frac {3 c^2 \left (2 b^2 c^4-21 a b c^2 d^2+12 a^2 d^4\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{9/2}}\right ) \] Input:

Integrate[x^4/((c + d*x)^3*(a + b*x^2)^(5/2)),x]
 

Output:

((Sqrt[a + b*x^2]*((-3*c^4*d*(b*c^2 + a*d^2))/(c + d*x)^2 + (3*(-3*b*c^5*d 
 + 8*a*c^3*d^3))/(c + d*x) + (2*a*(b*c^2 + a*d^2)*(-(a^2*d^3) + b^2*c^3*x 
+ 3*a*b*c*d*(c - d*x)))/(b*(a + b*x^2)^2) + (4*(-2*b^2*c^5*x + 3*a^2*c*d^3 
*(3*c - d*x) + a*b*c^3*d*(-9*c + 13*d*x)))/(a + b*x^2)))/(b*c^2 + a*d^2)^4 
 + (3*c^2*(2*b^2*c^4 - 21*a*b*c^2*d^2 + 12*a^2*d^4)*Log[c + d*x])/(b*c^2 + 
 a*d^2)^(9/2) - (3*c^2*(2*b^2*c^4 - 21*a*b*c^2*d^2 + 12*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 = 2.85 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {601, 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.

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

\(\Big \downarrow \) 601

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

\(\Big \downarrow \) 2178

\(\displaystyle \frac {a \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}-\frac {\frac {2 a c \left (x \left (3 a^2 d^4-13 a b c^2 d^2+2 b^2 c^4\right )+9 a c d \left (b c^2-a d^2\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {\int \frac {3 \left (\frac {a^2 b \left (b^2 c^4-8 a b d^2 c^2+3 a^2 d^4\right ) c^4}{\left (b c^2+a d^2\right )^4}-\frac {8 a^3 b d^3 \left (2 b c^2-a d^2\right ) x c^3}{\left (b c^2+a d^2\right )^4}-\frac {6 a^3 b d^4 \left (b c^2-a d^2\right ) x^2 c^2}{\left (b c^2+a d^2\right )^4}\right )}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a b}}{3 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}-\frac {\frac {2 a c \left (x \left (3 a^2 d^4-13 a b c^2 d^2+2 b^2 c^4\right )+9 a c d \left (b c^2-a d^2\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \int \frac {\frac {a^2 b \left (b^2 c^4-8 a b d^2 c^2+3 a^2 d^4\right ) c^4}{\left (b c^2+a d^2\right )^4}-\frac {8 a^3 b d^3 \left (2 b c^2-a d^2\right ) x c^3}{\left (b c^2+a d^2\right )^4}-\frac {6 a^3 b d^4 \left (b c^2-a d^2\right ) x^2 c^2}{\left (b c^2+a d^2\right )^4}}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a b}}{3 a}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {a \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}-\frac {\frac {2 a c \left (x \left (3 a^2 d^4-13 a b c^2 d^2+2 b^2 c^4\right )+9 a c d \left (b c^2-a d^2\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \left (-\frac {\int -\frac {a^2 b c^2 \left (2 c \left (b^2 c^4-9 a b d^2 c^2+2 a^2 d^4\right )-d \left (b^2 c^4+13 a b d^2 c^2-12 a^2 d^4\right ) x\right )}{\left (b c^2+a d^2\right )^3 (c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {a^2 b c^4 d \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}}{3 a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {a \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}-\frac {\frac {2 a c \left (x \left (3 a^2 d^4-13 a b c^2 d^2+2 b^2 c^4\right )+9 a c d \left (b c^2-a d^2\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \left (\frac {\int \frac {a^2 b c^2 \left (2 c \left (b^2 c^4-9 a b d^2 c^2+2 a^2 d^4\right )-d \left (b^2 c^4+13 a b d^2 c^2-12 a^2 d^4\right ) x\right )}{\left (b c^2+a d^2\right )^3 (c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {a^2 b c^4 d \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}}{3 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}-\frac {\frac {2 a c \left (x \left (3 a^2 d^4-13 a b c^2 d^2+2 b^2 c^4\right )+9 a c d \left (b c^2-a d^2\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \left (\frac {a^2 b c^2 \int \frac {2 c \left (b^2 c^4-9 a b d^2 c^2+2 a^2 d^4\right )-d \left (b^2 c^4+13 a b d^2 c^2-12 a^2 d^4\right ) x}{(c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )^4}-\frac {a^2 b c^4 d \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}}{3 a}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {a \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}-\frac {\frac {2 a c \left (x \left (3 a^2 d^4-13 a b c^2 d^2+2 b^2 c^4\right )+9 a c d \left (b c^2-a d^2\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \left (\frac {a^2 b c^2 \left (\left (12 a^2 d^4-21 a b c^2 d^2+2 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx-\frac {c d \sqrt {a+b x^2} \left (3 b c^2-8 a d^2\right )}{c+d x}\right )}{2 \left (a d^2+b c^2\right )^4}-\frac {a^2 b c^4 d \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}}{3 a}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {a \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}-\frac {\frac {2 a c \left (x \left (3 a^2 d^4-13 a b c^2 d^2+2 b^2 c^4\right )+9 a c d \left (b c^2-a d^2\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \left (\frac {a^2 b c^2 \left (-\left (12 a^2 d^4-21 a b c^2 d^2+2 b^2 c^4\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}-\frac {c d \sqrt {a+b x^2} \left (3 b c^2-8 a d^2\right )}{c+d x}\right )}{2 \left (a d^2+b c^2\right )^4}-\frac {a^2 b c^4 d \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}}{3 a}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {a \left (b c x \left (b c^2-3 a d^2\right )+a d \left (3 b c^2-a d^2\right )\right )}{3 b \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^3}-\frac {\frac {2 a c \left (x \left (3 a^2 d^4-13 a b c^2 d^2+2 b^2 c^4\right )+9 a c d \left (b c^2-a d^2\right )\right )}{\sqrt {a+b x^2} \left (a d^2+b c^2\right )^4}-\frac {3 \left (\frac {a^2 b c^2 \left (-\frac {\left (12 a^2 d^4-21 a b c^2 d^2+2 b^2 c^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a d^2+b c^2}}-\frac {c d \sqrt {a+b x^2} \left (3 b c^2-8 a d^2\right )}{c+d x}\right )}{2 \left (a d^2+b c^2\right )^4}-\frac {a^2 b c^4 d \sqrt {a+b x^2}}{2 (c+d x)^2 \left (a d^2+b c^2\right )^3}\right )}{a b}}{3 a}\)

Input:

Int[x^4/((c + d*x)^3*(a + b*x^2)^(5/2)),x]
 

Output:

(a*(a*d*(3*b*c^2 - a*d^2) + b*c*(b*c^2 - 3*a*d^2)*x))/(3*b*(b*c^2 + a*d^2) 
^3*(a + b*x^2)^(3/2)) - ((2*a*c*(9*a*c*d*(b*c^2 - a*d^2) + (2*b^2*c^4 - 13 
*a*b*c^2*d^2 + 3*a^2*d^4)*x))/((b*c^2 + a*d^2)^4*Sqrt[a + b*x^2]) - (3*(-1 
/2*(a^2*b*c^4*d*Sqrt[a + b*x^2])/((b*c^2 + a*d^2)^3*(c + d*x)^2) + (a^2*b* 
c^2*(-((c*d*(3*b*c^2 - 8*a*d^2)*Sqrt[a + b*x^2])/(c + d*x)) - ((2*b^2*c^4 
- 21*a*b*c^2*d^2 + 12*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. \(3166\) vs. \(2(295)=590\).

Time = 0.42 (sec) , antiderivative size = 3167, normalized size of antiderivative = 9.93

method	result	size
default	\(\text {Expression too large to display}\)	\(3167\)

Input:

int(x^4/(d*x+c)^3/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1342 vs. \(2 (296) = 592\).

Time = 1.47 (sec) , antiderivative size = 2711, normalized size of antiderivative = 8.50 \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^4/(d*x+c)^3/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/12*(3*(2*a^2*b^3*c^8 - 21*a^3*b^2*c^6*d^2 + 12*a^4*b*c^4*d^4 + (2*b^5*c 
^6*d^2 - 21*a*b^4*c^4*d^4 + 12*a^2*b^3*c^2*d^6)*x^6 + 2*(2*b^5*c^7*d - 21* 
a*b^4*c^5*d^3 + 12*a^2*b^3*c^3*d^5)*x^5 + (2*b^5*c^8 - 17*a*b^4*c^6*d^2 - 
30*a^2*b^3*c^4*d^4 + 24*a^3*b^2*c^2*d^6)*x^4 + 4*(2*a*b^4*c^7*d - 21*a^2*b 
^3*c^5*d^3 + 12*a^3*b^2*c^3*d^5)*x^3 + (4*a*b^4*c^8 - 40*a^2*b^3*c^6*d^2 + 
 3*a^3*b^2*c^4*d^4 + 12*a^4*b*c^2*d^6)*x^2 + 2*(2*a^2*b^3*c^7*d - 21*a^3*b 
^2*c^5*d^3 + 12*a^4*b*c^3*d^5)*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*(42*a^2*b^3*c^8 
*d - 19*a^3*b^2*c^6*d^3 - 59*a^4*b*c^4*d^5 + 2*a^5*c^2*d^7 + (17*b^5*c^7*d 
^2 - 59*a*b^4*c^5*d^4 - 64*a^2*b^3*c^3*d^6 + 12*a^3*b^2*c*d^8)*x^5 + (28*b 
^5*c^8*d - 61*a*b^4*c^6*d^3 - 101*a^2*b^3*c^4*d^5 - 12*a^3*b^2*c^2*d^7)*x^ 
4 + 2*(4*b^5*c^9 + 26*a*b^4*c^7*d^2 - 56*a^2*b^3*c^5*d^4 - 69*a^3*b^2*c^3* 
d^6 + 9*a^4*b*c*d^8)*x^3 + 2*(36*a*b^4*c^8*d - 36*a^2*b^3*c^6*d^3 - 74*a^3 
*b^2*c^4*d^5 - a^4*b*c^2*d^7 + a^5*d^9)*x^2 + (6*a*b^4*c^9 + 27*a^2*b^3*c^ 
7*d^2 - 65*a^3*b^2*c^5*d^4 - 82*a^4*b*c^3*d^6 + 4*a^5*c*d^8)*x)*sqrt(b*x^2 
 + a))/(a^2*b^6*c^12 + 5*a^3*b^5*c^10*d^2 + 10*a^4*b^4*c^8*d^4 + 10*a^5*b^ 
3*c^6*d^6 + 5*a^6*b^2*c^4*d^8 + a^7*b*c^2*d^10 + (b^8*c^10*d^2 + 5*a*b^7*c 
^8*d^4 + 10*a^2*b^6*c^6*d^6 + 10*a^3*b^5*c^4*d^8 + 5*a^4*b^4*c^2*d^10 + a^ 
5*b^3*d^12)*x^6 + 2*(b^8*c^11*d + 5*a*b^7*c^9*d^3 + 10*a^2*b^6*c^7*d^5 ...
 

Sympy [F]

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

integrate(x**4/(d*x+c)**3/(b*x**2+a)**(5/2),x)
 

Output:

Integral(x**4/((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. 1670 vs. \(2 (296) = 592\).

Time = 0.18 (sec) , antiderivative size = 1670, normalized size of antiderivative = 5.24 \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^4/(d*x+c)^3/(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

35/2*b^3*c^7*x/(sqrt(b*x^2 + a)*a*b^4*c^8*d^2 + 4*sqrt(b*x^2 + a)*a^2*b^3* 
c^6*d^4 + 6*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^6 + 4*sqrt(b*x^2 + a)*a^4*b*c^2* 
d^8 + sqrt(b*x^2 + a)*a^5*d^10) + 35/6*b^3*c^7*x/((b*x^2 + a)^(3/2)*a*b^3* 
c^6*d^4 + 3*(b*x^2 + a)^(3/2)*a^2*b^2*c^4*d^6 + 3*(b*x^2 + a)^(3/2)*a^3*b* 
c^2*d^8 + (b*x^2 + a)^(3/2)*a^4*d^10) + 35/3*b^3*c^7*x/(sqrt(b*x^2 + a)*a^ 
2*b^3*c^6*d^4 + 3*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^6 + 3*sqrt(b*x^2 + a)*a^4* 
b*c^2*d^8 + sqrt(b*x^2 + a)*a^5*d^10) + 35/2*b^2*c^6/(sqrt(b*x^2 + a)*b^4* 
c^8*d + 4*sqrt(b*x^2 + a)*a*b^3*c^6*d^3 + 6*sqrt(b*x^2 + a)*a^2*b^2*c^4*d^ 
5 + 4*sqrt(b*x^2 + a)*a^3*b*c^2*d^7 + sqrt(b*x^2 + a)*a^4*d^9) + 35/6*b^2* 
c^6/((b*x^2 + a)^(3/2)*b^3*c^6*d^3 + 3*(b*x^2 + a)^(3/2)*a*b^2*c^4*d^5 + 3 
*(b*x^2 + a)^(3/2)*a^2*b*c^2*d^7 + (b*x^2 + a)^(3/2)*a^3*d^9) - 45/2*b^2*c 
^5*x/(sqrt(b*x^2 + a)*a*b^3*c^6*d^2 + 3*sqrt(b*x^2 + a)*a^2*b^2*c^4*d^4 + 
3*sqrt(b*x^2 + a)*a^3*b*c^2*d^6 + sqrt(b*x^2 + a)*a^4*d^8) - 73/6*b^2*c^5* 
x/((b*x^2 + a)^(3/2)*a*b^2*c^4*d^4 + 2*(b*x^2 + a)^(3/2)*a^2*b*c^2*d^6 + ( 
b*x^2 + a)^(3/2)*a^3*d^8) - 73/3*b^2*c^5*x/(sqrt(b*x^2 + a)*a^2*b^2*c^4*d^ 
4 + 2*sqrt(b*x^2 + a)*a^3*b*c^2*d^6 + sqrt(b*x^2 + a)*a^4*d^8) - 7/2*b*c^5 
/((b*x^2 + a)^(3/2)*b^2*c^4*d^4*x + 2*(b*x^2 + a)^(3/2)*a*b*c^2*d^6*x + (b 
*x^2 + a)^(3/2)*a^2*d^8*x + (b*x^2 + a)^(3/2)*b^2*c^5*d^3 + 2*(b*x^2 + a)^ 
(3/2)*a*b*c^3*d^5 + (b*x^2 + a)^(3/2)*a^2*c*d^7) - 45/2*b*c^4/(sqrt(b*x^2 
+ a)*b^3*c^6*d + 3*sqrt(b*x^2 + a)*a*b^2*c^4*d^3 + 3*sqrt(b*x^2 + a)*a^...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2229 vs. \(2 (296) = 592\).

Time = 0.29 (sec) , antiderivative size = 2229, normalized size of antiderivative = 6.99 \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^4/(d*x+c)^3/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

-(2*b^2*c^6 - 21*a*b*c^4*d^2 + 12*a^2*c^2*d^4)*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*((2*a*b^16*c^29 + 11*a^2*b^15*c^27*d^2 - 21*a^3*b^14*c^25*d^4 - 3 
82*a^4*b^13*c^23*d^6 - 1672*a^5*b^12*c^21*d^8 - 4191*a^6*b^11*c^19*d^10 - 
6963*a^7*b^10*c^17*d^12 - 8052*a^8*b^9*c^15*d^14 - 6534*a^9*b^8*c^13*d^16 
- 3619*a^10*b^7*c^11*d^18 - 1243*a^11*b^6*c^9*d^20 - 174*a^12*b^5*c^7*d^22 
 + 44*a^13*b^4*c^5*d^24 + 23*a^14*b^3*c^3*d^26 + 3*a^15*b^2*c*d^28)*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^2 
6*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^14*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^1 
6*b^2*c^2*d^30 + a^17*b*d^32) + 9*(a^2*b^15*c^28*d + 11*a^3*b^14*c^26*d^3 
+ 54*a^4*b^13*c^24*d^5 + 154*a^5*b^12*c^22*d^7 + 275*a^6*b^11*c^20*d^9 + 2 
97*a^7*b^10*c^18*d^11 + 132*a^8*b^9*c^16*d^13 - 132*a^9*b^8*c^14*d^15 - 29 
7*a^10*b^7*c^12*d^17 - 275*a^11*b^6*c^10*d^19 - 154*a^12*b^5*c^8*d^21 - 54 
*a^13*b^4*c^6*d^23 - 11*a^14*b^3*c^4*d^25 - a^15*b^2*c^2*d^27)/(a*b^17*c^3 
2 + 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*...
 

Mupad [F(-1)]

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

int(x^4/((a + b*x^2)^(5/2)*(c + d*x)^3),x)
 

Output:

int(x^4/((a + b*x^2)^(5/2)*(c + d*x)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 3324, normalized size of antiderivative = 10.42 \[ \int \frac {x^4}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(x^4/(d*x+c)^3/(b*x^2+a)^(5/2),x)
 

Output:

(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*b*c**4*d**4 + 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**4*b*c**3*d**5*x + 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*b 
*c**2*d**6*x**2 - 63*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**2*c**6*d**2 - 126*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**2*c**5* 
d**3*x + 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**2*c**4*d**4*x**2 + 144*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**2*c**3*d**5* 
x**3 + 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**2*c**2*d**6*x**4 + 6*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**3*c**8 + 12*sqr 
t(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**3*c**7*d*x - 120*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqr 
t(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**3*c**6*d**2*x**2 - 252*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**3*c**5*d**3*x**3 - 90*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**3*c**4*d**4*x**4 + 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*...