\(\int x^4 (a+b x^2)^2 (c+d x^2)^{3/2} \, dx\) [613]

Optimal result

Integrand size = 24, antiderivative size = 281 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=-\frac {c^3 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) x \sqrt {c+d x^2}}{1024 d^4}+\frac {c^2 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) x^3 \sqrt {c+d x^2}}{1536 d^3}+\frac {c \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) x^5 \sqrt {c+d x^2}}{384 d^2}+\frac {\left (24 a^2 d^2+b c (7 b c-24 a d)\right ) x^5 \left (c+d x^2\right )^{3/2}}{192 d^2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {c^4 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{1024 d^{9/2}} \]

[Out]

1/192*(24*a^2*d^2+b*c*(-24*a*d+7*b*c))*x^5*(d*x^2+c)^(3/2)/d^2-1/120*b*(-24*a*d+7*b*c)*x^5*(d*x^2+c)^(5/2)/d^2
+1/12*b^2*x^7*(d*x^2+c)^(5/2)/d+1/1024*c^4*(24*a^2*d^2+b*c*(-24*a*d+7*b*c))*arctanh(x*d^(1/2)/(d*x^2+c)^(1/2))
/d^(9/2)-1/1024*c^3*(24*a^2*d^2+b*c*(-24*a*d+7*b*c))*x*(d*x^2+c)^(1/2)/d^4+1/1536*c^2*(24*a^2*d^2+b*c*(-24*a*d
+7*b*c))*x^3*(d*x^2+c)^(1/2)/d^3+1/384*c*(24*a^2*d^2+b*c*(-24*a*d+7*b*c))*x^5*(d*x^2+c)^(1/2)/d^2

Rubi [A] (verified)

Time = 0.19 (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} \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\frac {c^4 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{1024 d^{9/2}}-\frac {c^3 x \sqrt {c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1024 d^4}+\frac {c^2 x^3 \sqrt {c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1536 d^3}+\frac {1}{192} x^5 \left (c+d x^2\right )^{3/2} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right )+\frac {c x^5 \sqrt {c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{384 d^2}-\frac {b x^5 \left (c+d x^2\right )^{5/2} (7 b c-24 a d)}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d} \]

[In]

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

[Out]

-1/1024*(c^3*(24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*x*Sqrt[c + d*x^2])/d^4 + (c^2*(24*a^2*d^2 + b*c*(7*b*c - 24*a
*d))*x^3*Sqrt[c + d*x^2])/(1536*d^3) + (c*(24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*x^5*Sqrt[c + d*x^2])/(384*d^2) +
((24*a^2 + (b*c*(7*b*c - 24*a*d))/d^2)*x^5*(c + d*x^2)^(3/2))/192 - (b*(7*b*c - 24*a*d)*x^5*(c + d*x^2)^(5/2)
)/(120*d^2) + (b^2*x^7*(c + d*x^2)^(5/2))/(12*d) + (c^4*(24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*ArcTanh[(Sqrt[d]*x
)/Sqrt[c + d*x^2]])/(1024*d^(9/2))

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^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {\int x^4 \left (c+d x^2\right )^{3/2} \left (12 a^2 d-b (7 b c-24 a d) x^2\right ) \, dx}{12 d} \\ & = -\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {1}{24} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) \int x^4 \left (c+d x^2\right )^{3/2} \, dx \\ & = \frac {1}{192} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \left (c+d x^2\right )^{3/2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {1}{64} \left (c \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right )\right ) \int x^4 \sqrt {c+d x^2} \, dx \\ & = \frac {1}{384} c \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \sqrt {c+d x^2}+\frac {1}{192} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \left (c+d x^2\right )^{3/2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {1}{384} \left (c^2 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right )\right ) \int \frac {x^4}{\sqrt {c+d x^2}} \, dx \\ & = \frac {c^2 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}}{1536 d}+\frac {1}{384} c \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \sqrt {c+d x^2}+\frac {1}{192} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \left (c+d x^2\right )^{3/2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}-\frac {\left (c^3 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2}} \, dx}{512 d} \\ & = -\frac {c^3 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{1024 d^2}+\frac {c^2 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}}{1536 d}+\frac {1}{384} c \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \sqrt {c+d x^2}+\frac {1}{192} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \left (c+d x^2\right )^{3/2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {\left (c^4 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{1024 d^2} \\ & = -\frac {c^3 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{1024 d^2}+\frac {c^2 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}}{1536 d}+\frac {1}{384} c \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \sqrt {c+d x^2}+\frac {1}{192} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \left (c+d x^2\right )^{3/2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {\left (c^4 \left (24 a^2+\frac {b c (7 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^2} \\ & = -\frac {c^3 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{1024 d^2}+\frac {c^2 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}}{1536 d}+\frac {1}{384} c \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \sqrt {c+d x^2}+\frac {1}{192} \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) x^5 \left (c+d x^2\right )^{3/2}-\frac {b (7 b c-24 a d) x^5 \left (c+d x^2\right )^{5/2}}{120 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d}+\frac {c^4 \left (24 a^2+\frac {b c (7 b c-24 a d)}{d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{1024 d^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.92 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\frac {x \sqrt {c+d x^2} \left (-105 b^2 c^5+360 a b c^4 d-360 a^2 c^3 d^2+70 b^2 c^4 d x^2-240 a b c^3 d^2 x^2+240 a^2 c^2 d^3 x^2-56 b^2 c^3 d^2 x^4+192 a b c^2 d^3 x^4+2880 a^2 c d^4 x^4+48 b^2 c^2 d^3 x^6+4224 a b c d^4 x^6+1920 a^2 d^5 x^6+1664 b^2 c d^4 x^8+3072 a b d^5 x^8+1280 b^2 d^5 x^{10}\right )}{15360 d^4}+\frac {c^4 \left (7 b^2 c^2-24 a b c d+24 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{-\sqrt {c}+\sqrt {c+d x^2}}\right )}{512 d^{9/2}} \]

[In]

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

[Out]

(x*Sqrt[c + d*x^2]*(-105*b^2*c^5 + 360*a*b*c^4*d - 360*a^2*c^3*d^2 + 70*b^2*c^4*d*x^2 - 240*a*b*c^3*d^2*x^2 +
240*a^2*c^2*d^3*x^2 - 56*b^2*c^3*d^2*x^4 + 192*a*b*c^2*d^3*x^4 + 2880*a^2*c*d^4*x^4 + 48*b^2*c^2*d^3*x^6 + 422
4*a*b*c*d^4*x^6 + 1920*a^2*d^5*x^6 + 1664*b^2*c*d^4*x^8 + 3072*a*b*d^5*x^8 + 1280*b^2*d^5*x^10))/(15360*d^4) +
(c^4*(7*b^2*c^2 - 24*a*b*c*d + 24*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/(512*d^(9/2))

Maple [A] (verified)

Time = 3.05 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.72

method	result	size
pseudoelliptic	\(-\frac {3 \left (\left (-a^{2} c^{4} d^{2}+a b \,c^{5} d -\frac {7}{24} b^{2} c^{6}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )+x \left (-\frac {16 x^{6} \left (\frac {2}{3} b^{2} x^{4}+\frac {8}{5} a b \,x^{2}+a^{2}\right ) d^{\frac {11}{2}}}{3}+c \left (c^{2} \left (\frac {7}{45} b^{2} x^{4}+\frac {2}{3} a b \,x^{2}+a^{2}\right ) d^{\frac {5}{2}}-\frac {2 x^{2} \left (\frac {1}{5} b^{2} x^{4}+\frac {4}{5} a b \,x^{2}+a^{2}\right ) c \,d^{\frac {7}{2}}}{3}-8 x^{4} \left (\frac {26}{45} b^{2} x^{4}+\frac {22}{15} a b \,x^{2}+a^{2}\right ) d^{\frac {9}{2}}-\left (\left (\frac {7 b \,x^{2}}{36}+a \right ) d^{\frac {3}{2}}-\frac {7 b \sqrt {d}\, c}{24}\right ) b \,c^{3}\right )\right ) \sqrt {d \,x^{2}+c}\right )}{128 d^{\frac {9}{2}}}\)	\(201\)
risch	\(-\frac {x \left (-1280 b^{2} d^{5} x^{10}-3072 a b \,d^{5} x^{8}-1664 b^{2} c \,d^{4} x^{8}-1920 a^{2} d^{5} x^{6}-4224 a b c \,d^{4} x^{6}-48 b^{2} c^{2} d^{3} x^{6}-2880 a^{2} c \,d^{4} x^{4}-192 a b \,c^{2} d^{3} x^{4}+56 b^{2} c^{3} d^{2} x^{4}-240 a^{2} c^{2} d^{3} x^{2}+240 a b \,c^{3} d^{2} x^{2}-70 b^{2} c^{4} d \,x^{2}+360 a^{2} c^{3} d^{2}-360 a b \,c^{4} d +105 b^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{15360 d^{4}}+\frac {c^{4} \left (24 a^{2} d^{2}-24 a b c d +7 b^{2} c^{2}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{1024 d^{\frac {9}{2}}}\)	\(239\)
default	\(b^{2} \left (\frac {x^{7} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{12 d}-\frac {7 c \left (\frac {x^{5} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{10 d}-\frac {c \left (\frac {x^{3} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{8 d}-\frac {3 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6 d}\right )}{8 d}\right )}{2 d}\right )}{12 d}\right )+a^{2} \left (\frac {x^{3} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{8 d}-\frac {3 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6 d}\right )}{8 d}\right )+2 a b \left (\frac {x^{5} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{10 d}-\frac {c \left (\frac {x^{3} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{8 d}-\frac {3 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6 d}\right )}{8 d}\right )}{2 d}\right )\)	\(377\)

[In]

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

[Out]

-3/128/d^(9/2)*((-a^2*c^4*d^2+a*b*c^5*d-7/24*b^2*c^6)*arctanh((d*x^2+c)^(1/2)/x/d^(1/2))+x*(-16/3*x^6*(2/3*b^2
*x^4+8/5*a*b*x^2+a^2)*d^(11/2)+c*(c^2*(7/45*b^2*x^4+2/3*a*b*x^2+a^2)*d^(5/2)-2/3*x^2*(1/5*b^2*x^4+4/5*a*b*x^2+
a^2)*c*d^(7/2)-8*x^4*(26/45*b^2*x^4+22/15*a*b*x^2+a^2)*d^(9/2)-((7/36*b*x^2+a)*d^(3/2)-7/24*b*d^(1/2)*c)*b*c^3
))*(d*x^2+c)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.76 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\left [\frac {15 \, {\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, a^{2} c^{4} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (1280 \, b^{2} d^{6} x^{11} + 128 \, {\left (13 \, b^{2} c d^{5} + 24 \, a b d^{6}\right )} x^{9} + 48 \, {\left (b^{2} c^{2} d^{4} + 88 \, a b c d^{5} + 40 \, a^{2} d^{6}\right )} x^{7} - 8 \, {\left (7 \, b^{2} c^{3} d^{3} - 24 \, a b c^{2} d^{4} - 360 \, a^{2} c d^{5}\right )} x^{5} + 10 \, {\left (7 \, b^{2} c^{4} d^{2} - 24 \, a b c^{3} d^{3} + 24 \, a^{2} c^{2} d^{4}\right )} x^{3} - 15 \, {\left (7 \, b^{2} c^{5} d - 24 \, a b c^{4} d^{2} + 24 \, a^{2} c^{3} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{30720 \, d^{5}}, -\frac {15 \, {\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, a^{2} c^{4} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (1280 \, b^{2} d^{6} x^{11} + 128 \, {\left (13 \, b^{2} c d^{5} + 24 \, a b d^{6}\right )} x^{9} + 48 \, {\left (b^{2} c^{2} d^{4} + 88 \, a b c d^{5} + 40 \, a^{2} d^{6}\right )} x^{7} - 8 \, {\left (7 \, b^{2} c^{3} d^{3} - 24 \, a b c^{2} d^{4} - 360 \, a^{2} c d^{5}\right )} x^{5} + 10 \, {\left (7 \, b^{2} c^{4} d^{2} - 24 \, a b c^{3} d^{3} + 24 \, a^{2} c^{2} d^{4}\right )} x^{3} - 15 \, {\left (7 \, b^{2} c^{5} d - 24 \, a b c^{4} d^{2} + 24 \, a^{2} c^{3} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{15360 \, d^{5}}\right ] \]

[In]

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

[Out]

[1/30720*(15*(7*b^2*c^6 - 24*a*b*c^5*d + 24*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*(13*b^2*c*d^5 + 24*a*b*d^6)*x^9 + 48*(b^2*c^2*d^4 + 88*a*b*c*d^5 + 40*a^2*d^6)
*x^7 - 8*(7*b^2*c^3*d^3 - 24*a*b*c^2*d^4 - 360*a^2*c*d^5)*x^5 + 10*(7*b^2*c^4*d^2 - 24*a*b*c^3*d^3 + 24*a^2*c^
2*d^4)*x^3 - 15*(7*b^2*c^5*d - 24*a*b*c^4*d^2 + 24*a^2*c^3*d^3)*x)*sqrt(d*x^2 + c))/d^5, -1/15360*(15*(7*b^2*c
^6 - 24*a*b*c^5*d + 24*a^2*c^4*d^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - (1280*b^2*d^6*x^11 + 128*(13
*b^2*c*d^5 + 24*a*b*d^6)*x^9 + 48*(b^2*c^2*d^4 + 88*a*b*c*d^5 + 40*a^2*d^6)*x^7 - 8*(7*b^2*c^3*d^3 - 24*a*b*c^
2*d^4 - 360*a^2*c*d^5)*x^5 + 10*(7*b^2*c^4*d^2 - 24*a*b*c^3*d^3 + 24*a^2*c^2*d^4)*x^3 - 15*(7*b^2*c^5*d - 24*a
*b*c^4*d^2 + 24*a^2*c^3*d^3)*x)*sqrt(d*x^2 + c))/d^5]

Sympy [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.90 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\begin {cases} \frac {3 c^{2} \left (a^{2} c^{2} - \frac {5 c \left (2 a^{2} c d + 2 a b c^{2} - \frac {7 c \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2} - \frac {9 c \left (2 a b d^{2} + \frac {13 b^{2} c d}{12}\right )}{10 d}\right )}{8 d}\right )}{6 d}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {d} \sqrt {c + d x^{2}} + 2 d x \right )}}{\sqrt {d}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {d x^{2}}} & \text {otherwise} \end {cases}\right )}{8 d^{2}} + \sqrt {c + d x^{2}} \left (\frac {b^{2} d x^{11}}{12} - \frac {3 c x \left (a^{2} c^{2} - \frac {5 c \left (2 a^{2} c d + 2 a b c^{2} - \frac {7 c \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2} - \frac {9 c \left (2 a b d^{2} + \frac {13 b^{2} c d}{12}\right )}{10 d}\right )}{8 d}\right )}{6 d}\right )}{8 d^{2}} + \frac {x^{9} \cdot \left (2 a b d^{2} + \frac {13 b^{2} c d}{12}\right )}{10 d} + \frac {x^{7} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2} - \frac {9 c \left (2 a b d^{2} + \frac {13 b^{2} c d}{12}\right )}{10 d}\right )}{8 d} + \frac {x^{5} \cdot \left (2 a^{2} c d + 2 a b c^{2} - \frac {7 c \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2} - \frac {9 c \left (2 a b d^{2} + \frac {13 b^{2} c d}{12}\right )}{10 d}\right )}{8 d}\right )}{6 d} + \frac {x^{3} \left (a^{2} c^{2} - \frac {5 c \left (2 a^{2} c d + 2 a b c^{2} - \frac {7 c \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2} - \frac {9 c \left (2 a b d^{2} + \frac {13 b^{2} c d}{12}\right )}{10 d}\right )}{8 d}\right )}{6 d}\right )}{4 d}\right ) & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (\frac {a^{2} x^{5}}{5} + \frac {2 a b x^{7}}{7} + \frac {b^{2} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((3*c**2*(a**2*c**2 - 5*c*(2*a**2*c*d + 2*a*b*c**2 - 7*c*(a**2*d**2 + 4*a*b*c*d + b**2*c**2 - 9*c*(2*
a*b*d**2 + 13*b**2*c*d/12)/(10*d))/(8*d))/(6*d))*Piecewise((log(2*sqrt(d)*sqrt(c + d*x**2) + 2*d*x)/sqrt(d), N
e(c, 0)), (x*log(x)/sqrt(d*x**2), True))/(8*d**2) + sqrt(c + d*x**2)*(b**2*d*x**11/12 - 3*c*x*(a**2*c**2 - 5*c
*(2*a**2*c*d + 2*a*b*c**2 - 7*c*(a**2*d**2 + 4*a*b*c*d + b**2*c**2 - 9*c*(2*a*b*d**2 + 13*b**2*c*d/12)/(10*d))
/(8*d))/(6*d))/(8*d**2) + x**9*(2*a*b*d**2 + 13*b**2*c*d/12)/(10*d) + x**7*(a**2*d**2 + 4*a*b*c*d + b**2*c**2
- 9*c*(2*a*b*d**2 + 13*b**2*c*d/12)/(10*d))/(8*d) + x**5*(2*a**2*c*d + 2*a*b*c**2 - 7*c*(a**2*d**2 + 4*a*b*c*d
+ b**2*c**2 - 9*c*(2*a*b*d**2 + 13*b**2*c*d/12)/(10*d))/(8*d))/(6*d) + x**3*(a**2*c**2 - 5*c*(2*a**2*c*d + 2*
a*b*c**2 - 7*c*(a**2*d**2 + 4*a*b*c*d + b**2*c**2 - 9*c*(2*a*b*d**2 + 13*b**2*c*d/12)/(10*d))/(8*d))/(6*d))/(4
*d)), Ne(d, 0)), (c**(3/2)*(a**2*x**5/5 + 2*a*b*x**7/7 + b**2*x**9/9), True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.31 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} x^{7}}{12 \, d} - \frac {7 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c x^{5}}{120 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b x^{5}}{5 \, d} + \frac {7 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} x^{3}}{192 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c x^{3}}{8 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} x^{3}}{8 \, d} - \frac {7 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{3} x}{384 \, d^{4}} + \frac {7 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{4} x}{1536 \, d^{4}} + \frac {7 \, \sqrt {d x^{2} + c} b^{2} c^{5} x}{1024 \, d^{4}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c^{2} x}{16 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{3} x}{64 \, d^{3}} - \frac {3 \, \sqrt {d x^{2} + c} a b c^{4} x}{128 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} c x}{16 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c^{2} x}{64 \, d^{2}} + \frac {3 \, \sqrt {d x^{2} + c} a^{2} c^{3} x}{128 \, d^{2}} + \frac {7 \, b^{2} c^{6} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{1024 \, d^{\frac {9}{2}}} - \frac {3 \, a b c^{5} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {7}{2}}} + \frac {3 \, a^{2} c^{4} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {5}{2}}} \]

[In]

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

[Out]

1/12*(d*x^2 + c)^(5/2)*b^2*x^7/d - 7/120*(d*x^2 + c)^(5/2)*b^2*c*x^5/d^2 + 1/5*(d*x^2 + c)^(5/2)*a*b*x^5/d + 7
/192*(d*x^2 + c)^(5/2)*b^2*c^2*x^3/d^3 - 1/8*(d*x^2 + c)^(5/2)*a*b*c*x^3/d^2 + 1/8*(d*x^2 + c)^(5/2)*a^2*x^3/d
- 7/384*(d*x^2 + c)^(5/2)*b^2*c^3*x/d^4 + 7/1536*(d*x^2 + c)^(3/2)*b^2*c^4*x/d^4 + 7/1024*sqrt(d*x^2 + c)*b^2
*c^5*x/d^4 + 1/16*(d*x^2 + c)^(5/2)*a*b*c^2*x/d^3 - 1/64*(d*x^2 + c)^(3/2)*a*b*c^3*x/d^3 - 3/128*sqrt(d*x^2 +
c)*a*b*c^4*x/d^3 - 1/16*(d*x^2 + c)^(5/2)*a^2*c*x/d^2 + 1/64*(d*x^2 + c)^(3/2)*a^2*c^2*x/d^2 + 3/128*sqrt(d*x^
2 + c)*a^2*c^3*x/d^2 + 7/1024*b^2*c^6*arcsinh(d*x/sqrt(c*d))/d^(9/2) - 3/128*a*b*c^5*arcsinh(d*x/sqrt(c*d))/d^
(7/2) + 3/128*a^2*c^4*arcsinh(d*x/sqrt(c*d))/d^(5/2)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.94 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, b^{2} d x^{2} + \frac {13 \, b^{2} c d^{10} + 24 \, a b d^{11}}{d^{10}}\right )} x^{2} + \frac {3 \, {\left (b^{2} c^{2} d^{9} + 88 \, a b c d^{10} + 40 \, a^{2} d^{11}\right )}}{d^{10}}\right )} x^{2} - \frac {7 \, b^{2} c^{3} d^{8} - 24 \, a b c^{2} d^{9} - 360 \, a^{2} c d^{10}}{d^{10}}\right )} x^{2} + \frac {5 \, {\left (7 \, b^{2} c^{4} d^{7} - 24 \, a b c^{3} d^{8} + 24 \, a^{2} c^{2} d^{9}\right )}}{d^{10}}\right )} x^{2} - \frac {15 \, {\left (7 \, b^{2} c^{5} d^{6} - 24 \, a b c^{4} d^{7} + 24 \, a^{2} c^{3} d^{8}\right )}}{d^{10}}\right )} \sqrt {d x^{2} + c} x - \frac {{\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, a^{2} c^{4} d^{2}\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{1024 \, d^{\frac {9}{2}}} \]

[In]

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

[Out]

1/15360*(2*(4*(2*(8*(10*b^2*d*x^2 + (13*b^2*c*d^10 + 24*a*b*d^11)/d^10)*x^2 + 3*(b^2*c^2*d^9 + 88*a*b*c*d^10 +
40*a^2*d^11)/d^10)*x^2 - (7*b^2*c^3*d^8 - 24*a*b*c^2*d^9 - 360*a^2*c*d^10)/d^10)*x^2 + 5*(7*b^2*c^4*d^7 - 24*
a*b*c^3*d^8 + 24*a^2*c^2*d^9)/d^10)*x^2 - 15*(7*b^2*c^5*d^6 - 24*a*b*c^4*d^7 + 24*a^2*c^3*d^8)/d^10)*sqrt(d*x^
2 + c)*x - 1/1024*(7*b^2*c^6 - 24*a*b*c^5*d + 24*a^2*c^4*d^2)*log(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(9/2)

Mupad [F(-1)]

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

[In]

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

[Out]

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