3.7.94 \(\int \frac {x^4 (c+d x^2)^{5/2}}{a+b x^2} \, dx\) [694]

3.7.94.1 Optimal result

Integrand size = 24, antiderivative size = 291 \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {a^{3/2} (b c-a d)^{5/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^5}-\frac {\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 b^5 d^{3/2}} \]

output

1/8*d*x^5*(d*x^2+c)^(3/2)/b+a^(3/2)*(-a*d+b*c)^(5/2)*arctan(x*(-a*d+b*c)^( 
1/2)/a^(1/2)/(d*x^2+c)^(1/2))/b^5-1/128*(-128*a^4*d^4+320*a^3*b*c*d^3-240* 
a^2*b^2*c^2*d^2+40*a*b^3*c^3*d+5*b^4*c^4)*arctanh(x*d^(1/2)/(d*x^2+c)^(1/2 
))/b^5/d^(3/2)+1/128*(-64*a^3*d^3+144*a^2*b*c*d^2-88*a*b^2*c^2*d+5*b^3*c^3 
)*x*(d*x^2+c)^(1/2)/b^4/d+1/192*(48*a^2*d^2-104*a*b*c*d+59*b^2*c^2)*x^3*(d 
*x^2+c)^(1/2)/b^3+1/48*d*(-8*a*d+11*b*c)*x^5*(d*x^2+c)^(1/2)/b^2
 

3.7.94.2 Mathematica [A] (verified)

Time = 2.61 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.78 \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {b \sqrt {d} x \sqrt {c+d x^2} \left (-192 a^3 d^3+48 a^2 b d^2 \left (9 c+2 d x^2\right )-8 a b^2 d \left (33 c^2+26 c d x^2+8 d^2 x^4\right )+b^3 \left (15 c^3+118 c^2 d x^2+136 c d^2 x^4+48 d^3 x^6\right )\right )+384 \sqrt {a} \sqrt {d} (b c-a d)^2 \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \left (-b c+a d-\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )+384 \sqrt {a} \sqrt {d} (b c-a d)^2 \left (-b c+a d+\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )+6 \left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c}-\sqrt {c+d x^2}}\right )}{384 b^5 d^{3/2}} \]

input

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

output

(b*Sqrt[d]*x*Sqrt[c + d*x^2]*(-192*a^3*d^3 + 48*a^2*b*d^2*(9*c + 2*d*x^2) 
- 8*a*b^2*d*(33*c^2 + 26*c*d*x^2 + 8*d^2*x^4) + b^3*(15*c^3 + 118*c^2*d*x^ 
2 + 136*c*d^2*x^4 + 48*d^3*x^6)) + 384*Sqrt[a]*Sqrt[d]*(b*c - a*d)^2*Sqrt[ 
2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*(-(b*c) + a*d - Sqrt[b]*S 
qrt[c]*Sqrt[b*c - a*d])*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[ 
b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))] + 384*Sqrt[a]*Sqrt[ 
d]*(b*c - a*d)^2*(-(b*c) + a*d + Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d])*Sqrt[2*b 
*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[2*b*c - a*d + 2 
*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]) 
)] + 6*(5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 
 - 128*a^4*d^4)*ArcTanh[(Sqrt[d]*x)/(Sqrt[c] - Sqrt[c + d*x^2])])/(384*b^5 
*d^(3/2))
 

3.7.94.3 Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {379, 443, 444, 27, 444, 25, 398, 224, 219, 291, 218}

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.

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

\(\Big \downarrow \) 379

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

\(\Big \downarrow \) 443

\(\displaystyle \frac {\frac {\int \frac {x^4 \left (d \left (59 b^2 c^2-104 a b d c+48 a^2 d^2\right ) x^2+c \left (48 b^2 c^2-85 a b d c+40 a^2 d^2\right )\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{6 b}+\frac {d x^5 \sqrt {c+d x^2} (11 b c-8 a d)}{6 b}}{8 b}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\frac {\frac {x^3 \sqrt {c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{4 b}-\frac {\int \frac {3 d x^2 \left (a c \left (59 b^2 c^2-104 a b d c+48 a^2 d^2\right )-\left (5 b^3 c^3-88 a b^2 d c^2+144 a^2 b d^2 c-64 a^3 d^3\right ) x^2\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{4 b d}}{6 b}+\frac {d x^5 \sqrt {c+d x^2} (11 b c-8 a d)}{6 b}}{8 b}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {x^3 \sqrt {c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{4 b}-\frac {3 \int \frac {x^2 \left (a c \left (59 b^2 c^2-104 a b d c+48 a^2 d^2\right )-\left (5 b^3 c^3-88 a b^2 d c^2+144 a^2 b d^2 c-64 a^3 d^3\right ) x^2\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2} (11 b c-8 a d)}{6 b}}{8 b}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\frac {\frac {x^3 \sqrt {c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{4 b}-\frac {3 \left (-\frac {\int -\frac {\left (5 b^4 c^4+40 a b^3 d c^3-240 a^2 b^2 d^2 c^2+320 a^3 b d^3 c-128 a^4 d^4\right ) x^2+a c \left (5 b^3 c^3-88 a b^2 d c^2+144 a^2 b d^2 c-64 a^3 d^3\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 b d}-\frac {x \sqrt {c+d x^2} \left (-64 a^3 d^3+144 a^2 b c d^2-88 a b^2 c^2 d+5 b^3 c^3\right )}{2 b d}\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2} (11 b c-8 a d)}{6 b}}{8 b}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {x^3 \sqrt {c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{4 b}-\frac {3 \left (\frac {\int \frac {\left (5 b^4 c^4+40 a b^3 d c^3-240 a^2 b^2 d^2 c^2+320 a^3 b d^3 c-128 a^4 d^4\right ) x^2+a c \left (5 b^3 c^3-88 a b^2 d c^2+144 a^2 b d^2 c-64 a^3 d^3\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 b d}-\frac {x \sqrt {c+d x^2} \left (-64 a^3 d^3+144 a^2 b c d^2-88 a b^2 c^2 d+5 b^3 c^3\right )}{2 b d}\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2} (11 b c-8 a d)}{6 b}}{8 b}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {\frac {\frac {x^3 \sqrt {c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{4 b}-\frac {3 \left (\frac {\frac {\left (-128 a^4 d^4+320 a^3 b c d^3-240 a^2 b^2 c^2 d^2+40 a b^3 c^3 d+5 b^4 c^4\right ) \int \frac {1}{\sqrt {d x^2+c}}dx}{b}-\frac {128 a^2 d (b c-a d)^3 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}}{2 b d}-\frac {x \sqrt {c+d x^2} \left (-64 a^3 d^3+144 a^2 b c d^2-88 a b^2 c^2 d+5 b^3 c^3\right )}{2 b d}\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2} (11 b c-8 a d)}{6 b}}{8 b}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {x^3 \sqrt {c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{4 b}-\frac {3 \left (\frac {\frac {\left (-128 a^4 d^4+320 a^3 b c d^3-240 a^2 b^2 c^2 d^2+40 a b^3 c^3 d+5 b^4 c^4\right ) \int \frac {1}{1-\frac {d x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}-\frac {128 a^2 d (b c-a d)^3 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}}{2 b d}-\frac {x \sqrt {c+d x^2} \left (-64 a^3 d^3+144 a^2 b c d^2-88 a b^2 c^2 d+5 b^3 c^3\right )}{2 b d}\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2} (11 b c-8 a d)}{6 b}}{8 b}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {x^3 \sqrt {c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{4 b}-\frac {3 \left (\frac {\frac {\left (-128 a^4 d^4+320 a^3 b c d^3-240 a^2 b^2 c^2 d^2+40 a b^3 c^3 d+5 b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}-\frac {128 a^2 d (b c-a d)^3 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}}{2 b d}-\frac {x \sqrt {c+d x^2} \left (-64 a^3 d^3+144 a^2 b c d^2-88 a b^2 c^2 d+5 b^3 c^3\right )}{2 b d}\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2} (11 b c-8 a d)}{6 b}}{8 b}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {\frac {x^3 \sqrt {c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{4 b}-\frac {3 \left (\frac {\frac {\left (-128 a^4 d^4+320 a^3 b c d^3-240 a^2 b^2 c^2 d^2+40 a b^3 c^3 d+5 b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}-\frac {128 a^2 d (b c-a d)^3 \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}}{2 b d}-\frac {x \sqrt {c+d x^2} \left (-64 a^3 d^3+144 a^2 b c d^2-88 a b^2 c^2 d+5 b^3 c^3\right )}{2 b d}\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2} (11 b c-8 a d)}{6 b}}{8 b}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {x^3 \sqrt {c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{4 b}-\frac {3 \left (\frac {\frac {\left (-128 a^4 d^4+320 a^3 b c d^3-240 a^2 b^2 c^2 d^2+40 a b^3 c^3 d+5 b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}-\frac {128 a^{3/2} d (b c-a d)^{5/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b}}{2 b d}-\frac {x \sqrt {c+d x^2} \left (-64 a^3 d^3+144 a^2 b c d^2-88 a b^2 c^2 d+5 b^3 c^3\right )}{2 b d}\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2} (11 b c-8 a d)}{6 b}}{8 b}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}\)

input

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

output

(d*x^5*(c + d*x^2)^(3/2))/(8*b) + ((d*(11*b*c - 8*a*d)*x^5*Sqrt[c + d*x^2] 
)/(6*b) + (((59*b^2*c^2 - 104*a*b*c*d + 48*a^2*d^2)*x^3*Sqrt[c + d*x^2])/( 
4*b) - (3*(-1/2*((5*b^3*c^3 - 88*a*b^2*c^2*d + 144*a^2*b*c*d^2 - 64*a^3*d^ 
3)*x*Sqrt[c + d*x^2])/(b*d) + ((-128*a^(3/2)*d*(b*c - a*d)^(5/2)*ArcTan[(S 
qrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b + ((5*b^4*c^4 + 40*a*b^3*c 
^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*ArcTanh[(Sqrt[ 
d]*x)/Sqrt[c + d*x^2]])/(b*Sqrt[d]))/(2*b*d)))/(4*b))/(6*b))/(8*b)
 

3.7.94.3.1 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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

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 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 379

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 
1)/(b*e*(m + 2*(p + q) + 1))), x] + Simp[1/(b*(m + 2*(p + q) + 1))   Int[(e 
*x)^m*(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*((b*c - a*d)*(m + 1) + b*c*2 
*(p + q)) + (d*(b*c - a*d)*(m + 1) + d*2*(q - 1)*(b*c - a*d) + b*c*d*2*(p + 
 q))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0 
] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

rule 443

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

rule 444

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

3.7.94.4 Maple [A] (verified)

Time = 3.12 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.89

method	result	size
pseudoelliptic	\(-\frac {\frac {b \sqrt {d \,x^{2}+c}\, \left (-48 b^{3} d^{3} x^{6}+64 a \,b^{2} d^{3} x^{4}-136 b^{3} c \,d^{2} x^{4}-96 x^{2} a^{2} b \,d^{3}+208 x^{2} a \,b^{2} c \,d^{2}-118 x^{2} b^{3} c^{2} d +192 a^{3} d^{3}-432 a^{2} b c \,d^{2}+264 a \,b^{2} c^{2} d -15 b^{3} c^{3}\right ) x}{192 d}-\frac {\left (128 a^{4} d^{4}-320 a^{3} b c \,d^{3}+240 a^{2} b^{2} c^{2} d^{2}-40 a \,b^{3} c^{3} d -5 b^{4} c^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{64 d^{\frac {3}{2}}}+\frac {2 \left (a d -b c \right )^{3} a^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{2 b^{5}}\)	\(259\)
risch	\(-\frac {x \left (-48 b^{3} d^{3} x^{6}+64 a \,b^{2} d^{3} x^{4}-136 b^{3} c \,d^{2} x^{4}-96 x^{2} a^{2} b \,d^{3}+208 x^{2} a \,b^{2} c \,d^{2}-118 x^{2} b^{3} c^{2} d +192 a^{3} d^{3}-432 a^{2} b c \,d^{2}+264 a \,b^{2} c^{2} d -15 b^{3} c^{3}\right ) \sqrt {d \,x^{2}+c}}{384 d \,b^{4}}+\frac {\frac {\left (128 a^{4} d^{4}-320 a^{3} b c \,d^{3}+240 a^{2} b^{2} c^{2} d^{2}-40 a \,b^{3} c^{3} d -5 b^{4} c^{4}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b \sqrt {d}}-\frac {64 a^{2} d \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}+\frac {64 a^{2} d \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{128 d \,b^{4}}\)	\(602\)
default	\(\text {Expression too large to display}\)	\(2252\)

input

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

output

-1/2/b^5*(1/192*b*(d*x^2+c)^(1/2)*(-48*b^3*d^3*x^6+64*a*b^2*d^3*x^4-136*b^ 
3*c*d^2*x^4-96*a^2*b*d^3*x^2+208*a*b^2*c*d^2*x^2-118*b^3*c^2*d*x^2+192*a^3 
*d^3-432*a^2*b*c*d^2+264*a*b^2*c^2*d-15*b^3*c^3)/d*x-1/64*(128*a^4*d^4-320 
*a^3*b*c*d^3+240*a^2*b^2*c^2*d^2-40*a*b^3*c^3*d-5*b^4*c^4)/d^(3/2)*arctanh 
((d*x^2+c)^(1/2)/x/d^(1/2))+2*(a*d-b*c)^3*a^2/((a*d-b*c)*a)^(1/2)*arctanh( 
(d*x^2+c)^(1/2)/x*a/((a*d-b*c)*a)^(1/2)))
 

3.7.94.5 Fricas [A] (verification not implemented)

Time = 4.53 (sec) , antiderivative size = 1443, normalized size of antiderivative = 4.96 \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\text {Too large to display} \]

input

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

output

[-1/768*(3*(5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c 
*d^3 - 128*a^4*d^4)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c 
) - 192*(a*b^2*c^2*d^2 - 2*a^2*b*c*d^3 + a^3*d^4)*sqrt(-a*b*c + a^2*d)*log 
(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c 
*d)*x^2 + 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + 
c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 2*(48*b^4*d^4*x^7 + 8*(17*b^4*c*d^3 - 8 
*a*b^3*d^4)*x^5 + 2*(59*b^4*c^2*d^2 - 104*a*b^3*c*d^3 + 48*a^2*b^2*d^4)*x^ 
3 + 3*(5*b^4*c^3*d - 88*a*b^3*c^2*d^2 + 144*a^2*b^2*c*d^3 - 64*a^3*b*d^4)* 
x)*sqrt(d*x^2 + c))/(b^5*d^2), 1/384*(3*(5*b^4*c^4 + 40*a*b^3*c^3*d - 240* 
a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*sqrt(-d)*arctan(sqrt(-d)* 
x/sqrt(d*x^2 + c)) + 96*(a*b^2*c^2*d^2 - 2*a^2*b*c*d^3 + a^3*d^4)*sqrt(-a* 
b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a 
*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2* 
d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + (48*b^4*d^4*x^7 + 8*(17 
*b^4*c*d^3 - 8*a*b^3*d^4)*x^5 + 2*(59*b^4*c^2*d^2 - 104*a*b^3*c*d^3 + 48*a 
^2*b^2*d^4)*x^3 + 3*(5*b^4*c^3*d - 88*a*b^3*c^2*d^2 + 144*a^2*b^2*c*d^3 - 
64*a^3*b*d^4)*x)*sqrt(d*x^2 + c))/(b^5*d^2), 1/768*(384*(a*b^2*c^2*d^2 - 2 
*a^2*b*c*d^3 + a^3*d^4)*sqrt(a*b*c - a^2*d)*arctan(1/2*sqrt(a*b*c - a^2*d) 
*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b 
*c^2 - a^2*c*d)*x)) - 3*(5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d...
 

3.7.94.6 Sympy [F]

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

input

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

output

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

3.7.94.7 Maxima [F]

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

input

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

output

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

3.7.94.8 Giac [F(-2)]

Exception generated. \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

3.7.94.9 Mupad [F(-1)]

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

input

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

output

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