3.1453 \(\int \frac{(A+B x) (d+e x)^{5/2}}{(a-c x^2)^2} \, dx\)
Optimal. Leaf size=269 \[ -\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (3 \sqrt{a} A \sqrt{c} e-5 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \left (-3 \sqrt{a} A \sqrt{c} e-5 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} c^{9/4}}+\frac{e \sqrt{d+e x} (5 a B e+A c d)}{2 a c^2}+\frac{(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{2 a c \left (a-c x^2\right )} \]
[Out]
(e*(A*c*d + 5*a*B*e)*Sqrt[d + e*x])/(2*a*c^2) + ((d + e*x)^(3/2)*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(2*a*c*(
a - c*x^2)) - ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(2*A*c*d - 5*a*B*e + 3*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt
[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(9/4)) + ((Sqrt[c]*d + Sqrt[a]*e)^(3/2)*(2*A*c*d - 5*a*B
*e - 3*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(9/4))
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Rubi [A] time = 0.541456, antiderivative size = 269, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used =
{819, 825, 827, 1166, 208} \[ -\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (3 \sqrt{a} A \sqrt{c} e-5 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \left (-3 \sqrt{a} A \sqrt{c} e-5 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} c^{9/4}}+\frac{e \sqrt{d+e x} (5 a B e+A c d)}{2 a c^2}+\frac{(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{2 a c \left (a-c x^2\right )} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(d + e*x)^(5/2))/(a - c*x^2)^2,x]
[Out]
(e*(A*c*d + 5*a*B*e)*Sqrt[d + e*x])/(2*a*c^2) + ((d + e*x)^(3/2)*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(2*a*c*(
a - c*x^2)) - ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(2*A*c*d - 5*a*B*e + 3*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt
[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(9/4)) + ((Sqrt[c]*d + Sqrt[a]*e)^(3/2)*(2*A*c*d - 5*a*B
*e - 3*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(9/4))
Rule 819
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
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]
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]
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx &=\frac{(d+e x)^{3/2} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}-\frac{\int \frac{\sqrt{d+e x} \left (\frac{1}{2} \left (-2 A c d^2+a e (5 B d+3 A e)\right )+\frac{1}{2} e (A c d+5 a B e) x\right )}{a-c x^2} \, dx}{2 a c}\\ &=\frac{e (A c d+5 a B e) \sqrt{d+e x}}{2 a c^2}+\frac{(d+e x)^{3/2} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}+\frac{\int \frac{\frac{1}{2} \left (2 A c d \left (c d^2-2 a e^2\right )-5 a B e \left (c d^2+a e^2\right )\right )+\frac{1}{2} c e \left (A c d^2-10 a B d e-3 a A e^2\right ) x}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx}{2 a c^2}\\ &=\frac{e (A c d+5 a B e) \sqrt{d+e x}}{2 a c^2}+\frac{(d+e x)^{3/2} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2} c d e \left (A c d^2-10 a B d e-3 a A e^2\right )+\frac{1}{2} e \left (2 A c d \left (c d^2-2 a e^2\right )-5 a B e \left (c d^2+a e^2\right )\right )+\frac{1}{2} c e \left (A c d^2-10 a B d e-3 a 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^2}\\ &=\frac{e (A c d+5 a B e) \sqrt{d+e x}}{2 a c^2}+\frac{(d+e x)^{3/2} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}+\frac{\left (\left (\sqrt{c} d+\sqrt{a} e\right )^2 \left (2 A c d-5 a B e-3 \sqrt{a} A \sqrt{c} 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{\left (\left (\sqrt{c} d-\sqrt{a} e\right )^2 \left (2 A c d-5 a B e+3 \sqrt{a} A \sqrt{c} 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 (A c d+5 a B e) \sqrt{d+e x}}{2 a c^2}+\frac{(d+e x)^{3/2} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (2 A c d-5 a B e+3 \sqrt{a} A \sqrt{c} 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 (\sqrt{c} d+\sqrt{a} e\right )^{3/2} \left (2 A c d-5 a B e-3 \sqrt{a} A \sqrt{c} 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}}\\ \end{align*}
Mathematica [A] time = 0.60702, size = 279, normalized size = 1.04 \[ \frac{2 \sqrt{a} \sqrt [4]{c} \sqrt{d+e x} \left (5 a^2 B e^2+a c \left (A e (2 d+e x)+B \left (d^2+2 d e x-4 e^2 x^2\right )\right )+A c^2 d^2 x\right )+\left (c x^2-a\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (3 \sqrt{a} A \sqrt{c} e-5 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )-\left (c x^2-a\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \left (-3 \sqrt{a} A \sqrt{c} e-5 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} c^{9/4} \left (a-c x^2\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(d + e*x)^(5/2))/(a - c*x^2)^2,x]
[Out]
(2*Sqrt[a]*c^(1/4)*Sqrt[d + e*x]*(5*a^2*B*e^2 + A*c^2*d^2*x + a*c*(A*e*(2*d + e*x) + B*(d^2 + 2*d*e*x - 4*e^2*
x^2))) + (Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(2*A*c*d - 5*a*B*e + 3*Sqrt[a]*A*Sqrt[c]*e)*(-a + c*x^2)*ArcTanh[(c^(1/
4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] - (Sqrt[c]*d + Sqrt[a]*e)^(3/2)*(2*A*c*d - 5*a*B*e - 3*Sqrt[a]*
A*Sqrt[c]*e)*(-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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Maple [B] time = 0.043, size = 1055, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+a)^2,x)
[Out]
-1/2*e^3/c/(c*e^2*x^2-a*e^2)*(e*x+d)^(3/2)*A-1/2*e/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(3/2)*A*d^2-e^2/c/(c*e^2*x^2-a*
e^2)*(e*x+d)^(3/2)*B*d-1/2*e^3/c/(c*e^2*x^2-a*e^2)*(e*x+d)^(1/2)*A*d+1/2*e/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(1/2)*A
*d^3-1/2*e^4/c^2/(c*e^2*x^2-a*e^2)*a*(e*x+d)^(1/2)*B+1/2*e^2/c/(c*e^2*x^2-a*e^2)*(e*x+d)^(1/2)*B*d^2-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/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*d+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/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A
*d^3-5/4*e^4/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/((c*d+(a*c*e^2)^(1/2)
)*c)^(1/2))*B-5/4*e^2/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^
(1/2))*c)^(1/2))*B*d^2-3/4*e^3/c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2)
)*c)^(1/2))*A+1/4*e/a/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))
*A*d^2-5/2*e^2/c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*B*d-
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/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*
A*d+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/((-c*d+(a*c*e^2)^(1/2))*
c)^(1/2))*A*d^3-5/4*e^4/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/((-c*d+(a*
c*e^2)^(1/2))*c)^(1/2))*B-5/4*e^2/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c
*d+(a*c*e^2)^(1/2))*c)^(1/2))*B*d^2+3/4*e^3/c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(
a*c*e^2)^(1/2))*c)^(1/2))*A-1/4*e/a/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(
1/2))*c)^(1/2))*A*d^2+5/2*e^2/c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((-c*d+(a*c*e^2)^(1/2)
)*c)^(1/2))*B*d+2*e^2*B/c^2*(e*x+d)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c x^{2} - a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+a)^2,x, algorithm="maxima")
[Out]
integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 - a)^2, x)
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Fricas [B] time = 60.5898, size = 11829, normalized size = 43.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+a)^2,x, algorithm="fricas")
[Out]
-1/8*((a*c^3*x^2 - a^2*c^2)*sqrt((4*A^2*c^3*d^5 - 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3 + 30*A*B*a^3*e^5 +
a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*
A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A
^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*
B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) + 5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5*B^2*a^3 + A^2*a^2
*c)*d*e^4)/(a^3*c^4))*log(-(120*A^3*B*c^5*d^7*e^2 - 20*(45*A^2*B^2*a*c^4 - A^4*c^5)*d^6*e^3 + 10*(225*A*B^3*a^
2*c^3 - 77*A^3*B*a*c^4)*d^5*e^4 - (1875*B^4*a^3*c^2 - 3000*A^2*B^2*a^2*c^3 + 101*A^4*a*c^4)*d^4*e^5 - 20*(175*
A*B^3*a^3*c^2 - 73*A^3*B*a^2*c^3)*d^3*e^6 + 2*(625*B^4*a^4*c - 1050*A^2*B^2*a^3*c^2 + 81*A^4*a^2*c^3)*d^2*e^7
+ 10*(125*A*B^3*a^4*c - 81*A^3*B*a^3*c^2)*d*e^8 + (625*B^4*a^5 - 81*A^4*a^3*c^2)*e^9)*sqrt(e*x + d) + (30*A^2*
B*a^2*c^5*d^4*e^3 + 5*(15*A*B^2*a^3*c^4 + A^3*a^2*c^5)*d^3*e^4 - 15*(25*B^3*a^4*c^3 + 3*A^2*B*a^3*c^4)*d^2*e^5
- 3*(125*A*B^2*a^4*c^3 + 3*A^3*a^3*c^4)*d*e^6 - 5*(25*B^3*a^5*c^2 + 9*A^2*B*a^4*c^3)*e^7 - (2*A*a^3*c^8*d^2 -
5*B*a^4*c^7*d*e - 3*A*a^4*c^7*e^2)*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 +
25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7
+ 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^
9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)))*sqrt((4*A^2*c^3*d^5 - 20*A*B*a*c^2*d^
4*e + 30*A*B*a^2*c*d^2*e^3 + 30*A*B*a^3*e^5 + a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^
3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3
*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A
^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) + 5*(5*B^2*a^2*c - 3
*A^2*a*c^2)*d^3*e^2 + 15*(5*B^2*a^3 + A^2*a^2*c)*d*e^4)/(a^3*c^4))) - (a*c^3*x^2 - a^2*c^2)*sqrt((4*A^2*c^3*d^
5 - 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3 + 30*A*B*a^3*e^5 + a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(
15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B
^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(2
5*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) +
5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5*B^2*a^3 + A^2*a^2*c)*d*e^4)/(a^3*c^4))*log(-(120*A^3*B*c^5*d^7*
e^2 - 20*(45*A^2*B^2*a*c^4 - A^4*c^5)*d^6*e^3 + 10*(225*A*B^3*a^2*c^3 - 77*A^3*B*a*c^4)*d^5*e^4 - (1875*B^4*a^
3*c^2 - 3000*A^2*B^2*a^2*c^3 + 101*A^4*a*c^4)*d^4*e^5 - 20*(175*A*B^3*a^3*c^2 - 73*A^3*B*a^2*c^3)*d^3*e^6 + 2*
(625*B^4*a^4*c - 1050*A^2*B^2*a^3*c^2 + 81*A^4*a^2*c^3)*d^2*e^7 + 10*(125*A*B^3*a^4*c - 81*A^3*B*a^3*c^2)*d*e^
8 + (625*B^4*a^5 - 81*A^4*a^3*c^2)*e^9)*sqrt(e*x + d) - (30*A^2*B*a^2*c^5*d^4*e^3 + 5*(15*A*B^2*a^3*c^4 + A^3*
a^2*c^5)*d^3*e^4 - 15*(25*B^3*a^4*c^3 + 3*A^2*B*a^3*c^4)*d^2*e^5 - 3*(125*A*B^2*a^4*c^3 + 3*A^3*a^3*c^4)*d*e^6
- 5*(25*B^3*a^5*c^2 + 9*A^2*B*a^4*c^3)*e^7 - (2*A*a^3*c^8*d^2 - 5*B*a^4*c^7*d*e - 3*A*a^4*c^7*e^2)*sqrt((900*
A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4
*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*
A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*
a^2*c^2)*e^10)/(a^3*c^9)))*sqrt((4*A^2*c^3*d^5 - 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3 + 30*A*B*a^3*e^5 +
a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A
^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^
2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B
^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) + 5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5*B^2*a^3 + A^2*a^2*
c)*d*e^4)/(a^3*c^4))) + (a*c^3*x^2 - a^2*c^2)*sqrt((4*A^2*c^3*d^5 - 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3
+ 30*A*B*a^3*e^5 - a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*
B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125
*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*
B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) + 5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5
*B^2*a^3 + A^2*a^2*c)*d*e^4)/(a^3*c^4))*log(-(120*A^3*B*c^5*d^7*e^2 - 20*(45*A^2*B^2*a*c^4 - A^4*c^5)*d^6*e^3
+ 10*(225*A*B^3*a^2*c^3 - 77*A^3*B*a*c^4)*d^5*e^4 - (1875*B^4*a^3*c^2 - 3000*A^2*B^2*a^2*c^3 + 101*A^4*a*c^4)*
d^4*e^5 - 20*(175*A*B^3*a^3*c^2 - 73*A^3*B*a^2*c^3)*d^3*e^6 + 2*(625*B^4*a^4*c - 1050*A^2*B^2*a^3*c^2 + 81*A^4
*a^2*c^3)*d^2*e^7 + 10*(125*A*B^3*a^4*c - 81*A^3*B*a^3*c^2)*d*e^8 + (625*B^4*a^5 - 81*A^4*a^3*c^2)*e^9)*sqrt(e
*x + d) + (30*A^2*B*a^2*c^5*d^4*e^3 + 5*(15*A*B^2*a^3*c^4 + A^3*a^2*c^5)*d^3*e^4 - 15*(25*B^3*a^4*c^3 + 3*A^2*
B*a^3*c^4)*d^2*e^5 - 3*(125*A*B^2*a^4*c^3 + 3*A^3*a^3*c^4)*d*e^6 - 5*(25*B^3*a^5*c^2 + 9*A^2*B*a^4*c^3)*e^7 +
(2*A*a^3*c^8*d^2 - 5*B*a^4*c^7*d*e - 3*A*a^4*c^7*e^2)*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^
3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3
*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A
^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)))*sqrt((4*A^2*c^3*d^5
- 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3 + 30*A*B*a^3*e^5 - a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(1
5*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^
3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25
*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) +
5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5*B^2*a^3 + A^2*a^2*c)*d*e^4)/(a^3*c^4))) - (a*c^3*x^2 - a^2*c^2)*
sqrt((4*A^2*c^3*d^5 - 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3 + 30*A*B*a^3*e^5 - a^3*c^4*sqrt((900*A^2*B^2*c
^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4
*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3
)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*
e^10)/(a^3*c^9)) + 5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5*B^2*a^3 + A^2*a^2*c)*d*e^4)/(a^3*c^4))*log(-(
120*A^3*B*c^5*d^7*e^2 - 20*(45*A^2*B^2*a*c^4 - A^4*c^5)*d^6*e^3 + 10*(225*A*B^3*a^2*c^3 - 77*A^3*B*a*c^4)*d^5*
e^4 - (1875*B^4*a^3*c^2 - 3000*A^2*B^2*a^2*c^3 + 101*A^4*a*c^4)*d^4*e^5 - 20*(175*A*B^3*a^3*c^2 - 73*A^3*B*a^2
*c^3)*d^3*e^6 + 2*(625*B^4*a^4*c - 1050*A^2*B^2*a^3*c^2 + 81*A^4*a^2*c^3)*d^2*e^7 + 10*(125*A*B^3*a^4*c - 81*A
^3*B*a^3*c^2)*d*e^8 + (625*B^4*a^5 - 81*A^4*a^3*c^2)*e^9)*sqrt(e*x + d) - (30*A^2*B*a^2*c^5*d^4*e^3 + 5*(15*A*
B^2*a^3*c^4 + A^3*a^2*c^5)*d^3*e^4 - 15*(25*B^3*a^4*c^3 + 3*A^2*B*a^3*c^4)*d^2*e^5 - 3*(125*A*B^2*a^4*c^3 + 3*
A^3*a^3*c^4)*d*e^6 - 5*(25*B^3*a^5*c^2 + 9*A^2*B*a^4*c^3)*e^7 + (2*A*a^3*c^8*d^2 - 5*B*a^4*c^7*d*e - 3*A*a^4*c
^7*e^2)*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A
^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^
2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B
^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)))*sqrt((4*A^2*c^3*d^5 - 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3 +
30*A*B*a^3*e^5 - a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B
^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*
B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B
^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) + 5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5*
B^2*a^3 + A^2*a^2*c)*d*e^4)/(a^3*c^4))) - 4*(4*B*a*c*e^2*x^2 - B*a*c*d^2 - 2*A*a*c*d*e - 5*B*a^2*e^2 - (A*c^2*
d^2 + 2*B*a*c*d*e + A*a*c*e^2)*x)*sqrt(e*x + d))/(a*c^3*x^2 - a^2*c^2)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**(5/2)/(-c*x**2+a)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+a)^2,x, algorithm="giac")
[Out]
Timed out