3.8.76 \(\int \frac {x^4}{(a+b x^2)^2 (c+d x^2)^{5/2}} \, dx\) [776]

Optimal. Leaf size=174 \[ \frac {(2 b c+3 a d) x}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(4 b c+11 a d) x}{6 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\sqrt {a} (3 b c+2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 (b c-a d)^{7/2}} \]

[Out]

1/6*(3*a*d+2*b*c)*x/b/(-a*d+b*c)^2/(d*x^2+c)^(3/2)+1/2*a*x/b/(-a*d+b*c)/(b*x^2+a)/(d*x^2+c)^(3/2)-1/2*(2*a*d+3
*b*c)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))*a^(1/2)/(-a*d+b*c)^(7/2)+1/6*(11*a*d+4*b*c)*x/(-a*d+b
*c)^3/(d*x^2+c)^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {481, 541, 12, 385, 211} \begin {gather*} -\frac {\sqrt {a} (2 a d+3 b c) \text {ArcTan}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 (b c-a d)^{7/2}}+\frac {x (11 a d+4 b c)}{6 \sqrt {c+d x^2} (b c-a d)^3}+\frac {x (3 a d+2 b c)}{6 b \left (c+d x^2\right )^{3/2} (b c-a d)^2}+\frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*b*c + 3*a*d)*x)/(6*b*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + (a*x)/(2*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/
2)) + ((4*b*c + 11*a*d)*x)/(6*(b*c - a*d)^3*Sqrt[c + d*x^2]) - (Sqrt[a]*(3*b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d
]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*(b*c - a*d)^(7/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 211

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

Rule 385

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

Rule 481

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

Rule 541

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

Rubi steps

\begin {align*} \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx &=\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {a c-2 (b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx}{2 b (b c-a d)}\\ &=\frac {(2 b c+3 a d) x}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {5 a b c^2-2 b c (2 b c+3 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{6 b c (b c-a d)^2}\\ &=\frac {(2 b c+3 a d) x}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(4 b c+11 a d) x}{6 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\int \frac {3 a b c^2 (3 b c+2 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 b c^2 (b c-a d)^3}\\ &=\frac {(2 b c+3 a d) x}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(4 b c+11 a d) x}{6 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {(a (3 b c+2 a d)) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 (b c-a d)^3}\\ &=\frac {(2 b c+3 a d) x}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(4 b c+11 a d) x}{6 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {(a (3 b c+2 a d)) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 (b c-a d)^3}\\ &=\frac {(2 b c+3 a d) x}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(4 b c+11 a d) x}{6 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\sqrt {a} (3 b c+2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 (b c-a d)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 1.14, size = 176, normalized size = 1.01 \begin {gather*} \frac {1}{6} \left (\frac {x \left (2 b^2 c x^2 \left (3 c+2 d x^2\right )+2 a^2 d \left (3 c+4 d x^2\right )+a b \left (9 c^2+16 c d x^2+11 d^2 x^4\right )\right )}{(b c-a d)^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {3 \sqrt {a} (3 b c+2 a d) \tan ^{-1}\left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{(b c-a d)^{7/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x*(2*b^2*c*x^2*(3*c + 2*d*x^2) + 2*a^2*d*(3*c + 4*d*x^2) + a*b*(9*c^2 + 16*c*d*x^2 + 11*d^2*x^4)))/((b*c - a
*d)^3*(a + b*x^2)*(c + d*x^2)^(3/2)) + (3*Sqrt[a]*(3*b*c + 2*a*d)*ArcTan[(a*Sqrt[d] + b*x*(Sqrt[d]*x - Sqrt[c
+ d*x^2]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(b*c - a*d)^(7/2))/6

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3521\) vs. \(2(150)=300\).
time = 0.10, size = 3522, normalized size = 20.24

method	result	size
default	\(\text {Expression too large to display}\)	\(3522\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^2*(1/3*x/c/(d*x^2+c)^(3/2)+2/3*x/c^2/(d*x^2+c)^(1/2))-1/4*a/b^3*(1/(a*d-b*c)*b/(x+1/b*(-a*b)^(1/2))/(d*(x+
1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-5*d*(-a*b)^(1/2)/(a*d-b*c)*(-1/
3/(a*d-b*c)*b/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-d*(-a*b)^(1
/2)/(a*d-b*c)*(2/3*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x+1/b*(-a*b)
^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+16/3*d/(-4*d*(a*d-b*c)/b+4*d^2*a/b)^2*(2*
d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1/2)/b)/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(
a*d-b*c)/b)^(1/2))-1/(a*d-b*c)*b*(-1/(a*d-b*c)*b/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1
/2))-(a*d-b*c)/b)^(1/2)-2*d*(-a*b)^(1/2)/(a*d-b*c)*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*
c)/b+4*d^2*a/b)/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/(a*d-b*
c)*b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*(d
*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))))+4*
d/(a*d-b*c)*b*(2/3*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x+1/b*(-a*b)
^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+16/3*d/(-4*d*(a*d-b*c)/b+4*d^2*a/b)^2*(2*
d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1/2)/b)/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(
a*d-b*c)/b)^(1/2)))-3/4/b^2*a/(-a*b)^(1/2)*(-1/3/(a*d-b*c)*b/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1
/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+d*(-a*b)^(1/2)/(a*d-b*c)*(2/3*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b
)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^
(3/2)+16/3*d/(-4*d*(a*d-b*c)/b+4*d^2*a/b)^2*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/(d*(x-1/b*(-a*b)^(1/
2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))-1/(a*d-b*c)*b*(-1/(a*d-b*c)*b/(d*(x-1/b*(-a*
b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+2*d*(-a*b)^(1/2)/(a*d-b*c)*(2*d*(x-1/b*
(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x
-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/(a*d-b*c)*b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b
*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))
-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))))+3/4/b^2*a/(-a*b)^(1/2)*(-1/3/(a*d-b*c)*b/(d*(x+1/b*(-a*b)^(1/2))^
2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-d*(-a*b)^(1/2)/(a*d-b*c)*(2/3*(2*d*(x+1/b*(-a*b)^
(1/2))-2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-
a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+16/3*d/(-4*d*(a*d-b*c)/b+4*d^2*a/b)^2*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1/2
)/b)/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))-1/(a*d-b*c)*b*(-1/(
a*d-b*c)*b/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-2*d*(-a*b)^(1/
2)/(a*d-b*c)*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x+1/b*(-a*b)^(1/2)
)^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/(a*d-b*c)*b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-
b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/
2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))))-1/4*a/b^3*(1/(a*d-b*c)*b/(x-1/b*(-a*b)^(
1/2))/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+5*d*(-a*b)^(1/2)/(a
*d-b*c)*(-1/3/(a*d-b*c)*b/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)
+d*(-a*b)^(1/2)/(a*d-b*c)*(2/3*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(
x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+16/3*d/(-4*d*(a*d-b*c)/b+4*d^
2*a/b)^2*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a
*b)^(1/2))-(a*d-b*c)/b)^(1/2))-1/(a*d-b*c)*b*(-1/(a*d-b*c)*b/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1
/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+2*d*(-a*b)^(1/2)/(a*d-b*c)*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/(
-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/
2)+1/(a*d-b*c)*b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)
/b)^(1/2)*(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^
(1/2)))))+4*d/(a*d-b*c)*b*(2/3*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(
x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+16/3*d/(-4*d*(a*d-b*c)/b+4*d^
2*a/b)^2*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (150) = 300\).
time = 2.32, size = 1008, normalized size = 5.79 \begin {gather*} \left [-\frac {3 \, {\left ({\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{6} + 3 \, a b c^{3} + 2 \, a^{2} c^{2} d + {\left (6 \, b^{2} c^{2} d + 7 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x^{4} + {\left (3 \, b^{2} c^{3} + 8 \, a b c^{2} d + 4 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left ({\left (4 \, b^{2} c d + 11 \, a b d^{2}\right )} x^{5} + 2 \, {\left (3 \, b^{2} c^{2} + 8 \, a b c d + 4 \, a^{2} d^{2}\right )} x^{3} + 3 \, {\left (3 \, a b c^{2} + 2 \, a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c}}{24 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}, \frac {3 \, {\left ({\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{6} + 3 \, a b c^{3} + 2 \, a^{2} c^{2} d + {\left (6 \, b^{2} c^{2} d + 7 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x^{4} + {\left (3 \, b^{2} c^{3} + 8 \, a b c^{2} d + 4 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{3} + a c x\right )}}\right ) + 2 \, {\left ({\left (4 \, b^{2} c d + 11 \, a b d^{2}\right )} x^{5} + 2 \, {\left (3 \, b^{2} c^{2} + 8 \, a b c d + 4 \, a^{2} d^{2}\right )} x^{3} + 3 \, {\left (3 \, a b c^{2} + 2 \, a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/24*(3*((3*b^2*c*d^2 + 2*a*b*d^3)*x^6 + 3*a*b*c^3 + 2*a^2*c^2*d + (6*b^2*c^2*d + 7*a*b*c*d^2 + 2*a^2*d^3)*x
^4 + (3*b^2*c^3 + 8*a*b*c^2*d + 4*a^2*c*d^2)*x^2)*sqrt(-a/(b*c - a*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^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^3 - (a*b*c^2 - a^2*c*d)
*x)*sqrt(d*x^2 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*((4*b^2*c*d + 11*a*b*d^2)*x^5 + 2*(
3*b^2*c^2 + 8*a*b*c*d + 4*a^2*d^2)*x^3 + 3*(3*a*b*c^2 + 2*a^2*c*d)*x)*sqrt(d*x^2 + c))/(a*b^3*c^5 - 3*a^2*b^2*
c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2
*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2
*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x^2), 1/12*(3*((3*b^2*c*d^2 + 2*a*b*d^3)*x^6 + 3*a*b*c^3 + 2*a^2
*c^2*d + (6*b^2*c^2*d + 7*a*b*c*d^2 + 2*a^2*d^3)*x^4 + (3*b^2*c^3 + 8*a*b*c^2*d + 4*a^2*c*d^2)*x^2)*sqrt(a/(b*
c - a*d))*arctan(-1/2*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)*sqrt(a/(b*c - a*d))/(a*d*x^3 + a*c*x)) + 2*((4
*b^2*c*d + 11*a*b*d^2)*x^5 + 2*(3*b^2*c^2 + 8*a*b*c*d + 4*a^2*d^2)*x^3 + 3*(3*a*b*c^2 + 2*a^2*c*d)*x)*sqrt(d*x
^2 + c))/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2
*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4
+ (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (150) = 300\).
time = 1.34, size = 594, normalized size = 3.41 \begin {gather*} \frac {{\left (\frac {2 \, {\left (b^{4} c^{5} d^{2} - a b^{3} c^{4} d^{3} - 3 \, a^{2} b^{2} c^{3} d^{4} + 5 \, a^{3} b c^{2} d^{5} - 2 \, a^{4} c d^{6}\right )} x^{2}}{b^{6} c^{7} d - 6 \, a b^{5} c^{6} d^{2} + 15 \, a^{2} b^{4} c^{5} d^{3} - 20 \, a^{3} b^{3} c^{4} d^{4} + 15 \, a^{4} b^{2} c^{3} d^{5} - 6 \, a^{5} b c^{2} d^{6} + a^{6} c d^{7}} + \frac {3 \, {\left (b^{4} c^{6} d - 2 \, a b^{3} c^{5} d^{2} + 2 \, a^{3} b c^{3} d^{4} - a^{4} c^{2} d^{5}\right )}}{b^{6} c^{7} d - 6 \, a b^{5} c^{6} d^{2} + 15 \, a^{2} b^{4} c^{5} d^{3} - 20 \, a^{3} b^{3} c^{4} d^{4} + 15 \, a^{4} b^{2} c^{3} d^{5} - 6 \, a^{5} b c^{2} d^{6} + a^{6} c d^{7}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {{\left (3 \, a b c \sqrt {d} + 2 \, a^{2} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} d^{\frac {3}{2}} - a b c^{2} \sqrt {d}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*(b^4*c^5*d^2 - a*b^3*c^4*d^3 - 3*a^2*b^2*c^3*d^4 + 5*a^3*b*c^2*d^5 - 2*a^4*c*d^6)*x^2/(b^6*c^7*d - 6*a*
b^5*c^6*d^2 + 15*a^2*b^4*c^5*d^3 - 20*a^3*b^3*c^4*d^4 + 15*a^4*b^2*c^3*d^5 - 6*a^5*b*c^2*d^6 + a^6*c*d^7) + 3*
(b^4*c^6*d - 2*a*b^3*c^5*d^2 + 2*a^3*b*c^3*d^4 - a^4*c^2*d^5)/(b^6*c^7*d - 6*a*b^5*c^6*d^2 + 15*a^2*b^4*c^5*d^
3 - 20*a^3*b^3*c^4*d^4 + 15*a^4*b^2*c^3*d^5 - 6*a^5*b*c^2*d^6 + a^6*c*d^7))*x/(d*x^2 + c)^(3/2) + 1/2*(3*a*b*c
*sqrt(d) + 2*a^2*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2)
)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(a*b*c*d - a^2*d^2)) - ((sqrt(d)*x - sqrt(d*x^2 + c
))^2*a*b*c*sqrt(d) - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*d^(3/2) - a*b*c^2*sqrt(d))/((b^3*c^3 - 3*a*b^2*c^2*
d + 3*a^2*b*c*d^2 - a^3*d^3)*((sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c + 4*(s
qrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4}{{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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