\(\int \frac {x^4}{(c+d x)^2 \sqrt {a+b x^2}} \, dx\) [1191]

Optimal result

Integrand size = 22, antiderivative size = 198 \[ \int \frac {x^4}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=-\frac {2 c \sqrt {a+b x^2}}{b d^3}+\frac {x \sqrt {a+b x^2}}{2 b d^2}-\frac {c^4 \sqrt {a+b x^2}}{d^3 \left (b c^2+a d^2\right ) (c+d x)}+\frac {\left (6 b c^2-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2} d^4}+\frac {c^3 \left (3 b c^2+4 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^4 \left (b c^2+a d^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.68 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.13 \[ \int \frac {x^4}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (b c^2 \left (-6 c^2-3 c d x+d^2 x^2\right )+a d^2 \left (-4 c^2-3 c d x+d^2 x^2\right )\right )}{b \left (b c^2+a d^2\right ) (c+d x)}-\frac {4 c^3 \left (3 b c^2+4 a d^2\right ) \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+\frac {2 \left (6 b c^2-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{3/2}}}{2 d^4} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.30 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.29, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {603, 2185, 2185, 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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 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 603

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[x^m, c + d*x, x], R = PolynomialRemainde 
r[x^m, c + d*x, x]}, Simp[d*R*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 
1)*(b*c^2 + a*d^2))), x] + Simp[1/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x) 
^(n + 1)*(a + b*x^2)^p*ExpandToSum[(n + 1)*(b*c^2 + a*d^2)*Qx + b*c*R*(n + 
1) - b*d*R*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IGt 
Q[m, 1] && LtQ[n, -1] && 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 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(419\) vs. \(2(176)=352\).

Time = 0.40 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.12

method	result	size
risch	\(-\frac {\left (-d x +4 c \right ) \sqrt {b \,x^{2}+a}}{2 b \,d^{3}}-\frac {\frac {\left (a \,d^{2}-6 b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}-\frac {8 c^{3} b \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {2 c^{4} b \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}}{2 d^{3} b}\)	\(420\)
default	\(\frac {\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}}{d^{2}}+\frac {c^{4} \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{6}}+\frac {3 c^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d^{4} \sqrt {b}}-\frac {2 c \sqrt {b \,x^{2}+a}}{b \,d^{3}}+\frac {4 c^{3} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{5} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\)	\(435\)

Input:

int(x^4/(d*x+c)^2/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(-d*x+4*c)*(b*x^2+a)^(1/2)/b/d^3-1/2/d^3/b*((a*d^2-6*b*c^2)/d*ln(b^(1 
/2)*x+(b*x^2+a)^(1/2))/b^(1/2)-8*c^3/d^2*b/((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))-2*c^4/d^3*b*(-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) 
-b*c*d/(a*d^2+b*c^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. 431 vs. \(2 (177) = 354\).

Time = 37.89 (sec) , antiderivative size = 1786, normalized size of antiderivative = 9.02 \[ \int \frac {x^4}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

integrate(x^4/(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

[-1/4*((6*b^3*c^7 + 11*a*b^2*c^5*d^2 + 4*a^2*b*c^3*d^4 - a^3*c*d^6 + (6*b^ 
3*c^6*d + 11*a*b^2*c^4*d^3 + 4*a^2*b*c^2*d^5 - a^3*d^7)*x)*sqrt(b)*log(-2* 
b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(3*b^3*c^6 + 4*a*b^2*c^4*d^2 
+ (3*b^3*c^5*d + 4*a*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*(6*b^3*c^6*d 
 + 10*a*b^2*c^4*d^3 + 4*a^2*b*c^2*d^5 - (b^3*c^4*d^3 + 2*a*b^2*c^2*d^5 + a 
^2*b*d^7)*x^2 + 3*(b^3*c^5*d^2 + 2*a*b^2*c^3*d^4 + a^2*b*c*d^6)*x)*sqrt(b* 
x^2 + a))/(b^4*c^5*d^4 + 2*a*b^3*c^3*d^6 + a^2*b^2*c*d^8 + (b^4*c^4*d^5 + 
2*a*b^3*c^2*d^7 + a^2*b^2*d^9)*x), 1/4*(4*(3*b^3*c^6 + 4*a*b^2*c^4*d^2 + ( 
3*b^3*c^5*d + 4*a*b^2*c^3*d^3)*x)*sqrt(-b*c^2 - a*d^2)*arctan(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)) - (6*b^3*c^7 + 11*a*b^2*c^5*d^2 + 4*a^2*b*c^3*d^4 - a^3*c*d^6 
+ (6*b^3*c^6*d + 11*a*b^2*c^4*d^3 + 4*a^2*b*c^2*d^5 - a^3*d^7)*x)*sqrt(b)* 
log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(6*b^3*c^6*d + 10*a*b^ 
2*c^4*d^3 + 4*a^2*b*c^2*d^5 - (b^3*c^4*d^3 + 2*a*b^2*c^2*d^5 + a^2*b*d^7)* 
x^2 + 3*(b^3*c^5*d^2 + 2*a*b^2*c^3*d^4 + a^2*b*c*d^6)*x)*sqrt(b*x^2 + a))/ 
(b^4*c^5*d^4 + 2*a*b^3*c^3*d^6 + a^2*b^2*c*d^8 + (b^4*c^4*d^5 + 2*a*b^3*c^ 
2*d^7 + a^2*b^2*d^9)*x), -1/2*((6*b^3*c^7 + 11*a*b^2*c^5*d^2 + 4*a^2*b*c^3 
*d^4 - a^3*c*d^6 + (6*b^3*c^6*d + 11*a*b^2*c^4*d^3 + 4*a^2*b*c^2*d^5 - ...
 

Sympy [F]

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

integrate(x**4/(d*x+c)**2/(b*x**2+a)**(1/2),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.18 \[ \int \frac {x^4}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=-\frac {\sqrt {b x^{2} + a} c^{4}}{b c^{2} d^{4} x + a d^{6} x + b c^{3} d^{3} + a c d^{5}} + \frac {\sqrt {b x^{2} + a} x}{2 \, b d^{2}} + \frac {3 \, c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{4}} - \frac {a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}} d^{2}} + \frac {b c^{5} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{7}} - \frac {4 \, c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{5}} - \frac {2 \, \sqrt {b x^{2} + a} c}{b d^{3}} \] Input:

integrate(x^4/(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F(-1)]

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

integrate(x^4/(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

int(x^4/((a + b*x^2)^(1/2)*(c + d*x)^2),x)
 

Output:

int(x^4/((a + b*x^2)^(1/2)*(c + d*x)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1015, normalized size of antiderivative = 5.13 \[ \int \frac {x^4}{(c+d x)^2 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

int(x^4/(d*x+c)^2/(b*x^2+a)^(1/2),x)
 

Output:

(16*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**2*c**4*d**2 + 16*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**2*c**3*d**3*x + 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)*b**3*c**6 + 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)*b**3*c**5*d*x - 16*sqrt(a*d**2 + b*c**2)*log(c 
+ d*x)*a*b**2*c**4*d**2 - 16*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c** 
3*d**3*x - 12*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**3*c**6 - 12*sqrt(a*d** 
2 + b*c**2)*log(c + d*x)*b**3*c**5*d*x - 8*sqrt(a + b*x**2)*a**2*b*c**2*d* 
*5 - 6*sqrt(a + b*x**2)*a**2*b*c*d**6*x + 2*sqrt(a + b*x**2)*a**2*b*d**7*x 
**2 - 20*sqrt(a + b*x**2)*a*b**2*c**4*d**3 - 12*sqrt(a + b*x**2)*a*b**2*c* 
*3*d**4*x + 4*sqrt(a + b*x**2)*a*b**2*c**2*d**5*x**2 - 12*sqrt(a + b*x**2) 
*b**3*c**6*d - 6*sqrt(a + b*x**2)*b**3*c**5*d**2*x + 2*sqrt(a + b*x**2)*b* 
*3*c**4*d**3*x**2 + sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a**3*c*d**6 
+ sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a**3*d**7*x - 4*sqrt(b)*log(sq 
rt(a + b*x**2) - sqrt(b)*x)*a**2*b*c**3*d**4 - 4*sqrt(b)*log(sqrt(a + b*x* 
*2) - sqrt(b)*x)*a**2*b*c**2*d**5*x - 11*sqrt(b)*log(sqrt(a + b*x**2) - sq 
rt(b)*x)*a*b**2*c**5*d**2 - 11*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*a 
*b**2*c**4*d**3*x - 6*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*b**3*c**7 
- 6*sqrt(b)*log(sqrt(a + b*x**2) - sqrt(b)*x)*b**3*c**6*d*x - sqrt(b)*l...