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

Optimal. Leaf size=281 \[ \frac{c^2 x^3 \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1536 d^3}-\frac{c^3 x \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1024 d^4}+\frac{c^4 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{1024 d^{9/2}}+\frac{x^5 \left (c+d x^2\right )^{3/2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{192 d^2}+\frac{c x^5 \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{384 d^2}-\frac{b x^5 \left (c+d x^2\right )^{5/2} (7 b c-24 a d)}{120 d^2}+\frac{b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d} \]

[Out]

-(c^3*(24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*x*Sqrt[c + d*x^2])/(1024*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*d^2 + b*c*(7*b*c - 24*a*d))*x^5*(c + d*x^2)^(3/2))/(192*d^2) - (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))

________________________________________________________________________________________

Rubi [A]  time = 0.266162, antiderivative size = 278, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {464, 459, 279, 321, 217, 206} \[ \frac{c^2 x^3 \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1536 d^3}-\frac{c^3 x \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1024 d^4}+\frac{c^4 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{1024 d^{9/2}}+\frac{1}{192} x^5 \left (c+d x^2\right )^{3/2} \left (24 a^2+\frac{b c (7 b c-24 a d)}{d^2}\right )+\frac{c x^5 \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{384 d^2}-\frac{b x^5 \left (c+d x^2\right )^{5/2} (7 b c-24 a d)}{120 d^2}+\frac{b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^3*(24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*x*Sqrt[c + d*x^2])/(1024*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 464

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]

Rule 459

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 279

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 321

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 217

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx &=\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 ) \operatorname{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]  time = 0.142605, size = 225, normalized size = 0.8 \[ \frac{\sqrt{d} x \sqrt{c+d x^2} \left (120 a^2 d^2 \left (2 c^2 d x^2-3 c^3+24 c d^2 x^4+16 d^3 x^6\right )+24 a b d \left (8 c^2 d^2 x^4-10 c^3 d x^2+15 c^4+176 c d^3 x^6+128 d^4 x^8\right )+b^2 \left (48 c^2 d^3 x^6-56 c^3 d^2 x^4+70 c^4 d x^2-105 c^5+1664 c d^4 x^8+1280 d^5 x^{10}\right )\right )+15 c^4 \left (24 a^2 d^2-24 a b c d+7 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{15360 d^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d]*x*Sqrt[c + d*x^2]*(120*a^2*d^2*(-3*c^3 + 2*c^2*d*x^2 + 24*c*d^2*x^4 + 16*d^3*x^6) + 24*a*b*d*(15*c^4
- 10*c^3*d*x^2 + 8*c^2*d^2*x^4 + 176*c*d^3*x^6 + 128*d^4*x^8) + b^2*(-105*c^5 + 70*c^4*d*x^2 - 56*c^3*d^2*x^4
+ 48*c^2*d^3*x^6 + 1664*c*d^4*x^8 + 1280*d^5*x^10)) + 15*c^4*(7*b^2*c^2 - 24*a*b*c*d + 24*a^2*d^2)*Log[d*x + S
qrt[d]*Sqrt[c + d*x^2]])/(15360*d^(9/2))

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Maple [A]  time = 0.02, size = 389, normalized size = 1.4 \begin{align*}{\frac{{b}^{2}{x}^{7}}{12\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{b}^{2}c{x}^{5}}{120\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{x}^{3}{b}^{2}{c}^{2}}{192\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{b}^{2}{c}^{3}x}{384\,{d}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{b}^{2}{c}^{4}x}{1536\,{d}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{2}{c}^{5}x}{1024\,{d}^{4}}\sqrt{d{x}^{2}+c}}+{\frac{7\,{b}^{2}{c}^{6}}{1024}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{9}{2}}}}+{\frac{ab{x}^{5}}{5\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{abc{x}^{3}}{8\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{ab{c}^{2}x}{16\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{ab{c}^{3}x}{64\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,ab{c}^{4}x}{128\,{d}^{3}}\sqrt{d{x}^{2}+c}}-{\frac{3\,ab{c}^{5}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{7}{2}}}}+{\frac{{a}^{2}{x}^{3}}{8\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}cx}{16\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}{c}^{2}x}{64\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{2}{c}^{3}x}{128\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,{a}^{2}{c}^{4}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^2*x^7*(d*x^2+c)^(5/2)/d-7/120*b^2*c/d^2*x^5*(d*x^2+c)^(5/2)+7/192*b^2*c^2/d^3*x^3*(d*x^2+c)^(5/2)-7/384
*b^2*c^3/d^4*x*(d*x^2+c)^(5/2)+7/1536*b^2*c^4/d^4*x*(d*x^2+c)^(3/2)+7/1024*b^2*c^5/d^4*x*(d*x^2+c)^(1/2)+7/102
4*b^2*c^6/d^(9/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/5*a*b*x^5*(d*x^2+c)^(5/2)/d-1/8*a*b*c/d^2*x^3*(d*x^2+c)^(5/2
)+1/16*a*b*c^2/d^3*x*(d*x^2+c)^(5/2)-1/64*a*b*c^3/d^3*x*(d*x^2+c)^(3/2)-3/128*a*b*c^4/d^3*x*(d*x^2+c)^(1/2)-3/
128*a*b*c^5/d^(7/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/8*a^2*x^3*(d*x^2+c)^(5/2)/d-1/16*a^2*c/d^2*x*(d*x^2+c)^(5/
2)+1/64*a^2*c^2/d^2*x*(d*x^2+c)^(3/2)+3/128*a^2*c^3/d^2*x*(d*x^2+c)^(1/2)+3/128*a^2*c^4/d^(5/2)*ln(x*d^(1/2)+(
d*x^2+c)^(1/2))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.38041, size = 1116, normalized size = 3.97 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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Sympy [B]  time = 62.2183, size = 598, normalized size = 2.13 \begin{align*} - \frac{3 a^{2} c^{\frac{7}{2}} x}{128 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a^{2} c^{\frac{5}{2}} x^{3}}{128 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{13 a^{2} c^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 a^{2} \sqrt{c} d x^{7}}{16 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a^{2} c^{4} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{128 d^{\frac{5}{2}}} + \frac{a^{2} d^{2} x^{9}}{8 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a b c^{\frac{9}{2}} x}{128 d^{3} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{a b c^{\frac{7}{2}} x^{3}}{128 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a b c^{\frac{5}{2}} x^{5}}{320 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{23 a b c^{\frac{3}{2}} x^{7}}{80 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{19 a b \sqrt{c} d x^{9}}{40 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{3 a b c^{5} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{128 d^{\frac{7}{2}}} + \frac{a b d^{2} x^{11}}{5 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{7 b^{2} c^{\frac{11}{2}} x}{1024 d^{4} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{7 b^{2} c^{\frac{9}{2}} x^{3}}{3072 d^{3} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{7 b^{2} c^{\frac{7}{2}} x^{5}}{7680 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{\frac{5}{2}} x^{7}}{1920 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{107 b^{2} c^{\frac{3}{2}} x^{9}}{960 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{23 b^{2} \sqrt{c} d x^{11}}{120 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{7 b^{2} c^{6} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{1024 d^{\frac{9}{2}}} + \frac{b^{2} d^{2} x^{13}}{12 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a**2*c**(7/2)*x/(128*d**2*sqrt(1 + d*x**2/c)) - a**2*c**(5/2)*x**3/(128*d*sqrt(1 + d*x**2/c)) + 13*a**2*c**
(3/2)*x**5/(64*sqrt(1 + d*x**2/c)) + 5*a**2*sqrt(c)*d*x**7/(16*sqrt(1 + d*x**2/c)) + 3*a**2*c**4*asinh(sqrt(d)
*x/sqrt(c))/(128*d**(5/2)) + a**2*d**2*x**9/(8*sqrt(c)*sqrt(1 + d*x**2/c)) + 3*a*b*c**(9/2)*x/(128*d**3*sqrt(1
+ d*x**2/c)) + a*b*c**(7/2)*x**3/(128*d**2*sqrt(1 + d*x**2/c)) - a*b*c**(5/2)*x**5/(320*d*sqrt(1 + d*x**2/c))
+ 23*a*b*c**(3/2)*x**7/(80*sqrt(1 + d*x**2/c)) + 19*a*b*sqrt(c)*d*x**9/(40*sqrt(1 + d*x**2/c)) - 3*a*b*c**5*a
sinh(sqrt(d)*x/sqrt(c))/(128*d**(7/2)) + a*b*d**2*x**11/(5*sqrt(c)*sqrt(1 + d*x**2/c)) - 7*b**2*c**(11/2)*x/(1
024*d**4*sqrt(1 + d*x**2/c)) - 7*b**2*c**(9/2)*x**3/(3072*d**3*sqrt(1 + d*x**2/c)) + 7*b**2*c**(7/2)*x**5/(768
0*d**2*sqrt(1 + d*x**2/c)) - b**2*c**(5/2)*x**7/(1920*d*sqrt(1 + d*x**2/c)) + 107*b**2*c**(3/2)*x**9/(960*sqrt
(1 + d*x**2/c)) + 23*b**2*sqrt(c)*d*x**11/(120*sqrt(1 + d*x**2/c)) + 7*b**2*c**6*asinh(sqrt(d)*x/sqrt(c))/(102
4*d**(9/2)) + b**2*d**2*x**13/(12*sqrt(c)*sqrt(1 + d*x**2/c))

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Giac [A]  time = 1.15109, size = 355, normalized size = 1.26 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)