3.2.66 \(\int \frac {1}{(d+e x^2)^{3/2} (-c d^2+b d e+b e^2 x^2+c e^2 x^4)} \, dx\)

Optimal. Leaf size=149 \[ -\frac {c^2 \tanh ^{-1}\left (\frac {\sqrt {e} x \sqrt {2 c d-b e}}{\sqrt {d+e x^2} \sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^{5/2}}-\frac {x (7 c d-2 b e)}{3 d^2 \sqrt {d+e x^2} (2 c d-b e)^2}-\frac {x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)} \]

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Rubi [A]  time = 0.27, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {1149, 414, 527, 12, 377, 208} \begin {gather*} -\frac {c^2 \tanh ^{-1}\left (\frac {\sqrt {e} x \sqrt {2 c d-b e}}{\sqrt {d+e x^2} \sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^{5/2}}-\frac {x (7 c d-2 b e)}{3 d^2 \sqrt {d+e x^2} (2 c d-b e)^2}-\frac {x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x^2)^(3/2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

-x/(3*d*(2*c*d - b*e)*(d + e*x^2)^(3/2)) - ((7*c*d - 2*b*e)*x)/(3*d^2*(2*c*d - b*e)^2*Sqrt[d + e*x^2]) - (c^2*
ArcTanh[(Sqrt[e]*Sqrt[2*c*d - b*e]*x)/(Sqrt[c*d - b*e]*Sqrt[d + e*x^2])])/(Sqrt[e]*Sqrt[c*d - b*e]*(2*c*d - b*
e)^(5/2))

Rule 12

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

Rule 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 377

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 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

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]

Rule 1149

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p +
q)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2
, 0] && IntegerQ[p]

Rubi steps

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

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Mathematica [C]  time = 4.14, size = 1058, normalized size = 7.10 \begin {gather*} -\frac {x \left (-\frac {56 c^2 e^2 \left (\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{3/2} x^4}{(c d-b e)^2}+\frac {168 c^2 e^2 \tanh ^{-1}\left (\sqrt {\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}}\right ) x^4}{(c d-b e)^2}+\frac {36 c^2 e^2 \left (\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{7/2} \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right ) x^4}{(c d-b e)^2}+\frac {12 c^2 e^2 \left (\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{7/2} \, _3F_2\left (2,2,\frac {7}{2};1,\frac {9}{2};\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right ) x^4}{(c d-b e)^2}-\frac {168 c^2 e^2 \sqrt {\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}} x^4}{(c d-b e)^2}+\frac {140 c e \left (\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{3/2} x^2}{c d-b e}-\frac {420 c e \tanh ^{-1}\left (\sqrt {\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}}\right ) x^2}{c d-b e}-\frac {84 c e \left (\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{7/2} \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right ) x^2}{c d-b e}-\frac {24 c e \left (\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{7/2} \, _3F_2\left (2,2,\frac {7}{2};1,\frac {9}{2};\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right ) x^2}{c d-b e}+\frac {420 c e \sqrt {\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}} x^2}{c d-b e}-105 \left (\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{3/2}+315 \tanh ^{-1}\left (\sqrt {\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}}\right )+48 \left (\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{7/2} \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )+12 \left (\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{7/2} \, _3F_2\left (2,2,\frac {7}{2};1,\frac {9}{2};\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )-315 \sqrt {\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}}\right )}{63 (c d-b e) \left (\frac {e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{5/2} \left (e x^2+d\right )^{5/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((d + e*x^2)^(3/2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

-1/63*(x*(-315*Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))] + (420*c*e*x^2*Sqrt[(e*(-2*c*d + b*e)
*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e) - (168*c^2*e^2*x^4*Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e
)*(d + e*x^2))])/(c*d - b*e)^2 - 105*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(3/2) + (140*c*e*x^
2*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(3/2))/(c*d - b*e) - (56*c^2*e^2*x^4*((e*(-2*c*d + b*e
)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(3/2))/(c*d - b*e)^2 + 315*ArcTanh[Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) +
b*e)*(d + e*x^2))]] - (420*c*e*x^2*ArcTanh[Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))]])/(c*d -
b*e) + (168*c^2*e^2*x^4*ArcTanh[Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))]])/(c*d - b*e)^2 + 4
8*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, (e*(-2*c*d + b*e)
*x^2)/((-(c*d) + b*e)*(d + e*x^2))] - (84*c*e*x^2*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*
Hypergeometric2F1[2, 7/2, 9/2, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e) + (36*c^2*e^2
*x^4*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, (e*(-2*c*d + b
*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e)^2 + 12*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)
))^(7/2)*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))] - (24*c
*e*x^2*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, (e
*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e) + (12*c^2*e^2*x^4*((e*(-2*c*d + b*e)*x^2)/((-(
c*d) + b*e)*(d + e*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e
)*(d + e*x^2))])/(c*d - b*e)^2))/((c*d - b*e)*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(5/2)*(d +
e*x^2)^(5/2))

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IntegrateAlgebraic [A]  time = 0.57, size = 256, normalized size = 1.72 \begin {gather*} \frac {3 b d e x+2 b e^2 x^3-9 c d^2 x-7 c d e x^3}{3 d^2 \left (d+e x^2\right )^{3/2} (2 c d-b e)^2}-\frac {c^2 \sqrt {b^2 e^2-3 b c d e+2 c^2 d^2} \tanh ^{-1}\left (-\frac {c e x^2}{\sqrt {b^2 e^2-3 b c d e+2 c^2 d^2}}+\frac {c \sqrt {e} x \sqrt {d+e x^2}}{\sqrt {b^2 e^2-3 b c d e+2 c^2 d^2}}+\frac {c d}{\sqrt {b^2 e^2-3 b c d e+2 c^2 d^2}}-\frac {b e}{\sqrt {b^2 e^2-3 b c d e+2 c^2 d^2}}\right )}{\sqrt {e} (b e-2 c d)^3 (b e-c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d + e*x^2)^(3/2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

(-9*c*d^2*x + 3*b*d*e*x - 7*c*d*e*x^3 + 2*b*e^2*x^3)/(3*d^2*(2*c*d - b*e)^2*(d + e*x^2)^(3/2)) - (c^2*Sqrt[2*c
^2*d^2 - 3*b*c*d*e + b^2*e^2]*ArcTanh[(c*d)/Sqrt[2*c^2*d^2 - 3*b*c*d*e + b^2*e^2] - (b*e)/Sqrt[2*c^2*d^2 - 3*b
*c*d*e + b^2*e^2] - (c*e*x^2)/Sqrt[2*c^2*d^2 - 3*b*c*d*e + b^2*e^2] + (c*Sqrt[e]*x*Sqrt[d + e*x^2])/Sqrt[2*c^2
*d^2 - 3*b*c*d*e + b^2*e^2]])/(Sqrt[e]*(-2*c*d + b*e)^3*(-(c*d) + b*e))

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fricas [B]  time = 2.65, size = 1063, normalized size = 7.13 \begin {gather*} \left [\frac {3 \, {\left (c^{2} d^{2} e^{2} x^{4} + 2 \, c^{2} d^{3} e x^{2} + c^{2} d^{4}\right )} \sqrt {2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}} \log \left (\frac {c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + {\left (17 \, c^{2} d^{2} e^{2} - 24 \, b c d e^{3} + 8 \, b^{2} e^{4}\right )} x^{4} + 2 \, {\left (7 \, c^{2} d^{3} e - 11 \, b c d^{2} e^{2} + 4 \, b^{2} d e^{3}\right )} x^{2} - 4 \, \sqrt {2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}} {\left ({\left (3 \, c d e - 2 \, b e^{2}\right )} x^{3} + {\left (c d^{2} - b d e\right )} x\right )} \sqrt {e x^{2} + d}}{c^{2} e^{2} x^{4} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x^{2}}\right ) - 4 \, {\left ({\left (14 \, c^{3} d^{3} e^{2} - 25 \, b c^{2} d^{2} e^{3} + 13 \, b^{2} c d e^{4} - 2 \, b^{3} e^{5}\right )} x^{3} + 3 \, {\left (6 \, c^{3} d^{4} e - 11 \, b c^{2} d^{3} e^{2} + 6 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x\right )} \sqrt {e x^{2} + d}}{12 \, {\left (8 \, c^{4} d^{8} e - 20 \, b c^{3} d^{7} e^{2} + 18 \, b^{2} c^{2} d^{6} e^{3} - 7 \, b^{3} c d^{5} e^{4} + b^{4} d^{4} e^{5} + {\left (8 \, c^{4} d^{6} e^{3} - 20 \, b c^{3} d^{5} e^{4} + 18 \, b^{2} c^{2} d^{4} e^{5} - 7 \, b^{3} c d^{3} e^{6} + b^{4} d^{2} e^{7}\right )} x^{4} + 2 \, {\left (8 \, c^{4} d^{7} e^{2} - 20 \, b c^{3} d^{6} e^{3} + 18 \, b^{2} c^{2} d^{5} e^{4} - 7 \, b^{3} c d^{4} e^{5} + b^{4} d^{3} e^{6}\right )} x^{2}\right )}}, -\frac {3 \, {\left (c^{2} d^{2} e^{2} x^{4} + 2 \, c^{2} d^{3} e x^{2} + c^{2} d^{4}\right )} \sqrt {-2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}} \arctan \left (-\frac {\sqrt {-2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}} {\left (c d^{2} - b d e + {\left (3 \, c d e - 2 \, b e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d}}{2 \, {\left ({\left (2 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{3} + {\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )}}\right ) + 2 \, {\left ({\left (14 \, c^{3} d^{3} e^{2} - 25 \, b c^{2} d^{2} e^{3} + 13 \, b^{2} c d e^{4} - 2 \, b^{3} e^{5}\right )} x^{3} + 3 \, {\left (6 \, c^{3} d^{4} e - 11 \, b c^{2} d^{3} e^{2} + 6 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x\right )} \sqrt {e x^{2} + d}}{6 \, {\left (8 \, c^{4} d^{8} e - 20 \, b c^{3} d^{7} e^{2} + 18 \, b^{2} c^{2} d^{6} e^{3} - 7 \, b^{3} c d^{5} e^{4} + b^{4} d^{4} e^{5} + {\left (8 \, c^{4} d^{6} e^{3} - 20 \, b c^{3} d^{5} e^{4} + 18 \, b^{2} c^{2} d^{4} e^{5} - 7 \, b^{3} c d^{3} e^{6} + b^{4} d^{2} e^{7}\right )} x^{4} + 2 \, {\left (8 \, c^{4} d^{7} e^{2} - 20 \, b c^{3} d^{6} e^{3} + 18 \, b^{2} c^{2} d^{5} e^{4} - 7 \, b^{3} c d^{4} e^{5} + b^{4} d^{3} e^{6}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^(3/2)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="fricas")

[Out]

[1/12*(3*(c^2*d^2*e^2*x^4 + 2*c^2*d^3*e*x^2 + c^2*d^4)*sqrt(2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3)*log((c^2*d^4
- 2*b*c*d^3*e + b^2*d^2*e^2 + (17*c^2*d^2*e^2 - 24*b*c*d*e^3 + 8*b^2*e^4)*x^4 + 2*(7*c^2*d^3*e - 11*b*c*d^2*e^
2 + 4*b^2*d*e^3)*x^2 - 4*sqrt(2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3)*((3*c*d*e - 2*b*e^2)*x^3 + (c*d^2 - b*d*e)*
x)*sqrt(e*x^2 + d))/(c^2*e^2*x^4 + c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2*(c^2*d*e - b*c*e^2)*x^2)) - 4*((14*c^3*d^
3*e^2 - 25*b*c^2*d^2*e^3 + 13*b^2*c*d*e^4 - 2*b^3*e^5)*x^3 + 3*(6*c^3*d^4*e - 11*b*c^2*d^3*e^2 + 6*b^2*c*d^2*e
^3 - b^3*d*e^4)*x)*sqrt(e*x^2 + d))/(8*c^4*d^8*e - 20*b*c^3*d^7*e^2 + 18*b^2*c^2*d^6*e^3 - 7*b^3*c*d^5*e^4 + b
^4*d^4*e^5 + (8*c^4*d^6*e^3 - 20*b*c^3*d^5*e^4 + 18*b^2*c^2*d^4*e^5 - 7*b^3*c*d^3*e^6 + b^4*d^2*e^7)*x^4 + 2*(
8*c^4*d^7*e^2 - 20*b*c^3*d^6*e^3 + 18*b^2*c^2*d^5*e^4 - 7*b^3*c*d^4*e^5 + b^4*d^3*e^6)*x^2), -1/6*(3*(c^2*d^2*
e^2*x^4 + 2*c^2*d^3*e*x^2 + c^2*d^4)*sqrt(-2*c^2*d^2*e + 3*b*c*d*e^2 - b^2*e^3)*arctan(-1/2*sqrt(-2*c^2*d^2*e
+ 3*b*c*d*e^2 - b^2*e^3)*(c*d^2 - b*d*e + (3*c*d*e - 2*b*e^2)*x^2)*sqrt(e*x^2 + d)/((2*c^2*d^2*e^2 - 3*b*c*d*e
^3 + b^2*e^4)*x^3 + (2*c^2*d^3*e - 3*b*c*d^2*e^2 + b^2*d*e^3)*x)) + 2*((14*c^3*d^3*e^2 - 25*b*c^2*d^2*e^3 + 13
*b^2*c*d*e^4 - 2*b^3*e^5)*x^3 + 3*(6*c^3*d^4*e - 11*b*c^2*d^3*e^2 + 6*b^2*c*d^2*e^3 - b^3*d*e^4)*x)*sqrt(e*x^2
+ d))/(8*c^4*d^8*e - 20*b*c^3*d^7*e^2 + 18*b^2*c^2*d^6*e^3 - 7*b^3*c*d^5*e^4 + b^4*d^4*e^5 + (8*c^4*d^6*e^3 -
20*b*c^3*d^5*e^4 + 18*b^2*c^2*d^4*e^5 - 7*b^3*c*d^3*e^6 + b^4*d^2*e^7)*x^4 + 2*(8*c^4*d^7*e^2 - 20*b*c^3*d^6*
e^3 + 18*b^2*c^2*d^5*e^4 - 7*b^3*c*d^4*e^5 + b^4*d^3*e^6)*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^(3/2)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [b,c,d,exp(1),exp(2)]=[-21,-18,-46,11,70]Warning, need to choose a branch for the root of a poly
nomial with parameters. This might be wrong.The choice was done assuming [b,c,d,exp(1),exp(2)]=[72,91,-18,-31,
46]Evaluation time: 2.06Unable to transpose Error: Bad Argument Value

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maple [B]  time = 0.02, size = 1637, normalized size = 10.99

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)^(3/2)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)

[Out]

1/2*c^3*e/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1
/2)/(b*e-2*c*d)/((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)/c-
(b*e-2*c*d)/c)^(1/2)+1/2*c^2*e/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2)
)/(b*e-2*c*d)/d/((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)/c-
(b*e-2*c*d)/c)^(1/2)*x-1/2*c^3*e/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/
2))/(-(b*e-c*d)*c*e)^(1/2)/(b*e-2*c*d)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c-2*(-(b*e-c*d)*c*e)^(1/2)*(x
+(-(b*e-c*d)*c*e)^(1/2)/c/e)/c+2*(-(b*e-2*c*d)/c)^(1/2)*((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)
^(1/2)*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)/c-(b*e-2*c*d)/c)^(1/2))/(x+(-(b*e-c*d)*c*e)^(1/2)/c/e))-1/6*c/d/((-d*e)^
(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(x-(-d*e)^(1/2)/e)/((x-(-d*e)^(1/2)/e
)^2*e+2*(-d*e)^(1/2)*(x-(-d*e)^(1/2)/e))^(1/2)-1/3*c*e/d^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1
/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((x-(-d*e)^(1/2)/e)^2*e+2*(-d*e)^(1/2)*(x-(-d*e)^(1/2)/e))^(1/2)*x-1/6*c/d/((-d*
e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(x+(-d*e)^(1/2)/e)/((x+(-d*e)^(1/2
)/e)^2*e-2*(-d*e)^(1/2)*(x+(-d*e)^(1/2)/e))^(1/2)-1/3*c*e/d^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)
^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((x+(-d*e)^(1/2)/e)^2*e-2*(-d*e)^(1/2)*(x+(-d*e)^(1/2)/e))^(1/2)*x-1/2*c^3*e/
((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)/(b*e-2
*c*d)/((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)/c-(b*e-2*c*d
)/c)^(1/2)+1/2*c^2*e/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(b*e-2*c
*d)/d/((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)/c-(b*e-2*c*d
)/c)^(1/2)*x+1/2*c^3*e/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(b*e
-c*d)*c*e)^(1/2)/(b*e-2*c*d)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c+2*(-(b*e-c*d)*c*e)^(1/2)*(x-(-(b*e-c*
d)*c*e)^(1/2)/c/e)/c+2*(-(b*e-2*c*d)/c)^(1/2)*((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)*(x-
(-(b*e-c*d)*c*e)^(1/2)/c/e)/c-(b*e-2*c*d)/c)^(1/2))/(x-(-(b*e-c*d)*c*e)^(1/2)/c/e))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^(3/2)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="maxima")

[Out]

integrate(1/((c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e)*(e*x^2 + d)^(3/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (e\,x^2+d\right )}^{3/2}\,\left (-c\,d^2+b\,d\,e+c\,e^2\,x^4+b\,e^2\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x^2)^(3/2)*(b*e^2*x^2 - c*d^2 + c*e^2*x^4 + b*d*e)),x)

[Out]

int(1/((d + e*x^2)^(3/2)*(b*e^2*x^2 - c*d^2 + c*e^2*x^4 + b*d*e)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x^{2}\right )^{\frac {5}{2}} \left (b e - c d + c e x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)**(3/2)/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)

[Out]

Integral(1/((d + e*x**2)**(5/2)*(b*e - c*d + c*e*x**2)), x)

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