\(\int x^2 (a+b x^2)^2 (c+d x^2)^{5/2} \, dx\) [626]

Optimal result

Integrand size = 24, antiderivative size = 281 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {c^3 \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) x \sqrt {c+d x^2}}{1024 d^3}+\frac {c^2 \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) x^3 \sqrt {c+d x^2}}{512 d^2}+\frac {c \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) x^3 \left (c+d x^2\right )^{3/2}}{384 d^2}+\frac {\left (40 a^2 d^2+b c (5 b c-24 a d)\right ) x^3 \left (c+d x^2\right )^{5/2}}{320 d^2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}-\frac {c^4 \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{1024 d^{7/2}} \]

[Out]

1/384*c*(40*a^2*d^2+b*c*(-24*a*d+5*b*c))*x^3*(d*x^2+c)^(3/2)/d^2+1/320*(40*a^2*d^2+b*c*(-24*a*d+5*b*c))*x^3*(d
*x^2+c)^(5/2)/d^2-1/120*b*(-24*a*d+5*b*c)*x^3*(d*x^2+c)^(7/2)/d^2+1/12*b^2*x^5*(d*x^2+c)^(7/2)/d-1/1024*c^4*(4
0*a^2*d^2+b*c*(-24*a*d+5*b*c))*arctanh(x*d^(1/2)/(d*x^2+c)^(1/2))/d^(7/2)+1/1024*c^3*(40*a^2*d^2+b*c*(-24*a*d+
5*b*c))*x*(d*x^2+c)^(1/2)/d^3+1/512*c^2*(40*a^2*d^2+b*c*(-24*a*d+5*b*c))*x^3*(d*x^2+c)^(1/2)/d^2

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {475, 470, 285, 327, 223, 212} \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=-\frac {c^4 \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{1024 d^{7/2}}+\frac {c^3 x \sqrt {c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{1024 d^3}+\frac {c^2 x^3 \sqrt {c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{512 d^2}+\frac {1}{320} x^3 \left (c+d x^2\right )^{5/2} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right )+\frac {c x^3 \left (c+d x^2\right )^{3/2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{384 d^2}-\frac {b x^3 \left (c+d x^2\right )^{7/2} (5 b c-24 a d)}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d} \]

[In]

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

[Out]

(c^3*(40*a^2*d^2 + b*c*(5*b*c - 24*a*d))*x*Sqrt[c + d*x^2])/(1024*d^3) + (c^2*(40*a^2*d^2 + b*c*(5*b*c - 24*a*
d))*x^3*Sqrt[c + d*x^2])/(512*d^2) + (c*(40*a^2*d^2 + b*c*(5*b*c - 24*a*d))*x^3*(c + d*x^2)^(3/2))/(384*d^2) +
((40*a^2 + (b*c*(5*b*c - 24*a*d))/d^2)*x^3*(c + d*x^2)^(5/2))/320 - (b*(5*b*c - 24*a*d)*x^3*(c + d*x^2)^(7/2)
)/(120*d^2) + (b^2*x^5*(c + d*x^2)^(7/2))/(12*d) - (c^4*(40*a^2*d^2 + b*c*(5*b*c - 24*a*d))*ArcTanh[(Sqrt[d]*x
)/Sqrt[c + d*x^2]])/(1024*d^(7/2))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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 285

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + n
*p + 1))), x] + Dist[a*n*(p/(m + n*p + 1)), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 470

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

Rule 475

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

Rubi steps \begin{align*} \text {integral}& = \frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac {\int x^2 \left (c+d x^2\right )^{5/2} \left (12 a^2 d-b (5 b c-24 a d) x^2\right ) \, dx}{12 d} \\ & = -\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac {1}{40} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) \int x^2 \left (c+d x^2\right )^{5/2} \, dx \\ & = \frac {1}{320} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac {1}{64} \left (c \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right )\right ) \int x^2 \left (c+d x^2\right )^{3/2} \, dx \\ & = \frac {1}{384} c \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac {1}{320} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac {1}{128} \left (c^2 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right )\right ) \int x^2 \sqrt {c+d x^2} \, dx \\ & = \frac {1}{512} c^2 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}+\frac {1}{384} c \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac {1}{320} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}+\frac {1}{512} \left (c^3 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2}} \, dx \\ & = \frac {c^3 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{1024 d}+\frac {1}{512} c^2 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}+\frac {1}{384} c \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac {1}{320} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}-\frac {\left (c^4 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{1024 d} \\ & = \frac {c^3 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{1024 d}+\frac {1}{512} c^2 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}+\frac {1}{384} c \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac {1}{320} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}-\frac {\left (c^4 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{1024 d} \\ & = \frac {c^3 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{1024 d}+\frac {1}{512} c^2 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}+\frac {1}{384} c \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{3/2}+\frac {1}{320} \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) x^3 \left (c+d x^2\right )^{5/2}-\frac {b (5 b c-24 a d) x^3 \left (c+d x^2\right )^{7/2}}{120 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d}-\frac {c^4 \left (40 a^2+\frac {b c (5 b c-24 a d)}{d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{1024 d^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.92 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {x \sqrt {c+d x^2} \left (75 b^2 c^5-360 a b c^4 d+600 a^2 c^3 d^2-50 b^2 c^4 d x^2+240 a b c^3 d^2 x^2+4720 a^2 c^2 d^3 x^2+40 b^2 c^3 d^2 x^4+5952 a b c^2 d^3 x^4+5440 a^2 c d^4 x^4+2160 b^2 c^2 d^3 x^6+8064 a b c d^4 x^6+1920 a^2 d^5 x^6+3200 b^2 c d^4 x^8+3072 a b d^5 x^8+1280 b^2 d^5 x^{10}\right )}{15360 d^3}-\frac {c^4 \left (5 b^2 c^2-24 a b c d+40 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{-\sqrt {c}+\sqrt {c+d x^2}}\right )}{512 d^{7/2}} \]

[In]

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

[Out]

(x*Sqrt[c + d*x^2]*(75*b^2*c^5 - 360*a*b*c^4*d + 600*a^2*c^3*d^2 - 50*b^2*c^4*d*x^2 + 240*a*b*c^3*d^2*x^2 + 47
20*a^2*c^2*d^3*x^2 + 40*b^2*c^3*d^2*x^4 + 5952*a*b*c^2*d^3*x^4 + 5440*a^2*c*d^4*x^4 + 2160*b^2*c^2*d^3*x^6 + 8
064*a*b*c*d^4*x^6 + 1920*a^2*d^5*x^6 + 3200*b^2*c*d^4*x^8 + 3072*a*b*d^5*x^8 + 1280*b^2*d^5*x^10))/(15360*d^3)
- (c^4*(5*b^2*c^2 - 24*a*b*c*d + 40*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/(512*d^(7/2))

Maple [A] (verified)

Time = 3.03 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.72

method	result	size
pseudoelliptic	\(\frac {\frac {5 \left (-a^{2} c^{4} d^{2}+\frac {3}{5} a b \,c^{5} d -\frac {1}{8} b^{2} c^{6}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{128}+\frac {5 x \left (\frac {16 x^{6} \left (\frac {2}{3} b^{2} x^{4}+\frac {8}{5} a b \,x^{2}+a^{2}\right ) d^{\frac {11}{2}}}{5}+\left (c^{2} \left (\frac {1}{15} b^{2} x^{4}+\frac {2}{5} a b \,x^{2}+a^{2}\right ) d^{\frac {5}{2}}+\frac {118 x^{2} \left (\frac {27}{59} b^{2} x^{4}+\frac {372}{295} a b \,x^{2}+a^{2}\right ) c \,d^{\frac {7}{2}}}{15}+\left (\frac {16}{3} b^{2} x^{8}+\frac {336}{25} a b \,x^{6}+\frac {136}{15} a^{2} x^{4}\right ) d^{\frac {9}{2}}-\frac {3 \left (\left (\frac {5 b \,x^{2}}{36}+a \right ) d^{\frac {3}{2}}-\frac {5 b \sqrt {d}\, c}{24}\right ) b \,c^{3}}{5}\right ) c \right ) \sqrt {d \,x^{2}+c}}{128}}{d^{\frac {7}{2}}}\)	\(203\)
risch	\(\frac {x \left (1280 b^{2} d^{5} x^{10}+3072 a b \,d^{5} x^{8}+3200 b^{2} c \,d^{4} x^{8}+1920 a^{2} d^{5} x^{6}+8064 a b c \,d^{4} x^{6}+2160 b^{2} c^{2} d^{3} x^{6}+5440 a^{2} c \,d^{4} x^{4}+5952 a b \,c^{2} d^{3} x^{4}+40 b^{2} c^{3} d^{2} x^{4}+4720 a^{2} c^{2} d^{3} x^{2}+240 a b \,c^{3} d^{2} x^{2}-50 b^{2} c^{4} d \,x^{2}+600 a^{2} c^{3} d^{2}-360 a b \,c^{4} d +75 b^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{15360 d^{3}}-\frac {c^{4} \left (40 a^{2} d^{2}-24 a b c d +5 b^{2} c^{2}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{1024 d^{\frac {7}{2}}}\)	\(239\)
default	\(b^{2} \left (\frac {x^{5} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{12 d}-\frac {5 c \left (\frac {x^{3} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{10 d}-\frac {3 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{8 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{8 d}\right )}{10 d}\right )}{12 d}\right )+a^{2} \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{8 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{8 d}\right )+2 a b \left (\frac {x^{3} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{10 d}-\frac {3 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{8 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{8 d}\right )}{10 d}\right )\)	\(353\)

[In]

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

[Out]

5/128/d^(7/2)*((-a^2*c^4*d^2+3/5*a*b*c^5*d-1/8*b^2*c^6)*arctanh((d*x^2+c)^(1/2)/x/d^(1/2))+x*(16/5*x^6*(2/3*b^
2*x^4+8/5*a*b*x^2+a^2)*d^(11/2)+(c^2*(1/15*b^2*x^4+2/5*a*b*x^2+a^2)*d^(5/2)+118/15*x^2*(27/59*b^2*x^4+372/295*
a*b*x^2+a^2)*c*d^(7/2)+(16/3*b^2*x^8+336/25*a*b*x^6+136/15*a^2*x^4)*d^(9/2)-3/5*((5/36*b*x^2+a)*d^(3/2)-5/24*b
*d^(1/2)*c)*b*c^3)*c)*(d*x^2+c)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.76 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\left [\frac {15 \, {\left (5 \, b^{2} c^{6} - 24 \, a b c^{5} d + 40 \, a^{2} c^{4} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (1280 \, b^{2} d^{6} x^{11} + 128 \, {\left (25 \, b^{2} c d^{5} + 24 \, a b d^{6}\right )} x^{9} + 48 \, {\left (45 \, b^{2} c^{2} d^{4} + 168 \, a b c d^{5} + 40 \, a^{2} d^{6}\right )} x^{7} + 8 \, {\left (5 \, b^{2} c^{3} d^{3} + 744 \, a b c^{2} d^{4} + 680 \, a^{2} c d^{5}\right )} x^{5} - 10 \, {\left (5 \, b^{2} c^{4} d^{2} - 24 \, a b c^{3} d^{3} - 472 \, a^{2} c^{2} d^{4}\right )} x^{3} + 15 \, {\left (5 \, b^{2} c^{5} d - 24 \, a b c^{4} d^{2} + 40 \, a^{2} c^{3} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{30720 \, d^{4}}, \frac {15 \, {\left (5 \, b^{2} c^{6} - 24 \, a b c^{5} d + 40 \, a^{2} c^{4} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (1280 \, b^{2} d^{6} x^{11} + 128 \, {\left (25 \, b^{2} c d^{5} + 24 \, a b d^{6}\right )} x^{9} + 48 \, {\left (45 \, b^{2} c^{2} d^{4} + 168 \, a b c d^{5} + 40 \, a^{2} d^{6}\right )} x^{7} + 8 \, {\left (5 \, b^{2} c^{3} d^{3} + 744 \, a b c^{2} d^{4} + 680 \, a^{2} c d^{5}\right )} x^{5} - 10 \, {\left (5 \, b^{2} c^{4} d^{2} - 24 \, a b c^{3} d^{3} - 472 \, a^{2} c^{2} d^{4}\right )} x^{3} + 15 \, {\left (5 \, b^{2} c^{5} d - 24 \, a b c^{4} d^{2} + 40 \, a^{2} c^{3} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{15360 \, d^{4}}\right ] \]

[In]

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

[Out]

[1/30720*(15*(5*b^2*c^6 - 24*a*b*c^5*d + 40*a^2*c^4*d^2)*sqrt(d)*log(-2*d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(d)*x -
c) + 2*(1280*b^2*d^6*x^11 + 128*(25*b^2*c*d^5 + 24*a*b*d^6)*x^9 + 48*(45*b^2*c^2*d^4 + 168*a*b*c*d^5 + 40*a^2*
d^6)*x^7 + 8*(5*b^2*c^3*d^3 + 744*a*b*c^2*d^4 + 680*a^2*c*d^5)*x^5 - 10*(5*b^2*c^4*d^2 - 24*a*b*c^3*d^3 - 472*
a^2*c^2*d^4)*x^3 + 15*(5*b^2*c^5*d - 24*a*b*c^4*d^2 + 40*a^2*c^3*d^3)*x)*sqrt(d*x^2 + c))/d^4, 1/15360*(15*(5*
b^2*c^6 - 24*a*b*c^5*d + 40*a^2*c^4*d^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + (1280*b^2*d^6*x^11 + 12
8*(25*b^2*c*d^5 + 24*a*b*d^6)*x^9 + 48*(45*b^2*c^2*d^4 + 168*a*b*c*d^5 + 40*a^2*d^6)*x^7 + 8*(5*b^2*c^3*d^3 +
744*a*b*c^2*d^4 + 680*a^2*c*d^5)*x^5 - 10*(5*b^2*c^4*d^2 - 24*a*b*c^3*d^3 - 472*a^2*c^2*d^4)*x^3 + 15*(5*b^2*c
^5*d - 24*a*b*c^4*d^2 + 40*a^2*c^3*d^3)*x)*sqrt(d*x^2 + c))/d^4]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 665 vs. \(2 (270) = 540\).

Time = 0.57 (sec) , antiderivative size = 665, normalized size of antiderivative = 2.37 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\begin {cases} - \frac {c \left (a^{2} c^{3} - \frac {3 c \left (3 a^{2} c^{2} d + 2 a b c^{3} - \frac {5 c \left (3 a^{2} c d^{2} + 6 a b c^{2} d + b^{2} c^{3} - \frac {7 c \left (a^{2} d^{3} + 6 a b c d^{2} + 3 b^{2} c^{2} d - \frac {9 c \left (2 a b d^{3} + \frac {25 b^{2} c d^{2}}{12}\right )}{10 d}\right )}{8 d}\right )}{6 d}\right )}{4 d}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {d} \sqrt {c + d x^{2}} + 2 d x \right )}}{\sqrt {d}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {d x^{2}}} & \text {otherwise} \end {cases}\right )}{2 d} + \sqrt {c + d x^{2}} \left (\frac {b^{2} d^{2} x^{11}}{12} + \frac {x^{9} \cdot \left (2 a b d^{3} + \frac {25 b^{2} c d^{2}}{12}\right )}{10 d} + \frac {x^{7} \left (a^{2} d^{3} + 6 a b c d^{2} + 3 b^{2} c^{2} d - \frac {9 c \left (2 a b d^{3} + \frac {25 b^{2} c d^{2}}{12}\right )}{10 d}\right )}{8 d} + \frac {x^{5} \cdot \left (3 a^{2} c d^{2} + 6 a b c^{2} d + b^{2} c^{3} - \frac {7 c \left (a^{2} d^{3} + 6 a b c d^{2} + 3 b^{2} c^{2} d - \frac {9 c \left (2 a b d^{3} + \frac {25 b^{2} c d^{2}}{12}\right )}{10 d}\right )}{8 d}\right )}{6 d} + \frac {x^{3} \cdot \left (3 a^{2} c^{2} d + 2 a b c^{3} - \frac {5 c \left (3 a^{2} c d^{2} + 6 a b c^{2} d + b^{2} c^{3} - \frac {7 c \left (a^{2} d^{3} + 6 a b c d^{2} + 3 b^{2} c^{2} d - \frac {9 c \left (2 a b d^{3} + \frac {25 b^{2} c d^{2}}{12}\right )}{10 d}\right )}{8 d}\right )}{6 d}\right )}{4 d} + \frac {x \left (a^{2} c^{3} - \frac {3 c \left (3 a^{2} c^{2} d + 2 a b c^{3} - \frac {5 c \left (3 a^{2} c d^{2} + 6 a b c^{2} d + b^{2} c^{3} - \frac {7 c \left (a^{2} d^{3} + 6 a b c d^{2} + 3 b^{2} c^{2} d - \frac {9 c \left (2 a b d^{3} + \frac {25 b^{2} c d^{2}}{12}\right )}{10 d}\right )}{8 d}\right )}{6 d}\right )}{4 d}\right )}{2 d}\right ) & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (\frac {a^{2} x^{3}}{3} + \frac {2 a b x^{5}}{5} + \frac {b^{2} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-c*(a**2*c**3 - 3*c*(3*a**2*c**2*d + 2*a*b*c**3 - 5*c*(3*a**2*c*d**2 + 6*a*b*c**2*d + b**2*c**3 - 7
*c*(a**2*d**3 + 6*a*b*c*d**2 + 3*b**2*c**2*d - 9*c*(2*a*b*d**3 + 25*b**2*c*d**2/12)/(10*d))/(8*d))/(6*d))/(4*d
))*Piecewise((log(2*sqrt(d)*sqrt(c + d*x**2) + 2*d*x)/sqrt(d), Ne(c, 0)), (x*log(x)/sqrt(d*x**2), True))/(2*d)
+ sqrt(c + d*x**2)*(b**2*d**2*x**11/12 + x**9*(2*a*b*d**3 + 25*b**2*c*d**2/12)/(10*d) + x**7*(a**2*d**3 + 6*a
*b*c*d**2 + 3*b**2*c**2*d - 9*c*(2*a*b*d**3 + 25*b**2*c*d**2/12)/(10*d))/(8*d) + x**5*(3*a**2*c*d**2 + 6*a*b*c
**2*d + b**2*c**3 - 7*c*(a**2*d**3 + 6*a*b*c*d**2 + 3*b**2*c**2*d - 9*c*(2*a*b*d**3 + 25*b**2*c*d**2/12)/(10*d
))/(8*d))/(6*d) + x**3*(3*a**2*c**2*d + 2*a*b*c**3 - 5*c*(3*a**2*c*d**2 + 6*a*b*c**2*d + b**2*c**3 - 7*c*(a**2
*d**3 + 6*a*b*c*d**2 + 3*b**2*c**2*d - 9*c*(2*a*b*d**3 + 25*b**2*c*d**2/12)/(10*d))/(8*d))/(6*d))/(4*d) + x*(a
**2*c**3 - 3*c*(3*a**2*c**2*d + 2*a*b*c**3 - 5*c*(3*a**2*c*d**2 + 6*a*b*c**2*d + b**2*c**3 - 7*c*(a**2*d**3 +
6*a*b*c*d**2 + 3*b**2*c**2*d - 9*c*(2*a*b*d**3 + 25*b**2*c*d**2/12)/(10*d))/(8*d))/(6*d))/(4*d))/(2*d)), Ne(d,
0)), (c**(5/2)*(a**2*x**3/3 + 2*a*b*x**5/5 + b**2*x**7/7), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.28 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} x^{5}}{12 \, d} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c x^{3}}{24 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a b x^{3}}{5 \, d} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{2} x}{64 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{3} x}{384 \, d^{3}} - \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{4} x}{1536 \, d^{3}} - \frac {5 \, \sqrt {d x^{2} + c} b^{2} c^{5} x}{1024 \, d^{3}} - \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c x}{40 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c^{2} x}{80 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{3} x}{64 \, d^{2}} + \frac {3 \, \sqrt {d x^{2} + c} a b c^{4} x}{128 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} x}{8 \, d} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} c x}{48 \, d} - \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c^{2} x}{192 \, d} - \frac {5 \, \sqrt {d x^{2} + c} a^{2} c^{3} x}{128 \, d} - \frac {5 \, b^{2} c^{6} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{1024 \, d^{\frac {7}{2}}} + \frac {3 \, a b c^{5} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {5}{2}}} - \frac {5 \, a^{2} c^{4} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {3}{2}}} \]

[In]

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

[Out]

1/12*(d*x^2 + c)^(7/2)*b^2*x^5/d - 1/24*(d*x^2 + c)^(7/2)*b^2*c*x^3/d^2 + 1/5*(d*x^2 + c)^(7/2)*a*b*x^3/d + 1/
64*(d*x^2 + c)^(7/2)*b^2*c^2*x/d^3 - 1/384*(d*x^2 + c)^(5/2)*b^2*c^3*x/d^3 - 5/1536*(d*x^2 + c)^(3/2)*b^2*c^4*
x/d^3 - 5/1024*sqrt(d*x^2 + c)*b^2*c^5*x/d^3 - 3/40*(d*x^2 + c)^(7/2)*a*b*c*x/d^2 + 1/80*(d*x^2 + c)^(5/2)*a*b
*c^2*x/d^2 + 1/64*(d*x^2 + c)^(3/2)*a*b*c^3*x/d^2 + 3/128*sqrt(d*x^2 + c)*a*b*c^4*x/d^2 + 1/8*(d*x^2 + c)^(7/2
)*a^2*x/d - 1/48*(d*x^2 + c)^(5/2)*a^2*c*x/d - 5/192*(d*x^2 + c)^(3/2)*a^2*c^2*x/d - 5/128*sqrt(d*x^2 + c)*a^2
*c^3*x/d - 5/1024*b^2*c^6*arcsinh(d*x/sqrt(c*d))/d^(7/2) + 3/128*a*b*c^5*arcsinh(d*x/sqrt(c*d))/d^(5/2) - 5/12
8*a^2*c^4*arcsinh(d*x/sqrt(c*d))/d^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.94 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, b^{2} d^{2} x^{2} + \frac {25 \, b^{2} c d^{11} + 24 \, a b d^{12}}{d^{10}}\right )} x^{2} + \frac {3 \, {\left (45 \, b^{2} c^{2} d^{10} + 168 \, a b c d^{11} + 40 \, a^{2} d^{12}\right )}}{d^{10}}\right )} x^{2} + \frac {5 \, b^{2} c^{3} d^{9} + 744 \, a b c^{2} d^{10} + 680 \, a^{2} c d^{11}}{d^{10}}\right )} x^{2} - \frac {5 \, {\left (5 \, b^{2} c^{4} d^{8} - 24 \, a b c^{3} d^{9} - 472 \, a^{2} c^{2} d^{10}\right )}}{d^{10}}\right )} x^{2} + \frac {15 \, {\left (5 \, b^{2} c^{5} d^{7} - 24 \, a b c^{4} d^{8} + 40 \, a^{2} c^{3} d^{9}\right )}}{d^{10}}\right )} \sqrt {d x^{2} + c} x + \frac {{\left (5 \, b^{2} c^{6} - 24 \, a b c^{5} d + 40 \, a^{2} c^{4} d^{2}\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{1024 \, d^{\frac {7}{2}}} \]

[In]

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

[Out]

1/15360*(2*(4*(2*(8*(10*b^2*d^2*x^2 + (25*b^2*c*d^11 + 24*a*b*d^12)/d^10)*x^2 + 3*(45*b^2*c^2*d^10 + 168*a*b*c
*d^11 + 40*a^2*d^12)/d^10)*x^2 + (5*b^2*c^3*d^9 + 744*a*b*c^2*d^10 + 680*a^2*c*d^11)/d^10)*x^2 - 5*(5*b^2*c^4*
d^8 - 24*a*b*c^3*d^9 - 472*a^2*c^2*d^10)/d^10)*x^2 + 15*(5*b^2*c^5*d^7 - 24*a*b*c^4*d^8 + 40*a^2*c^3*d^9)/d^10
)*sqrt(d*x^2 + c)*x + 1/1024*(5*b^2*c^6 - 24*a*b*c^5*d + 40*a^2*c^4*d^2)*log(abs(-sqrt(d)*x + sqrt(d*x^2 + c))
)/d^(7/2)

Mupad [F(-1)]

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

[In]

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

[Out]

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