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

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

[Out]

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]  time = 0.27, antiderivative size = 278, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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])

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 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 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 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]

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 = 0.21, size = 225, normalized size = 0.80 \[ \frac {15 c^4 \left (24 a^2 d^2-24 a b c d+7 b^2 c^2\right ) \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )+\sqrt {d} x \sqrt {c+d x^2} \left (120 a^2 d^2 \left (-3 c^3+2 c^2 d x^2+24 c d^2 x^4+16 d^3 x^6\right )+24 a b d \left (15 c^4-10 c^3 d x^2+8 c^2 d^2 x^4+176 c d^3 x^6+128 d^4 x^8\right )+b^2 \left (-105 c^5+70 c^4 d x^2-56 c^3 d^2 x^4+48 c^2 d^3 x^6+1664 c d^4 x^8+1280 d^5 x^{10}\right )\right )}{15360 d^{9/2}} \]

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))

________________________________________________________________________________________

fricas [A]  time = 1.18, size = 494, normalized size = 1.76 \[ \left [\frac {15 \, {\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, a^{2} c^{4} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (1280 \, b^{2} d^{6} x^{11} + 128 \, {\left (13 \, b^{2} c d^{5} + 24 \, a b d^{6}\right )} x^{9} + 48 \, {\left (b^{2} c^{2} d^{4} + 88 \, a b c d^{5} + 40 \, a^{2} d^{6}\right )} x^{7} - 8 \, {\left (7 \, b^{2} c^{3} d^{3} - 24 \, a b c^{2} d^{4} - 360 \, a^{2} c d^{5}\right )} x^{5} + 10 \, {\left (7 \, b^{2} c^{4} d^{2} - 24 \, a b c^{3} d^{3} + 24 \, a^{2} c^{2} d^{4}\right )} x^{3} - 15 \, {\left (7 \, b^{2} c^{5} d - 24 \, a b c^{4} d^{2} + 24 \, a^{2} c^{3} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{30720 \, d^{5}}, -\frac {15 \, {\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, a^{2} c^{4} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (1280 \, b^{2} d^{6} x^{11} + 128 \, {\left (13 \, b^{2} c d^{5} + 24 \, a b d^{6}\right )} x^{9} + 48 \, {\left (b^{2} c^{2} d^{4} + 88 \, a b c d^{5} + 40 \, a^{2} d^{6}\right )} x^{7} - 8 \, {\left (7 \, b^{2} c^{3} d^{3} - 24 \, a b c^{2} d^{4} - 360 \, a^{2} c d^{5}\right )} x^{5} + 10 \, {\left (7 \, b^{2} c^{4} d^{2} - 24 \, a b c^{3} d^{3} + 24 \, a^{2} c^{2} d^{4}\right )} x^{3} - 15 \, {\left (7 \, b^{2} c^{5} d - 24 \, a b c^{4} d^{2} + 24 \, a^{2} c^{3} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{15360 \, d^{5}}\right ] \]

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]

________________________________________________________________________________________

giac [A]  time = 0.49, size = 263, normalized size = 0.94 \[ \frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, b^{2} d x^{2} + \frac {13 \, b^{2} c d^{10} + 24 \, a b d^{11}}{d^{10}}\right )} x^{2} + \frac {3 \, {\left (b^{2} c^{2} d^{9} + 88 \, a b c d^{10} + 40 \, a^{2} d^{11}\right )}}{d^{10}}\right )} x^{2} - \frac {7 \, b^{2} c^{3} d^{8} - 24 \, a b c^{2} d^{9} - 360 \, a^{2} c d^{10}}{d^{10}}\right )} x^{2} + \frac {5 \, {\left (7 \, b^{2} c^{4} d^{7} - 24 \, a b c^{3} d^{8} + 24 \, a^{2} c^{2} d^{9}\right )}}{d^{10}}\right )} x^{2} - \frac {15 \, {\left (7 \, b^{2} c^{5} d^{6} - 24 \, a b c^{4} d^{7} + 24 \, a^{2} c^{3} d^{8}\right )}}{d^{10}}\right )} \sqrt {d x^{2} + c} x - \frac {{\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, a^{2} c^{4} d^{2}\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{1024 \, d^{\frac {9}{2}}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 389, normalized size = 1.38 \[ \frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} b^{2} x^{7}}{12 d}+\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} a b \,x^{5}}{5 d}-\frac {7 \left (d \,x^{2}+c \right )^{\frac {5}{2}} b^{2} c \,x^{5}}{120 d^{2}}+\frac {3 a^{2} c^{4} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{128 d^{\frac {5}{2}}}-\frac {3 a b \,c^{5} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{128 d^{\frac {7}{2}}}+\frac {7 b^{2} c^{6} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{1024 d^{\frac {9}{2}}}+\frac {3 \sqrt {d \,x^{2}+c}\, a^{2} c^{3} x}{128 d^{2}}+\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} a^{2} x^{3}}{8 d}-\frac {3 \sqrt {d \,x^{2}+c}\, a b \,c^{4} x}{128 d^{3}}-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} a b c \,x^{3}}{8 d^{2}}+\frac {7 \sqrt {d \,x^{2}+c}\, b^{2} c^{5} x}{1024 d^{4}}+\frac {7 \left (d \,x^{2}+c \right )^{\frac {5}{2}} b^{2} c^{2} x^{3}}{192 d^{3}}+\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a^{2} c^{2} x}{64 d^{2}}-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a b \,c^{3} x}{64 d^{3}}+\frac {7 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b^{2} c^{4} x}{1536 d^{4}}-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} a^{2} c x}{16 d^{2}}+\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} a b \,c^{2} x}{16 d^{3}}-\frac {7 \left (d \,x^{2}+c \right )^{\frac {5}{2}} b^{2} c^{3} x}{384 d^{4}} \]

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(d^(1/2)*x+(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(d^(1/2)*x+(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(d^(1/2)*x+(
d*x^2+c)^(1/2))

________________________________________________________________________________________

maxima [A]  time = 1.14, size = 367, normalized size = 1.31 \[ \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} x^{7}}{12 \, d} - \frac {7 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c x^{5}}{120 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b x^{5}}{5 \, d} + \frac {7 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} x^{3}}{192 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c x^{3}}{8 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} x^{3}}{8 \, d} - \frac {7 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{3} x}{384 \, d^{4}} + \frac {7 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{4} x}{1536 \, d^{4}} + \frac {7 \, \sqrt {d x^{2} + c} b^{2} c^{5} x}{1024 \, d^{4}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c^{2} x}{16 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{3} x}{64 \, d^{3}} - \frac {3 \, \sqrt {d x^{2} + c} a b c^{4} x}{128 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} c x}{16 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c^{2} x}{64 \, d^{2}} + \frac {3 \, \sqrt {d x^{2} + c} a^{2} c^{3} x}{128 \, d^{2}} + \frac {7 \, b^{2} c^{6} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{1024 \, d^{\frac {9}{2}}} - \frac {3 \, a b c^{5} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {7}{2}}} + \frac {3 \, a^{2} c^{4} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {5}{2}}} \]

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]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [B]  time = 83.44, size = 598, normalized size = 2.13 \[ - \frac {3 a^{2} c^{\frac {7}{2}} x}{128 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a^{2} c^{\frac {5}{2}} x^{3}}{128 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {13 a^{2} c^{\frac {3}{2}} x^{5}}{64 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {5 a^{2} \sqrt {c} d x^{7}}{16 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a^{2} c^{4} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{128 d^{\frac {5}{2}}} + \frac {a^{2} d^{2} x^{9}}{8 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a b c^{\frac {9}{2}} x}{128 d^{3} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {a b c^{\frac {7}{2}} x^{3}}{128 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a b c^{\frac {5}{2}} x^{5}}{320 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {23 a b c^{\frac {3}{2}} x^{7}}{80 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {19 a b \sqrt {c} d x^{9}}{40 \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {3 a b c^{5} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{128 d^{\frac {7}{2}}} + \frac {a b d^{2} x^{11}}{5 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {7 b^{2} c^{\frac {11}{2}} x}{1024 d^{4} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {7 b^{2} c^{\frac {9}{2}} x^{3}}{3072 d^{3} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {7 b^{2} c^{\frac {7}{2}} x^{5}}{7680 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} c^{\frac {5}{2}} x^{7}}{1920 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {107 b^{2} c^{\frac {3}{2}} x^{9}}{960 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {23 b^{2} \sqrt {c} d x^{11}}{120 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {7 b^{2} c^{6} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{1024 d^{\frac {9}{2}}} + \frac {b^{2} d^{2} x^{13}}{12 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} \]

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))

________________________________________________________________________________________