3.6.44 \(\int \frac {(d+e x)^{7/2}}{(a-c x^2)^2} \, dx\)
Optimal. Leaf size=263 \[ -\frac {\left (5 \sqrt {a} e+2 \sqrt {c} d\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac {\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{9/4}}+\frac {e \sqrt {d+e x} \left (5 a e^2+c d^2\right )}{2 a c^2}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}+\frac {d e (d+e x)^{3/2}}{2 a c} \]
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Rubi [A] time = 0.53, antiderivative size = 263, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used =
{739, 825, 827, 1166, 208} \begin {gather*} -\frac {\left (5 \sqrt {a} e+2 \sqrt {c} d\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac {\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{9/4}}+\frac {e \sqrt {d+e x} \left (5 a e^2+c d^2\right )}{2 a c^2}+\frac {(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}+\frac {d e (d+e x)^{3/2}}{2 a c} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^(7/2)/(a - c*x^2)^2,x]
[Out]
(e*(c*d^2 + 5*a*e^2)*Sqrt[d + e*x])/(2*a*c^2) + (d*e*(d + e*x)^(3/2))/(2*a*c) + ((a*e + c*d*x)*(d + e*x)^(5/2)
)/(2*a*c*(a - c*x^2)) - ((Sqrt[c]*d - Sqrt[a]*e)^(5/2)*(2*Sqrt[c]*d + 5*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e
*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(9/4)) + ((2*Sqrt[c]*d - 5*Sqrt[a]*e)*(Sqrt[c]*d + Sqrt[a]*e)^
(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(9/4))
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 739
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(a*e - c*d*x)*(a
+ c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
Rule 825
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g*(d + e*x)^m)/
(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x])/(a + c*x^2), x], x] /
; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && GtQ[m, 0]
Rule 827
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]
Rule 1166
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]
Rubi steps
\begin {align*} \int \frac {(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx &=\frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (-2 c d^2+5 a e^2\right )+\frac {3}{2} c d e x\right )}{a-c x^2} \, dx}{2 a c}\\ &=\frac {d e (d+e x)^{3/2}}{2 a c}+\frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}+\frac {\int \frac {\sqrt {d+e x} \left (c d \left (c d^2-4 a e^2\right )-\frac {1}{2} c e \left (c d^2+5 a e^2\right ) x\right )}{a-c x^2} \, dx}{2 a c^2}\\ &=\frac {e \left (c d^2+5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}+\frac {d e (d+e x)^{3/2}}{2 a c}+\frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {-\frac {1}{2} c \left (c d^2-5 a e^2\right ) \left (2 c d^2+a e^2\right )-\frac {1}{2} c^2 d e \left (c d^2-13 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{2 a c^3}\\ &=\frac {e \left (c d^2+5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}+\frac {d e (d+e x)^{3/2}}{2 a c}+\frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} c^2 d^2 e \left (c d^2-13 a e^2\right )-\frac {1}{2} c e \left (c d^2-5 a e^2\right ) \left (2 c d^2+a e^2\right )-\frac {1}{2} c^2 d e \left (c d^2-13 a e^2\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{a c^3}\\ &=\frac {e \left (c d^2+5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}+\frac {d e (d+e x)^{3/2}}{2 a c}+\frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}+\frac {\left (\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^3\right ) \operatorname {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} c^{3/2}}-\frac {\left (\left (\sqrt {c} d-\sqrt {a} e\right )^3 \left (2 \sqrt {c} d+5 \sqrt {a} e\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} c^{3/2}}\\ &=\frac {e \left (c d^2+5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}+\frac {d e (d+e x)^{3/2}}{2 a c}+\frac {(a e+c d x) (d+e x)^{5/2}}{2 a c \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \left (2 \sqrt {c} d+5 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac {\left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 251, normalized size = 0.95 \begin {gather*} \frac {2 \sqrt {a} \sqrt [4]{c} \sqrt {d+e x} \left (5 a^2 e^3+a c e \left (3 d^2+3 d e x-4 e^2 x^2\right )+c^2 d^3 x\right )+\left (c x^2-a\right ) \left (5 \sqrt {a} e+2 \sqrt {c} d\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )-\left (c x^2-a\right ) \left (2 \sqrt {c} d-5 \sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{9/4} \left (a-c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^(7/2)/(a - c*x^2)^2,x]
[Out]
(2*Sqrt[a]*c^(1/4)*Sqrt[d + e*x]*(5*a^2*e^3 + c^2*d^3*x + a*c*e*(3*d^2 + 3*d*e*x - 4*e^2*x^2)) + (Sqrt[c]*d -
Sqrt[a]*e)^(5/2)*(2*Sqrt[c]*d + 5*Sqrt[a]*e)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqr
t[a]*e]] - (2*Sqrt[c]*d - 5*Sqrt[a]*e)*(Sqrt[c]*d + Sqrt[a]*e)^(5/2)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*
x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(9/4)*(a - c*x^2))
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IntegrateAlgebraic [A] time = 1.19, size = 413, normalized size = 1.57 \begin {gather*} \frac {e \sqrt {d+e x} \left (5 a^2 e^4-4 a c d^2 e^2+11 a c d e^2 (d+e x)-4 a c e^2 (d+e x)^2-c^2 d^4+c^2 d^3 (d+e x)\right )}{2 a c^2 \left (a e^2-c d^2+2 c d (d+e x)-c (d+e x)^2\right )}+\frac {\left (-13 a^{3/2} \sqrt {c} d e^3-5 a^2 e^4+\sqrt {a} c^{3/2} d^3 e-9 a c d^2 e^2+2 c^2 d^4\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{4 a^{3/2} c^2 \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}+\frac {\left (-13 a^{3/2} \sqrt {c} d e^3+5 a^2 e^4+\sqrt {a} c^{3/2} d^3 e+9 a c d^2 e^2-2 c^2 d^4\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{4 a^{3/2} c^2 \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(d + e*x)^(7/2)/(a - c*x^2)^2,x]
[Out]
(e*Sqrt[d + e*x]*(-(c^2*d^4) - 4*a*c*d^2*e^2 + 5*a^2*e^4 + c^2*d^3*(d + e*x) + 11*a*c*d*e^2*(d + e*x) - 4*a*c*
e^2*(d + e*x)^2))/(2*a*c^2*(-(c*d^2) + a*e^2 + 2*c*d*(d + e*x) - c*(d + e*x)^2)) + ((2*c^2*d^4 + Sqrt[a]*c^(3/
2)*d^3*e - 9*a*c*d^2*e^2 - 13*a^(3/2)*Sqrt[c]*d*e^3 - 5*a^2*e^4)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt
[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(4*a^(3/2)*c^2*Sqrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))]) + ((-2*c^2*d^4 +
Sqrt[a]*c^(3/2)*d^3*e + 9*a*c*d^2*e^2 - 13*a^(3/2)*Sqrt[c]*d*e^3 + 5*a^2*e^4)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*S
qrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(4*a^(3/2)*c^2*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))])
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fricas [B] time = 0.63, size = 2073, normalized size = 7.88
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(7/2)/(-c*x^2+a)^2,x, algorithm="fricas")
[Out]
1/8*((a*c^3*x^2 - a^2*c^2)*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^3*c^4*sqr
t((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*
c^9)))/(a^3*c^4))*log((140*c^5*d^10*e^3 - 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 - 8366*a^3*c^2*d^4*e^9 + 2
500*a^4*c*d^2*e^11 + 625*a^5*e^13)*sqrt(e*x + d) + (35*a^2*c^5*d^6*e^4 + 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*
e^8 - 125*a^5*c^2*e^10 - 2*(a^3*c^8*d^3 - 4*a^4*c^7*d*e^2)*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 2196
6*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2 + 70*a
^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^3*c^4*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 +
7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))) - (a*c^3*x^2 - a^2*c^2)*sqrt((4*c^3*d^7 - 35*a*c^
2*d^5*e^2 + 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^3*c^4*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^
2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))*log((140*c^5*d^10*e^3 - 1771*a*c^4
*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 - 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 + 625*a^5*e^13)*sqrt(e*x + d) - (
35*a^2*c^5*d^6*e^4 + 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*e^8 - 125*a^5*c^2*e^10 - 2*(a^3*c^8*d^3 - 4*a^4*c^7*
d*e^2)*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e
^14)/(a^3*c^9)))*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^3*c^4*sqrt((1225*c^
4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^
3*c^4))) + (a*c^3*x^2 - a^2*c^2)*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^3*c
^4*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)
/(a^3*c^9)))/(a^3*c^4))*log((140*c^5*d^10*e^3 - 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 - 8366*a^3*c^2*d^4*e
^9 + 2500*a^4*c*d^2*e^11 + 625*a^5*e^13)*sqrt(e*x + d) + (35*a^2*c^5*d^6*e^4 + 21*a^3*c^4*d^4*e^6 - 795*a^4*c^
3*d^2*e^8 - 125*a^5*c^2*e^10 + 2*(a^3*c^8*d^3 - 4*a^4*c^7*d*e^2)*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8
+ 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2
+ 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^3*c^4*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*
e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))) - (a*c^3*x^2 - a^2*c^2)*sqrt((4*c^3*d^7 - 3
5*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^3*c^4*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21
966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))*log((140*c^5*d^10*e^3 - 1771
*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 - 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 + 625*a^5*e^13)*sqrt(e*x +
d) - (35*a^2*c^5*d^6*e^4 + 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*e^8 - 125*a^5*c^2*e^10 + 2*(a^3*c^8*d^3 - 4*a^
4*c^7*d*e^2)*sqrt((1225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625
*a^4*e^14)/(a^3*c^9)))*sqrt((4*c^3*d^7 - 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^3*c^4*sqrt((1
225*c^4*d^8*e^6 - 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 + 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)
))/(a^3*c^4))) + 4*(4*a*c*e^3*x^2 - 3*a*c*d^2*e - 5*a^2*e^3 - (c^2*d^3 + 3*a*c*d*e^2)*x)*sqrt(e*x + d))/(a*c^3
*x^2 - a^2*c^2)
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giac [B] time = 0.66, size = 572, normalized size = 2.17 \begin {gather*} -\frac {{\left ({\left (c^{2} d^{3} e^{2} - 13 \, a c d e^{4}\right )} a^{2} {\left | c \right |} + {\left (\sqrt {a c} c^{2} d^{4} e + 4 \, \sqrt {a c} a c d^{2} e^{3} - 5 \, \sqrt {a c} a^{2} e^{5}\right )} {\left | a \right |} {\left | c \right |} - {\left (2 \, a c^{3} d^{5} - 9 \, a^{2} c^{2} d^{3} e^{2} - 5 \, a^{3} c d e^{4}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{3} d + \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} - a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} e - \sqrt {a c} a c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |}} - \frac {{\left ({\left (\sqrt {a c} c d^{3} e^{2} - 13 \, \sqrt {a c} a d e^{4}\right )} a^{2} {\left | c \right |} - {\left (a c^{2} d^{4} e + 4 \, a^{2} c d^{2} e^{3} - 5 \, a^{3} e^{5}\right )} {\left | a \right |} {\left | c \right |} - {\left (2 \, \sqrt {a c} a c^{2} d^{5} - 9 \, \sqrt {a c} a^{2} c d^{3} e^{2} - 5 \, \sqrt {a c} a^{3} d e^{4}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{3} d - \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} - a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} d + \sqrt {a c} a^{2} c^{2} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |}} + \frac {2 \, \sqrt {x e + d} e^{3}}{c^{2}} - \frac {{\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{3} e - \sqrt {x e + d} c^{2} d^{4} e + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} a c d e^{3} + \sqrt {x e + d} a^{2} e^{5}}{2 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )} a c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(7/2)/(-c*x^2+a)^2,x, algorithm="giac")
[Out]
-1/4*((c^2*d^3*e^2 - 13*a*c*d*e^4)*a^2*abs(c) + (sqrt(a*c)*c^2*d^4*e + 4*sqrt(a*c)*a*c*d^2*e^3 - 5*sqrt(a*c)*a
^2*e^5)*abs(a)*abs(c) - (2*a*c^3*d^5 - 9*a^2*c^2*d^3*e^2 - 5*a^3*c*d*e^4)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(
a*c^3*d + sqrt(a^2*c^6*d^2 - (a*c^3*d^2 - a^2*c^2*e^2)*a*c^3))/(a*c^3)))/((a^2*c^3*e - sqrt(a*c)*a*c^3*d)*sqrt
(-c^2*d - sqrt(a*c)*c*e)*abs(a)) - 1/4*((sqrt(a*c)*c*d^3*e^2 - 13*sqrt(a*c)*a*d*e^4)*a^2*abs(c) - (a*c^2*d^4*e
+ 4*a^2*c*d^2*e^3 - 5*a^3*e^5)*abs(a)*abs(c) - (2*sqrt(a*c)*a*c^2*d^5 - 9*sqrt(a*c)*a^2*c*d^3*e^2 - 5*sqrt(a*
c)*a^3*d*e^4)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^3*d - sqrt(a^2*c^6*d^2 - (a*c^3*d^2 - a^2*c^2*e^2)*a*c^3
))/(a*c^3)))/((a^2*c^3*d + sqrt(a*c)*a^2*c^2*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(a)) + 2*sqrt(x*e + d)*e^3/c^2
- 1/2*((x*e + d)^(3/2)*c^2*d^3*e - sqrt(x*e + d)*c^2*d^4*e + 3*(x*e + d)^(3/2)*a*c*d*e^3 + sqrt(x*e + d)*a^2*
e^5)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 - a*e^2)*a*c^2)
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maple [B] time = 0.10, size = 717, normalized size = 2.73 \begin {gather*} -\frac {5 a \,e^{5} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, c}-\frac {5 a \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, c}+\frac {c \,d^{4} e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}+\frac {c \,d^{4} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {9 d^{2} e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {9 d^{2} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\sqrt {e x +d}\, a \,e^{5}}{2 \left (c \,e^{2} x^{2}-a \,e^{2}\right ) c^{2}}+\frac {\sqrt {e x +d}\, d^{4} e}{2 \left (c \,e^{2} x^{2}-a \,e^{2}\right ) a}+\frac {d^{3} e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {13 d \,e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, c}+\frac {13 d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, c}-\frac {\left (e x +d \right )^{\frac {3}{2}} d^{3} e}{2 \left (c \,e^{2} x^{2}-a \,e^{2}\right ) a}-\frac {3 \left (e x +d \right )^{\frac {3}{2}} d \,e^{3}}{2 \left (c \,e^{2} x^{2}-a \,e^{2}\right ) c}+\frac {2 \sqrt {e x +d}\, e^{3}}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^(7/2)/(-c*x^2+a)^2,x)
[Out]
2*e^3/c^2*(e*x+d)^(1/2)-3/2*e^3/c/(c*e^2*x^2-a*e^2)*d*(e*x+d)^(3/2)-1/2*e/(c*e^2*x^2-a*e^2)*d^3/a*(e*x+d)^(3/2
)-1/2*e^5/c^2/(c*e^2*x^2-a*e^2)*a*(e*x+d)^(1/2)+1/2*e/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(1/2)*d^4-5/4*e^5/c*a/(a*c*e
^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)-9/4*e^3/(a*
c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d^2+1/2*
e*c/a/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)
*d^4-13/4*e^3/c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d+1/4
*e/a/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d^3-5/4*e^5/c*a/
(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)-9/4*
e^3/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*
d^2+1/2*e*c/a/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)
^(1/2)*c)*d^4+13/4*e^3/c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2
)*c)*d-1/4*e/a/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d^3
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} - a\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(7/2)/(-c*x^2+a)^2,x, algorithm="maxima")
[Out]
integrate((e*x + d)^(7/2)/(c*x^2 - a)^2, x)
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mupad [B] time = 1.21, size = 4090, normalized size = 15.55
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x)^(7/2)/(a - c*x^2)^2,x)
[Out]
atan((a^2*e^10*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)
/(64*a^2*c^2) + (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^
3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/((491*a*d^3*e^11)/(2*c) - (885*d^5*e^9)/2 + (329*c*d^7*e^7)/(2*a)
+ (50*a^2*d*e^13)/c^2 - (35*c^2*d^9*e^5)/(2*a^2) + (125*e^14*(a^9*c^9)^(1/2))/(4*a^2*c^7) + (335*d^2*e^12*(a^9
*c^9)^(1/2))/(2*a^3*c^6) - (204*d^4*e^10*(a^9*c^9)^(1/2))/(a^4*c^5) - (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a^5*c^4)
+ (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^6*c^3)) - (d^3*e^7*(a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) +
d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9)
+ (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*308i)/((329*a^3
*c^4*d^7*e^7)/2 - (35*a^2*c^5*d^9*e^5)/2 - (885*a^4*c^3*d^5*e^9)/2 + (491*a^5*c^2*d^3*e^11)/2 + 50*a^6*c*d*e^1
3 + (125*a^2*e^14*(a^9*c^9)^(1/2))/(4*c^4) + (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^2) - (204*d^4*e^10*(a^9*c^9)^(1
/2))/c^2 + (335*a*d^2*e^12*(a^9*c^9)^(1/2))/(2*c^3) - (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a*c)) + (d^5*e^5*(a^9*c^9
)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^
2*c^2) + (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(a^9*
c^9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/(50*a^7*d*e^13 + (491*a^6*c*d^3*e^11)/2 - (35*a^3*c^4*d^9*e^5)/2 + (329*a
^4*c^3*d^7*e^7)/2 - (885*a^5*c^2*d^5*e^9)/2 + (125*a^3*e^14*(a^9*c^9)^(1/2))/(4*c^5) - (7*d^6*e^8*(a^9*c^9)^(1
/2))/(2*c^2) - (204*a*d^4*e^10*(a^9*c^9)^(1/2))/c^3 + (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a*c) + (335*a^2*d^2*e^12
*(a^9*c^9)^(1/2))/(2*c^4)) + (a*d^2*e^8*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/
(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(a^9*c^9)^(1/2))/
(32*a^5*c^8) - (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*308i)/((329*d^7*e^7)/(2*a) - (885*d^5*e^9)/(2*
c) + (491*a*d^3*e^11)/(2*c^2) - (35*c*d^9*e^5)/(2*a^2) + (50*a^2*d*e^13)/c^3 + (125*e^14*(a^9*c^9)^(1/2))/(4*a
^2*c^8) + (335*d^2*e^12*(a^9*c^9)^(1/2))/(2*a^3*c^7) - (204*d^4*e^10*(a^9*c^9)^(1/2))/(a^4*c^6) - (7*d^6*e^8*(
a^9*c^9)^(1/2))/(2*a^5*c^5) + (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^6*c^4)) - (c*d^4*e^6*(d + e*x)^(1/2)*((105*d*e
^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(a^9*c^9)^(1/2))
/(64*a^4*c^9) + (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*7
0i)/((329*d^7*e^7)/(2*a) - (885*d^5*e^9)/(2*c) + (491*a*d^3*e^11)/(2*c^2) - (35*c*d^9*e^5)/(2*a^2) + (50*a^2*d
*e^13)/c^3 + (125*e^14*(a^9*c^9)^(1/2))/(4*a^2*c^8) + (335*d^2*e^12*(a^9*c^9)^(1/2))/(2*a^3*c^7) - (204*d^4*e^
10*(a^9*c^9)^(1/2))/(a^4*c^6) - (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a^5*c^5) + (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^6*
c^4)) - (d*e^9*(a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3
) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c
^8) - (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/(50*a^5*c^2*d*e^13 - (35*a*c^6*d^9*e^5)/2 + (329*a
^2*c^5*d^7*e^7)/2 - (885*a^3*c^4*d^5*e^9)/2 + (491*a^4*c^3*d^3*e^11)/2 + (125*a*e^14*(a^9*c^9)^(1/2))/(4*c^3)
- (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a^2) + (335*d^2*e^12*(a^9*c^9)^(1/2))/(2*c^2) + (35*c*d^8*e^6*(a^9*c^9)^(1/2)
)/(4*a^3) - (204*d^4*e^10*(a^9*c^9)^(1/2))/(a*c)))*((25*a^2*e^7*(a^9*c^9)^(1/2) + 4*a^3*c^8*d^7 + 105*a^6*c^5*
d*e^6 - 35*a^4*c^7*d^5*e^2 + 70*a^5*c^6*d^3*e^4 - 35*c^2*d^4*e^3*(a^9*c^9)^(1/2) + 154*a*c*d^2*e^5*(a^9*c^9)^(
1/2))/(64*a^6*c^9))^(1/2)*2i - atan((a^2*e^10*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3
*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(a^9*c^9)^(
1/2))/(32*a^5*c^8) + (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/((885*d^5*e^9)/2 - (491*a*d^3*e^11)
/(2*c) - (329*c*d^7*e^7)/(2*a) - (50*a^2*d*e^13)/c^2 + (35*c^2*d^9*e^5)/(2*a^2) + (125*e^14*(a^9*c^9)^(1/2))/(
4*a^2*c^7) + (335*d^2*e^12*(a^9*c^9)^(1/2))/(2*a^3*c^6) - (204*d^4*e^10*(a^9*c^9)^(1/2))/(a^4*c^5) - (7*d^6*e^
8*(a^9*c^9)^(1/2))/(2*a^5*c^4) + (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^6*c^3)) + (d^3*e^7*(a^9*c^9)^(1/2)*(d + e*x
)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7
*(a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*
a^6*c^7))^(1/2)*308i)/((35*a^2*c^5*d^9*e^5)/2 - (329*a^3*c^4*d^7*e^7)/2 + (885*a^4*c^3*d^5*e^9)/2 - (491*a^5*c
^2*d^3*e^11)/2 - 50*a^6*c*d*e^13 + (125*a^2*e^14*(a^9*c^9)^(1/2))/(4*c^4) + (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^
2) - (204*d^4*e^10*(a^9*c^9)^(1/2))/c^2 + (335*a*d^2*e^12*(a^9*c^9)^(1/2))/(2*c^3) - (7*d^6*e^8*(a^9*c^9)^(1/2
))/(2*a*c)) + (d^5*e^5*(a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(
32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(a^9*c^9)^(1/2))/(
32*a^5*c^8) + (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/(50*a^7*d*e^13 + (491*a^6*c*d^3*e^11)/2 -
(35*a^3*c^4*d^9*e^5)/2 + (329*a^4*c^3*d^7*e^7)/2 - (885*a^5*c^2*d^5*e^9)/2 - (125*a^3*e^14*(a^9*c^9)^(1/2))/(4
*c^5) + (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*c^2) + (204*a*d^4*e^10*(a^9*c^9)^(1/2))/c^3 - (35*d^8*e^6*(a^9*c^9)^(1/
2))/(4*a*c) - (335*a^2*d^2*e^12*(a^9*c^9)^(1/2))/(2*c^4)) - (a*d^2*e^8*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) +
d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9)
- (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*308i)/((329*d^7
*e^7)/(2*a) - (885*d^5*e^9)/(2*c) + (491*a*d^3*e^11)/(2*c^2) - (35*c*d^9*e^5)/(2*a^2) + (50*a^2*d*e^13)/c^3 -
(125*e^14*(a^9*c^9)^(1/2))/(4*a^2*c^8) - (335*d^2*e^12*(a^9*c^9)^(1/2))/(2*a^3*c^7) + (204*d^4*e^10*(a^9*c^9)^
(1/2))/(a^4*c^6) + (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a^5*c^5) - (35*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^6*c^4)) + (c*d^
4*e^6*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*
c^2) - (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(a^9*c^
9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/((329*d^7*e^7)/(2*a) - (885*d^5*e^9)/(2*c) + (491*a*d^3*e^11)/(2*c^2) - (35
*c*d^9*e^5)/(2*a^2) + (50*a^2*d*e^13)/c^3 - (125*e^14*(a^9*c^9)^(1/2))/(4*a^2*c^8) - (335*d^2*e^12*(a^9*c^9)^(
1/2))/(2*a^3*c^7) + (204*d^4*e^10*(a^9*c^9)^(1/2))/(a^4*c^6) + (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a^5*c^5) - (35*d
^8*e^6*(a^9*c^9)^(1/2))/(4*a^6*c^4)) + (d*e^9*(a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) + d^7/(16*
a^3*c) + (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2
*e^5*(a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/((35*a*c^6*d^9*e^5)
/2 - 50*a^5*c^2*d*e^13 - (329*a^2*c^5*d^7*e^7)/2 + (885*a^3*c^4*d^5*e^9)/2 - (491*a^4*c^3*d^3*e^11)/2 + (125*a
*e^14*(a^9*c^9)^(1/2))/(4*c^3) - (7*d^6*e^8*(a^9*c^9)^(1/2))/(2*a^2) + (335*d^2*e^12*(a^9*c^9)^(1/2))/(2*c^2)
+ (35*c*d^8*e^6*(a^9*c^9)^(1/2))/(4*a^3) - (204*d^4*e^10*(a^9*c^9)^(1/2))/(a*c)))*((4*a^3*c^8*d^7 - 25*a^2*e^7
*(a^9*c^9)^(1/2) + 105*a^6*c^5*d*e^6 - 35*a^4*c^7*d^5*e^2 + 70*a^5*c^6*d^3*e^4 + 35*c^2*d^4*e^3*(a^9*c^9)^(1/2
) - 154*a*c*d^2*e^5*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2)*2i - (((a^2*e^5 - c^2*d^4*e)*(d + e*x)^(1/2))/(2*a) +
((c^2*d^3*e + 3*a*c*d*e^3)*(d + e*x)^(3/2))/(2*a))/(c^3*(d + e*x)^2 + c^3*d^2 - a*c^2*e^2 - 2*c^3*d*(d + e*x)
) + (2*e^3*(d + e*x)^(1/2))/c^2
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**(7/2)/(-c*x**2+a)**2,x)
[Out]
Timed out
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