3.307 \(\int \frac {x^5}{(\frac {e (a+b x^2)}{c+d x^2})^{3/2}} \, dx\)

Optimal. Leaf size=354 \[ \frac {\left (c+d x^2\right )^3 \left (7 a^2 d^2-2 a b c d+b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{6 b^2 d e^2 (b c-a d)^2}-\frac {a^2 \left (c+d x^2\right )^3}{b e (b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(b c-a d) \left (5 a d (2 b c-7 a d)+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{16 b^{9/2} d^{3/2} e^{3/2}}-\frac {\left (c+d x^2\right ) \left (5 a d (2 b c-7 a d)+b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 b^4 d e^2}-\frac {\left (c+d x^2\right )^2 \left (5 a d (2 b c-7 a d)+b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 b^3 d e^2 (b c-a d)} \]

[Out]

-1/16*(-a*d+b*c)*(b^2*c^2+5*a*d*(-7*a*d+2*b*c))*arctanh(d^(1/2)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/b^(1/2)/e^(1/2))
/b^(9/2)/d^(3/2)/e^(3/2)-a^2*(d*x^2+c)^3/b/(-a*d+b*c)^2/e/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/16*(b^2*c^2+5*a*d*(-
7*a*d+2*b*c))*(d*x^2+c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/b^4/d/e^2-1/24*(b^2*c^2+5*a*d*(-7*a*d+2*b*c))*(d*x^2+c)^
2*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/b^3/d/(-a*d+b*c)/e^2+1/6*(7*a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x^2+c)^3*(e*(b*x^2+a
)/(d*x^2+c))^(1/2)/b^2/d/(-a*d+b*c)^2/e^2

________________________________________________________________________________________

Rubi [A]  time = 0.38, antiderivative size = 348, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1960, 462, 385, 199, 208} \[ -\frac {a^2 \left (c+d x^2\right )^3}{b e (b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(b c-a d) \left (5 a d (2 b c-7 a d)+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{16 b^{9/2} d^{3/2} e^{3/2}}+\frac {\left (c+d x^2\right )^3 \left (\frac {c^2}{d}-\frac {a (2 b c-7 a d)}{b^2}\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{6 e^2 (b c-a d)^2}-\frac {\left (c+d x^2\right )^2 \left (\frac {5 a (2 b c-7 a d)}{b^2}+\frac {c^2}{d}\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 b e^2 (b c-a d)}-\frac {\left (c+d x^2\right ) \left (5 a d (2 b c-7 a d)+b^2 c^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 b^4 d e^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5/((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]

[Out]

-((b^2*c^2 + 5*a*d*(2*b*c - 7*a*d))*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))/(16*b^4*d*e^2) - ((c^2/d +
(5*a*(2*b*c - 7*a*d))/b^2)*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2)^2)/(24*b*(b*c - a*d)*e^2) - (a^2*(c +
d*x^2)^3)/(b*(b*c - a*d)^2*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]) + ((c^2/d - (a*(2*b*c - 7*a*d))/b^2)*Sqrt[(e*
(a + b*x^2))/(c + d*x^2)]*(c + d*x^2)^3)/(6*(b*c - a*d)^2*e^2) - ((b*c - a*d)*(b^2*c^2 + 5*a*d*(2*b*c - 7*a*d)
)*ArcTanh[(Sqrt[d]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqrt[b]*Sqrt[e])])/(16*b^(9/2)*d^(3/2)*e^(3/2))

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 208

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

Rule 385

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

Rule 462

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

Rule 1960

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Den
ominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(Simplify[(m + 1)/n] - 1
))/(b*e - d*x^q)^(Simplify[(m + 1)/n] + 1), x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, m, n}, x] && FractionQ[p] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx &=((b c-a d) e) \operatorname {Subst}\left (\int \frac {\left (-a e+c x^2\right )^2}{x^2 \left (b e-d x^2\right )^4} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=-\frac {a^2 \left (c+d x^2\right )^3}{b (b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {-a (2 b c-7 a d) e^2+b c^2 e x^2}{\left (b e-d x^2\right )^4} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{b}\\ &=-\frac {a^2 \left (c+d x^2\right )^3}{b (b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3}{6 b^2 d (b c-a d)^2 e^2}-\frac {\left ((b c-a d) \left (b^2 c^2+5 a d (2 b c-7 a d)\right ) e\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b e-d x^2\right )^3} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{6 b^2 d}\\ &=-\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 b^3 d (b c-a d) e^2}-\frac {a^2 \left (c+d x^2\right )^3}{b (b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3}{6 b^2 d (b c-a d)^2 e^2}-\frac {\left ((b c-a d) \left (b^2 c^2+5 a d (2 b c-7 a d)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b e-d x^2\right )^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{8 b^3 d}\\ &=-\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{16 b^4 d e^2}-\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 b^3 d (b c-a d) e^2}-\frac {a^2 \left (c+d x^2\right )^3}{b (b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3}{6 b^2 d (b c-a d)^2 e^2}-\frac {\left ((b c-a d) \left (b^2 c^2+5 a d (2 b c-7 a d)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b e-d x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{16 b^4 d e}\\ &=-\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{16 b^4 d e^2}-\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 b^3 d (b c-a d) e^2}-\frac {a^2 \left (c+d x^2\right )^3}{b (b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3}{6 b^2 d (b c-a d)^2 e^2}-\frac {(b c-a d) \left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{16 b^{9/2} d^{3/2} e^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.48, size = 247, normalized size = 0.70 \[ \frac {\sqrt {d} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \left (105 a^3 d^2+5 a^2 b d \left (7 d x^2-20 c\right )+a b^2 \left (3 c^2-38 c d x^2-14 d^2 x^4\right )+b^3 x^2 \left (3 c^2+14 c d x^2+8 d^2 x^4\right )\right )-3 \sqrt {a+b x^2} \sqrt {b c-a d} \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{48 b^4 d^{3/2} e \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/((e*(a + b*x^2))/(c + d*x^2))^(3/2),x]

[Out]

(Sqrt[d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*(105*a^3*d^2 + 5*a^2*b*d*(-20*c + 7*d*x^2) + a*b^2*(3*c^2 - 38*c*d*
x^2 - 14*d^2*x^4) + b^3*x^2*(3*c^2 + 14*c*d*x^2 + 8*d^2*x^4)) - 3*Sqrt[b*c - a*d]*(b^2*c^2 + 10*a*b*c*d - 35*a
^2*d^2)*Sqrt[a + b*x^2]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/(48*b^4*d^(3/2)*e*Sqrt[(e*(a + b*x
^2))/(c + d*x^2)]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)])

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fricas [A]  time = 1.81, size = 781, normalized size = 2.21 \[ \left [\frac {3 \, {\left (a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 45 \, a^{3} b c d^{2} + 35 \, a^{4} d^{3} + {\left (b^{4} c^{3} + 9 \, a b^{3} c^{2} d - 45 \, a^{2} b^{2} c d^{2} + 35 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {b d e} \log \left (8 \, b^{2} d^{2} e x^{4} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} e x^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e - 4 \, {\left (2 \, b d^{2} x^{4} + b c^{2} + a c d + {\left (3 \, b c d + a d^{2}\right )} x^{2}\right )} \sqrt {b d e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}\right ) + 4 \, {\left (8 \, b^{4} d^{4} x^{8} + 3 \, a b^{3} c^{3} d - 100 \, a^{2} b^{2} c^{2} d^{2} + 105 \, a^{3} b c d^{3} + 2 \, {\left (11 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{6} + {\left (17 \, b^{4} c^{2} d^{2} - 52 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x^{4} + {\left (3 \, b^{4} c^{3} d - 35 \, a b^{3} c^{2} d^{2} - 65 \, a^{2} b^{2} c d^{3} + 105 \, a^{3} b d^{4}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{192 \, {\left (b^{6} d^{2} e^{2} x^{2} + a b^{5} d^{2} e^{2}\right )}}, \frac {3 \, {\left (a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 45 \, a^{3} b c d^{2} + 35 \, a^{4} d^{3} + {\left (b^{4} c^{3} + 9 \, a b^{3} c^{2} d - 45 \, a^{2} b^{2} c d^{2} + 35 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {-b d e} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {-b d e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{2 \, {\left (b^{2} d e x^{2} + a b d e\right )}}\right ) + 2 \, {\left (8 \, b^{4} d^{4} x^{8} + 3 \, a b^{3} c^{3} d - 100 \, a^{2} b^{2} c^{2} d^{2} + 105 \, a^{3} b c d^{3} + 2 \, {\left (11 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{6} + {\left (17 \, b^{4} c^{2} d^{2} - 52 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x^{4} + {\left (3 \, b^{4} c^{3} d - 35 \, a b^{3} c^{2} d^{2} - 65 \, a^{2} b^{2} c d^{3} + 105 \, a^{3} b d^{4}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{96 \, {\left (b^{6} d^{2} e^{2} x^{2} + a b^{5} d^{2} e^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="fricas")

[Out]

[1/192*(3*(a*b^3*c^3 + 9*a^2*b^2*c^2*d - 45*a^3*b*c*d^2 + 35*a^4*d^3 + (b^4*c^3 + 9*a*b^3*c^2*d - 45*a^2*b^2*c
*d^2 + 35*a^3*b*d^3)*x^2)*sqrt(b*d*e)*log(8*b^2*d^2*e*x^4 + 8*(b^2*c*d + a*b*d^2)*e*x^2 + (b^2*c^2 + 6*a*b*c*d
+ a^2*d^2)*e - 4*(2*b*d^2*x^4 + b*c^2 + a*c*d + (3*b*c*d + a*d^2)*x^2)*sqrt(b*d*e)*sqrt((b*e*x^2 + a*e)/(d*x^
2 + c))) + 4*(8*b^4*d^4*x^8 + 3*a*b^3*c^3*d - 100*a^2*b^2*c^2*d^2 + 105*a^3*b*c*d^3 + 2*(11*b^4*c*d^3 - 7*a*b^
3*d^4)*x^6 + (17*b^4*c^2*d^2 - 52*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x^4 + (3*b^4*c^3*d - 35*a*b^3*c^2*d^2 - 65*a^2
*b^2*c*d^3 + 105*a^3*b*d^4)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b^6*d^2*e^2*x^2 + a*b^5*d^2*e^2), 1/96*(3
*(a*b^3*c^3 + 9*a^2*b^2*c^2*d - 45*a^3*b*c*d^2 + 35*a^4*d^3 + (b^4*c^3 + 9*a*b^3*c^2*d - 45*a^2*b^2*c*d^2 + 35
*a^3*b*d^3)*x^2)*sqrt(-b*d*e)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(-b*d*e)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)
)/(b^2*d*e*x^2 + a*b*d*e)) + 2*(8*b^4*d^4*x^8 + 3*a*b^3*c^3*d - 100*a^2*b^2*c^2*d^2 + 105*a^3*b*c*d^3 + 2*(11*
b^4*c*d^3 - 7*a*b^3*d^4)*x^6 + (17*b^4*c^2*d^2 - 52*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x^4 + (3*b^4*c^3*d - 35*a*b^
3*c^2*d^2 - 65*a^2*b^2*c*d^3 + 105*a^3*b*d^4)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b^6*d^2*e^2*x^2 + a*b^5
*d^2*e^2)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep*d+c)]Unable to divide, perhaps due to rounding error%%%{%%%{2,[1,2,2]%%%},[2,1,2,0]%%%}+%%%{%%%{-4,
[2,1,2]%%%},[2,1,1,1]%%%}+%%%{%%%{2,[3,0,2]%%%},[2,1,0,2]%%%}+%%%{%%{[%%%{-4,[0,2,2]%%%},0]:[1,0,%%%{-1,[1,1,1
]%%%}]%%},[1,1,3,0]%%%}+%%%{%%{[%%%{8,[1,1,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,1,2,1]%%%}+%%%{%%{[%%%{-4,
[2,0,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,1,1,2]%%%}+%%%{%%%{2,[0,3,3]%%%},[0,1,4,0]%%%}+%%%{%%%{-4,[1,2,3
]%%%},[0,1,3,1]%%%}+%%%{%%%{2,[2,1,3]%%%},[0,1,2,2]%%%} / %%%{%%%{1,[2,0,2]%%%},[2,0,0,0]%%%}+%%%{%%{[%%%{-2,[
1,0,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,0,1,0]%%%}+%%%{%%%{1,[1,1,3]%%%},[0,0,2,0]%%%} Error: Bad Argumen
t Value

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maple [B]  time = 0.07, size = 1027, normalized size = 2.90 \[ \frac {\left (-105 a^{3} b \,d^{3} x^{2} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+135 a^{2} b^{2} c \,d^{2} x^{2} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-27 a \,b^{3} c^{2} d \,x^{2} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-3 b^{4} c^{3} x^{2} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-60 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a \,b^{2} d^{2} x^{4}+12 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, b^{3} c d \,x^{4}-105 a^{4} d^{3} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+135 a^{3} b c \,d^{2} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-27 a^{2} b^{2} c^{2} d \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-3 a \,b^{3} c^{3} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+54 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a^{2} b \,d^{2} x^{2}-108 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a \,b^{2} c d \,x^{2}+6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, b^{3} c^{2} x^{2}+114 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a^{3} d^{2}+96 \sqrt {b d}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{3} d^{2}-120 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a^{2} b c d -96 \sqrt {b d}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{2} b c d +6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, a \,b^{2} c^{2}+16 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {b d}\, b^{2} d \,x^{2}+16 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {b d}\, a b d \right ) \left (b \,x^{2}+a \right )}{96 \sqrt {b d}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \left (d \,x^{2}+c \right ) \left (\frac {\left (b \,x^{2}+a \right ) e}{d \,x^{2}+c}\right )^{\frac {3}{2}} b^{4} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/((b*x^2+a)/(d*x^2+c)*e)^(3/2),x)

[Out]

1/96*(-60*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)*x^4*a*b^2*d^2+12*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)
*(b*d)^(1/2)*x^4*b^3*c*d-105*ln(1/2*(2*b*d*x^2+a*d+b*c+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2))/(b*d
)^(1/2))*x^2*a^3*b*d^3+135*ln(1/2*(2*b*d*x^2+a*d+b*c+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^
(1/2))*x^2*a^2*b^2*c*d^2-27*ln(1/2*(2*b*d*x^2+a*d+b*c+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2))/(b*d)
^(1/2))*x^2*a*b^3*c^2*d-3*ln(1/2*(2*b*d*x^2+a*d+b*c+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(
1/2))*x^2*b^4*c^3+16*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*(b*d)^(1/2)*x^2*b^2*d+54*(b*d*x^4+a*d*x^2+b*c*x^2+a*c
)^(1/2)*(b*d)^(1/2)*x^2*a^2*b*d^2-108*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)*x^2*a*b^2*c*d+6*(b*d*x^4
+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)*x^2*b^3*c^2-105*ln(1/2*(2*b*d*x^2+a*d+b*c+2*(b*d*x^4+a*d*x^2+b*c*x^2+a
*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))*a^4*d^3+135*ln(1/2*(2*b*d*x^2+a*d+b*c+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2
)*(b*d)^(1/2))/(b*d)^(1/2))*a^3*b*c*d^2-27*ln(1/2*(2*b*d*x^2+a*d+b*c+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*
d)^(1/2))/(b*d)^(1/2))*a^2*b^2*c^2*d-3*ln(1/2*(2*b*d*x^2+a*d+b*c+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(
1/2))/(b*d)^(1/2))*a*b^3*c^3+16*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*(b*d)^(1/2)*a*b*d+114*(b*d*x^4+a*d*x^2+b*c
*x^2+a*c)^(1/2)*(b*d)^(1/2)*a^3*d^2-120*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)*a^2*b*c*d+6*(b*d*x^4+a
*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)*a*b^2*c^2+96*(b*d)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*a^3*d^2-96*(b*d)^(1
/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*a^2*b*c*d)/d/b^4*(b*x^2+a)/(b*d)^(1/2)/((d*x^2+c)*(b*x^2+a))^(1/2)/(d*x^2+c)/(
(b*x^2+a)/(d*x^2+c)*e)^(3/2)

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maxima [A]  time = 1.83, size = 465, normalized size = 1.31 \[ \frac {1}{96} \, e {\left (\frac {2 \, {\left (48 \, {\left (a^{2} b^{4} c d - a^{3} b^{3} d^{2}\right )} e^{3} + \frac {3 \, {\left (b^{3} c^{3} d^{2} + 9 \, a b^{2} c^{2} d^{3} - 45 \, a^{2} b c d^{4} + 35 \, a^{3} d^{5}\right )} {\left (b x^{2} + a\right )}^{3} e^{3}}{{\left (d x^{2} + c\right )}^{3}} - \frac {8 \, {\left (b^{4} c^{3} d + 9 \, a b^{3} c^{2} d^{2} - 45 \, a^{2} b^{2} c d^{3} + 35 \, a^{3} b d^{4}\right )} {\left (b x^{2} + a\right )}^{2} e^{3}}{{\left (d x^{2} + c\right )}^{2}} - \frac {3 \, {\left (b^{5} c^{3} - 23 \, a b^{4} c^{2} d + 99 \, a^{2} b^{3} c d^{2} - 77 \, a^{3} b^{2} d^{3}\right )} {\left (b x^{2} + a\right )} e^{3}}{d x^{2} + c}\right )}}{b^{4} d^{4} \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {7}{2}} e^{2} - 3 \, b^{5} d^{3} \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {5}{2}} e^{3} + 3 \, b^{6} d^{2} \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} e^{4} - b^{7} d \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} e^{5}} + \frac {3 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3}\right )} \log \left (\frac {d \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} - \sqrt {b d e}}{d \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} + \sqrt {b d e}}\right )}{\sqrt {b d e} b^{4} d e^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="maxima")

[Out]

1/96*e*(2*(48*(a^2*b^4*c*d - a^3*b^3*d^2)*e^3 + 3*(b^3*c^3*d^2 + 9*a*b^2*c^2*d^3 - 45*a^2*b*c*d^4 + 35*a^3*d^5
)*(b*x^2 + a)^3*e^3/(d*x^2 + c)^3 - 8*(b^4*c^3*d + 9*a*b^3*c^2*d^2 - 45*a^2*b^2*c*d^3 + 35*a^3*b*d^4)*(b*x^2 +
a)^2*e^3/(d*x^2 + c)^2 - 3*(b^5*c^3 - 23*a*b^4*c^2*d + 99*a^2*b^3*c*d^2 - 77*a^3*b^2*d^3)*(b*x^2 + a)*e^3/(d*
x^2 + c))/(b^4*d^4*((b*x^2 + a)*e/(d*x^2 + c))^(7/2)*e^2 - 3*b^5*d^3*((b*x^2 + a)*e/(d*x^2 + c))^(5/2)*e^3 + 3
*b^6*d^2*((b*x^2 + a)*e/(d*x^2 + c))^(3/2)*e^4 - b^7*d*sqrt((b*x^2 + a)*e/(d*x^2 + c))*e^5) + 3*(b^3*c^3 + 9*a
*b^2*c^2*d - 45*a^2*b*c*d^2 + 35*a^3*d^3)*log((d*sqrt((b*x^2 + a)*e/(d*x^2 + c)) - sqrt(b*d*e))/(d*sqrt((b*x^2
+ a)*e/(d*x^2 + c)) + sqrt(b*d*e)))/(sqrt(b*d*e)*b^4*d*e^2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5}{{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/((e*(a + b*x^2))/(c + d*x^2))^(3/2),x)

[Out]

int(x^5/((e*(a + b*x^2))/(c + d*x^2))^(3/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(e*(b*x**2+a)/(d*x**2+c))**(3/2),x)

[Out]

Timed out

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