\(\int \frac {x^4}{(c+d x)^2 (a+b x^2)^{3/2}} \, dx\) [1239]

Optimal result

Integrand size = 22, antiderivative size = 188 \[ \int \frac {x^4}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {a \left (2 a c d+\left (b c^2-a d^2\right ) x\right )}{b \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}-\frac {c^4 \sqrt {a+b x^2}}{d \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2} d^2}+\frac {c^3 \left (b c^2+4 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^2 \left (b c^2+a d^2\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.16 \[ \int \frac {x^4}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {d \left (-b^2 c^4 x^2+a^2 d^2 \left (2 c^2+c d x-d^2 x^2\right )+a b c^2 \left (-c^2+c d x+d^2 x^2\right )\right )}{b \left (b c^2+a d^2\right )^2 (c+d x) \sqrt {a+b x^2}}+\frac {2 c^3 \left (b c^2+4 a d^2\right ) \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{3/2}}}{d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {601, 2182, 25, 27, 719, 224, 219, 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.

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

\(\Big \downarrow \) 601

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(a*(2*a*c*d + (b*c^2 - a*d^2)*x))/(b*(b*c^2 + a*d^2)^2*Sqrt[a + b*x^2]) - 
((a*c^4*Sqrt[a + b*x^2])/(d*(b*c^2 + a*d^2)^2*(c + d*x)) + (a*(-(((b*c^2 + 
 a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - (b*c^3*(b*c^2 
 + 4*a*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/ 
(d*(b*c^2 + a*d^2)^(3/2))))/(b*d*(b*c^2 + a*d^2)))/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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 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 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[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. \(915\) vs. \(2(172)=344\).

Time = 0.43 (sec) , antiderivative size = 916, normalized size of antiderivative = 4.87

method	result	size
default	\(\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{d^{2}}+\frac {c^{4} \left (-\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {3 b c d \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{a \,d^{2}+b \,c^{2}}-\frac {4 b \,d^{2} \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{6}}+\frac {3 c^{2} x}{d^{4} \sqrt {b \,x^{2}+a}\, a}+\frac {2 c}{d^{3} b \sqrt {b \,x^{2}+a}}-\frac {4 c^{3} \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{5}}\)	\(916\)

Input:

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

Output:

1/d^2*(-x/b/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))+c^4/d 
^6*(-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)^(1/2)+3*b*c*d/(a*d^2+b*c^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*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))+3*c^2/d^4/(b*x^2+a)^(1/2)/a*x+ 
2*c/d^3/b/(b*x^2+a)^(1/2)-4/d^5*c^3*(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)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 705 vs. \(2 (173) = 346\).

Time = 51.82 (sec) , antiderivative size = 2888, normalized size of antiderivative = 15.36 \[ \int \frac {x^4}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*((a*b^3*c^7 + 3*a^2*b^2*c^5*d^2 + 3*a^3*b*c^3*d^4 + a^4*c*d^6 + (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)*x^3 + (b^4*c^7 + 
3*a*b^3*c^5*d^2 + 3*a^2*b^2*c^3*d^4 + a^3*b*c*d^6)*x^2 + (a*b^3*c^6*d + 3* 
a^2*b^2*c^4*d^3 + 3*a^3*b*c^2*d^5 + a^4*d^7)*x)*sqrt(b)*log(-2*b*x^2 - 2*s 
qrt(b*x^2 + a)*sqrt(b)*x - a) + (a*b^3*c^6 + 4*a^2*b^2*c^4*d^2 + (b^4*c^5* 
d + 4*a*b^3*c^3*d^3)*x^3 + (b^4*c^6 + 4*a*b^3*c^4*d^2)*x^2 + (a*b^3*c^5*d 
+ 4*a^2*b^2*c^3*d^3)*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*(a*b^3*c^6*d - a^2*b^2*c^ 
4*d^3 - 2*a^3*b*c^2*d^5 + (b^4*c^6*d + a^3*b*d^7)*x^2 - (a*b^3*c^5*d^2 + 2 
*a^2*b^2*c^3*d^4 + a^3*b*c*d^6)*x)*sqrt(b*x^2 + a))/(a*b^5*c^7*d^2 + 3*a^2 
*b^4*c^5*d^4 + 3*a^3*b^3*c^3*d^6 + a^4*b^2*c*d^8 + (b^6*c^6*d^3 + 3*a*b^5* 
c^4*d^5 + 3*a^2*b^4*c^2*d^7 + a^3*b^3*d^9)*x^3 + (b^6*c^7*d^2 + 3*a*b^5*c^ 
5*d^4 + 3*a^2*b^4*c^3*d^6 + a^3*b^3*c*d^8)*x^2 + (a*b^5*c^6*d^3 + 3*a^2*b^ 
4*c^4*d^5 + 3*a^3*b^3*c^2*d^7 + a^4*b^2*d^9)*x), 1/2*(2*(a*b^3*c^6 + 4*a^2 
*b^2*c^4*d^2 + (b^4*c^5*d + 4*a*b^3*c^3*d^3)*x^3 + (b^4*c^6 + 4*a*b^3*c^4* 
d^2)*x^2 + (a*b^3*c^5*d + 4*a^2*b^2*c^3*d^3)*x)*sqrt(-b*c^2 - a*d^2)*arcta 
n(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + 
(b^2*c^2 + a*b*d^2)*x^2)) + (a*b^3*c^7 + 3*a^2*b^2*c^5*d^2 + 3*a^3*b*c^3*d 
^4 + a^4*c*d^6 + (b^4*c^6*d + 3*a*b^3*c^4*d^3 + 3*a^2*b^2*c^2*d^5 + a^3...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (173) = 346\).

Time = 0.09 (sec) , antiderivative size = 475, normalized size of antiderivative = 2.53 \[ \int \frac {x^4}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {3 \, b^{2} c^{6} x}{\sqrt {b x^{2} + a} a b^{2} c^{4} d^{4} + 2 \, \sqrt {b x^{2} + a} a^{2} b c^{2} d^{6} + \sqrt {b x^{2} + a} a^{3} d^{8}} + \frac {3 \, b c^{5}}{\sqrt {b x^{2} + a} b^{2} c^{4} d^{3} + 2 \, \sqrt {b x^{2} + a} a b c^{2} d^{5} + \sqrt {b x^{2} + a} a^{2} d^{7}} - \frac {6 \, b c^{4} x}{\sqrt {b x^{2} + a} a b c^{2} d^{4} + \sqrt {b x^{2} + a} a^{2} d^{6}} - \frac {c^{4}}{\sqrt {b x^{2} + a} b c^{2} d^{4} x + \sqrt {b x^{2} + a} a d^{6} x + \sqrt {b x^{2} + a} b c^{3} d^{3} + \sqrt {b x^{2} + a} a c d^{5}} - \frac {4 \, c^{3}}{\sqrt {b x^{2} + a} b c^{2} d^{3} + \sqrt {b x^{2} + a} a d^{5}} + \frac {3 \, c^{2} x}{\sqrt {b x^{2} + a} a d^{4}} - \frac {x}{\sqrt {b x^{2} + a} b d^{2}} + \frac {\operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}} d^{2}} + \frac {3 \, b c^{5} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} d^{7}} - \frac {4 \, c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{5}} + \frac {2 \, c}{\sqrt {b x^{2} + a} b d^{3}} \] Input:

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

Output:

3*b^2*c^6*x/(sqrt(b*x^2 + a)*a*b^2*c^4*d^4 + 2*sqrt(b*x^2 + a)*a^2*b*c^2*d 
^6 + sqrt(b*x^2 + a)*a^3*d^8) + 3*b*c^5/(sqrt(b*x^2 + a)*b^2*c^4*d^3 + 2*s 
qrt(b*x^2 + a)*a*b*c^2*d^5 + sqrt(b*x^2 + a)*a^2*d^7) - 6*b*c^4*x/(sqrt(b* 
x^2 + a)*a*b*c^2*d^4 + sqrt(b*x^2 + a)*a^2*d^6) - c^4/(sqrt(b*x^2 + a)*b*c 
^2*d^4*x + sqrt(b*x^2 + a)*a*d^6*x + sqrt(b*x^2 + a)*b*c^3*d^3 + sqrt(b*x^ 
2 + a)*a*c*d^5) - 4*c^3/(sqrt(b*x^2 + a)*b*c^2*d^3 + sqrt(b*x^2 + a)*a*d^5 
) + 3*c^2*x/(sqrt(b*x^2 + a)*a*d^4) - x/(sqrt(b*x^2 + a)*b*d^2) + arcsinh( 
b*x/sqrt(a*b))/(b^(3/2)*d^2) + 3*b*c^5*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + 
c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(5/2)*d^7) - 4*c^3*ar 
csinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + 
 b*c^2/d^2)^(3/2)*d^5) + 2*c/(sqrt(b*x^2 + a)*b*d^3)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^4}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Recursive assumption sageVARa>=((-s 
ageVARb*sageVARc^2*sageVARd^2*t_nostep^2-2*sageVARb*sageVARc*sageVARd*t_no 
step-sage
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 1999, normalized size of antiderivative = 10.63 \[ \int \frac {x^4}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(8*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 + 8*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**3*d**3*x + 2*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*b**3*c**6 + 2*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 + 8*sqrt(a*d**2 + b*c**2)*l 
og( - 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**2 + 8*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**3 + 2*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**6*x**2 + 2 
*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**3 - 8*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b** 
2*c**4*d**2 - 8*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b**2*c**3*d**3*x - 
 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**3*c**6 - 2*sqrt(a*d**2 + b*c**2 
)*log(c + d*x)*a*b**3*c**5*d*x - 8*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b* 
*3*c**4*d**2*x**2 - 8*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**3*c**3*d**3* 
x**3 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**4*c**6*x**2 - 2*sqrt(a*d**2 
 + b*c**2)*log(c + d*x)*b**4*c**5*d*x**3 + 4*sqrt(a + b*x**2)*a**3*b*c**2* 
d**5 + 2*sqrt(a + b*x**2)*a**3*b*c*d**6*x - 2*sqrt(a + b*x**2)*a**3*b*d**7 
*x**2 + 2*sqrt(a + b*x**2)*a**2*b**2*c**4*d**3 + 4*sqrt(a + b*x**2)*a**...