3.1447 \(\int \frac{(A+B x) (d+e x)^{5/2}}{a-c x^2} \, dx\)

Optimal. Leaf size=237 \[ -\frac{2 \sqrt{d+e x} \left (a B e^2+2 A c d e+B c d^2\right )}{c^2}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{9/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{9/4}}-\frac{2 (d+e x)^{3/2} (A e+B d)}{3 c}-\frac{2 B (d+e x)^{5/2}}{5 c} \]

[Out]

(-2*(B*c*d^2 + 2*A*c*d*e + a*B*e^2)*Sqrt[d + e*x])/c^2 - (2*(B*d + A*e)*(d + e*x)^(3/2))/(3*c) - (2*B*(d + e*x
)^(5/2))/(5*c) + ((Sqrt[a]*B - A*Sqrt[c])*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[S
qrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(9/4)) + ((Sqrt[a]*B + A*Sqrt[c])*(Sqrt[c]*d + Sqrt[a]*e)^(5/2)*ArcTanh[(c^
(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(9/4))

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Rubi [A]  time = 0.641833, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {825, 827, 1166, 208} \[ -\frac{2 \sqrt{d+e x} \left (a B e^2+2 A c d e+B c d^2\right )}{c^2}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{9/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{9/4}}-\frac{2 (d+e x)^{3/2} (A e+B d)}{3 c}-\frac{2 B (d+e x)^{5/2}}{5 c} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(5/2))/(a - c*x^2),x]

[Out]

(-2*(B*c*d^2 + 2*A*c*d*e + a*B*e^2)*Sqrt[d + e*x])/c^2 - (2*(B*d + A*e)*(d + e*x)^(3/2))/(3*c) - (2*B*(d + e*x
)^(5/2))/(5*c) + ((Sqrt[a]*B - A*Sqrt[c])*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[S
qrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(9/4)) + ((Sqrt[a]*B + A*Sqrt[c])*(Sqrt[c]*d + Sqrt[a]*e)^(5/2)*ArcTanh[(c^
(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(9/4))

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}}{a-c x^2} \, dx &=-\frac{2 B (d+e x)^{5/2}}{5 c}-\frac{\int \frac{(d+e x)^{3/2} (-A c d-a B e-c (B d+A e) x)}{a-c x^2} \, dx}{c}\\ &=-\frac{2 (B d+A e) (d+e x)^{3/2}}{3 c}-\frac{2 B (d+e x)^{5/2}}{5 c}+\frac{\int \frac{\sqrt{d+e x} \left (c \left (A c d^2+2 a B d e+a A e^2\right )+c \left (B c d^2+2 A c d e+a B e^2\right ) x\right )}{a-c x^2} \, dx}{c^2}\\ &=-\frac{2 \left (B c d^2+2 A c d e+a B e^2\right ) \sqrt{d+e x}}{c^2}-\frac{2 (B d+A e) (d+e x)^{3/2}}{3 c}-\frac{2 B (d+e x)^{5/2}}{5 c}-\frac{\int \frac{-c \left (a B e \left (3 c d^2+a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right )-c^2 \left (B c d^3+3 A c d^2 e+3 a B d e^2+a A e^3\right ) x}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx}{c^3}\\ &=-\frac{2 \left (B c d^2+2 A c d e+a B e^2\right ) \sqrt{d+e x}}{c^2}-\frac{2 (B d+A e) (d+e x)^{3/2}}{3 c}-\frac{2 B (d+e x)^{5/2}}{5 c}-\frac{2 \operatorname{Subst}\left (\int \frac{c^2 d \left (B c d^3+3 A c d^2 e+3 a B d e^2+a A e^3\right )-c e \left (a B e \left (3 c d^2+a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right )-c^2 \left (B c d^3+3 A c d^2 e+3 a B d e^2+a A e^3\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt{d+e x}\right )}{c^3}\\ &=-\frac{2 \left (B c d^2+2 A c d e+a B e^2\right ) \sqrt{d+e x}}{c^2}-\frac{2 (B d+A e) (d+e x)^{3/2}}{3 c}-\frac{2 B (d+e x)^{5/2}}{5 c}+\frac{\left (\left (\sqrt{a} B-A \sqrt{c}\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 )}{\sqrt{a} c^{3/2}}+\frac{\left (\left (\sqrt{a} B+A \sqrt{c}\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 )}{\sqrt{a} c^{3/2}}\\ &=-\frac{2 \left (B c d^2+2 A c d e+a B e^2\right ) \sqrt{d+e x}}{c^2}-\frac{2 (B d+A e) (d+e x)^{3/2}}{3 c}-\frac{2 B (d+e x)^{5/2}}{5 c}+\frac{\left (\sqrt{a} B-A \sqrt{c}\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 )}{\sqrt{a} c^{9/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\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 )}{\sqrt{a} c^{9/4}}\\ \end{align*}

Mathematica [A]  time = 0.498983, size = 223, normalized size = 0.94 \[ \frac{-2 \sqrt{a} \sqrt [4]{c} \sqrt{d+e x} \left (15 a B e^2+5 A c e (7 d+e x)+B c \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )-15 \left (A \sqrt{c}-\sqrt{a} B\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )+15 \left (\sqrt{a} B+A \sqrt{c}\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{15 \sqrt{a} c^{9/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(5/2))/(a - c*x^2),x]

[Out]

(-2*Sqrt[a]*c^(1/4)*Sqrt[d + e*x]*(15*a*B*e^2 + 5*A*c*e*(7*d + e*x) + B*c*(23*d^2 + 11*d*e*x + 3*e^2*x^2)) - 1
5*(-(Sqrt[a]*B) + A*Sqrt[c])*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sq
rt[a]*e]] + 15*(Sqrt[a]*B + A*Sqrt[c])*(Sqrt[c]*d + Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt
[c]*d + Sqrt[a]*e]])/(15*Sqrt[a]*c^(9/4))

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Maple [B]  time = 0.059, size = 981, normalized size = 4.1 \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),x)

[Out]

-2/5*B*(e*x+d)^(5/2)/c-2/3/c*A*(e*x+d)^(3/2)*e-2/3/c*B*(e*x+d)^(3/2)*d-4/c*A*d*e*(e*x+d)^(1/2)-2/c^2*a*B*e^2*(
e*x+d)^(1/2)-2/c*B*d^2*(e*x+d)^(1/2)+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*a*d*e^3+c/(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*e+1/c/(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*a^2*e^4+3/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*ar
ctanh((e*x+d)^(1/2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*B*a*d^2*e^2+1/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*a*e^3+3/((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*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))*B*a*d*e^2+1/((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^3+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*a*d*e^3+c/(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*e+1/c/(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*a^2*e^4+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))*B*a*d^2*e^2-1/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*a*e^3-3/((-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*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))*B*a*d*e^2-1/((-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^3

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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}}}{c x^{2} - a}\,{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),x, algorithm="maxima")

[Out]

-integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 - a), x)

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Fricas [B]  time = 58.0375, size = 14797, normalized size = 62.43 \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),x, algorithm="fricas")

[Out]

-1/30*(15*c^2*sqrt((10*A*B*a*c^2*d^4*e + 20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (B^2*a*c^2 + A^2*c^3)*d^5 + 10
*(B^2*a^2*c + A^2*a*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^2*a^2*c)*d*e^4 + a*c^4*sqrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*
a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3
*B*a*c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B
*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A
^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4
*c^2)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c^4))*log(-(2*(A*B^3*a*c^5 - A^3*B*
c^6)*d^9 + 5*(B^4*a^2*c^4 - A^4*c^6)*d^8*e + 16*(A*B^3*a^2*c^4 - A^3*B*a*c^5)*d^7*e^2 - 28*(A*B^3*a^3*c^3 - A^
3*B*a^2*c^4)*d^5*e^4 - 14*(B^4*a^4*c^2 - A^4*a^2*c^4)*d^4*e^5 + 8*(B^4*a^5*c - A^4*a^3*c^3)*d^2*e^7 + 10*(A*B^
3*a^5*c - A^3*B*a^4*c^2)*d*e^8 + (B^4*a^6 - A^4*a^4*c^2)*e^9)*sqrt(e*x + d) + (2*A*B^2*a*c^6*d^7 + (5*B^3*a^2*
c^5 + 9*A^2*B*a*c^6)*d^6*e + 2*(16*A*B^2*a^2*c^5 + 5*A^3*a*c^6)*d^5*e^2 + 5*(3*B^3*a^3*c^4 + 11*A^2*B*a^2*c^5)
*d^4*e^3 + 10*(5*A*B^2*a^3*c^4 + 2*A^3*a^2*c^5)*d^3*e^4 + (11*B^3*a^4*c^3 + 31*A^2*B*a^3*c^4)*d^2*e^5 + 2*(6*A
*B^2*a^4*c^3 + A^3*a^3*c^4)*d*e^6 + (B^3*a^5*c^2 + A^2*B*a^4*c^3)*e^7 - (A*a*c^8*d^2 + 2*B*a^2*c^7*d*e + A*a^2
*c^7*e^2)*sqrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5
+ 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*
A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 1
1*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2 + A^4
*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(
a*c^9)))*sqrt((10*A*B*a*c^2*d^4*e + 20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (B^2*a*c^2 + A^2*c^3)*d^5 + 10*(B^2
*a^2*c + A^2*a*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^2*a^2*c)*d*e^4 + a*c^4*sqrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5
+ A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*
c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*a^2*
c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*
a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)
*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c^4))) - 15*c^2*sqrt((10*A*B*a*c^2*d^4*e
+ 20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (B^2*a*c^2 + A^2*c^3)*d^5 + 10*(B^2*a^2*c + A^2*a*c^2)*d^3*e^2 + 5*(
B^2*a^3 + A^2*a^2*c)*d*e^4 + a*c^4*sqrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^
2*c^4 + 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3
+ 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2
+ 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c
+ 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5
*c + A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c^4))*log(-(2*(A*B^3*a*c^5 - A^3*B*c^6)*d^9 + 5*(B^4*a^2*c^4 - A^4*c^6)*d
^8*e + 16*(A*B^3*a^2*c^4 - A^3*B*a*c^5)*d^7*e^2 - 28*(A*B^3*a^3*c^3 - A^3*B*a^2*c^4)*d^5*e^4 - 14*(B^4*a^4*c^2
- A^4*a^2*c^4)*d^4*e^5 + 8*(B^4*a^5*c - A^4*a^3*c^3)*d^2*e^7 + 10*(A*B^3*a^5*c - A^3*B*a^4*c^2)*d*e^8 + (B^4*
a^6 - A^4*a^4*c^2)*e^9)*sqrt(e*x + d) - (2*A*B^2*a*c^6*d^7 + (5*B^3*a^2*c^5 + 9*A^2*B*a*c^6)*d^6*e + 2*(16*A*B
^2*a^2*c^5 + 5*A^3*a*c^6)*d^5*e^2 + 5*(3*B^3*a^3*c^4 + 11*A^2*B*a^2*c^5)*d^4*e^3 + 10*(5*A*B^2*a^3*c^4 + 2*A^3
*a^2*c^5)*d^3*e^4 + (11*B^3*a^4*c^3 + 31*A^2*B*a^3*c^4)*d^2*e^5 + 2*(6*A*B^2*a^4*c^3 + A^3*a^3*c^4)*d*e^6 + (B
^3*a^5*c^2 + A^2*B*a^4*c^3)*e^7 - (A*a*c^8*d^2 + 2*B*a^2*c^7*d*e + A*a^2*c^7*e^2)*sqrt((4*A^2*B^2*c^6*d^10 + 2
0*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*
c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^
3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^
4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c +
A^3*B*a^4*c^2)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))*sqrt((10*A*B*a*c^2*d^4*e + 20
*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (B^2*a*c^2 + A^2*c^3)*d^5 + 10*(B^2*a^2*c + A^2*a*c^2)*d^3*e^2 + 5*(B^2*a
^3 + A^2*a^2*c)*d*e^4 + a*c^4*sqrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4
+ 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3 + 32*
A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62
*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A
^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c +
A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c^4))) + 15*c^2*sqrt((10*A*B*a*c^2*d^4*e + 20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^
5 + (B^2*a*c^2 + A^2*c^3)*d^5 + 10*(B^2*a^2*c + A^2*a*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^2*a^2*c)*d*e^4 - a*c^4*sqr
t((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*
d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^
6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4
)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2
*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c
^4))*log(-(2*(A*B^3*a*c^5 - A^3*B*c^6)*d^9 + 5*(B^4*a^2*c^4 - A^4*c^6)*d^8*e + 16*(A*B^3*a^2*c^4 - A^3*B*a*c^5
)*d^7*e^2 - 28*(A*B^3*a^3*c^3 - A^3*B*a^2*c^4)*d^5*e^4 - 14*(B^4*a^4*c^2 - A^4*a^2*c^4)*d^4*e^5 + 8*(B^4*a^5*c
- A^4*a^3*c^3)*d^2*e^7 + 10*(A*B^3*a^5*c - A^3*B*a^4*c^2)*d*e^8 + (B^4*a^6 - A^4*a^4*c^2)*e^9)*sqrt(e*x + d)
+ (2*A*B^2*a*c^6*d^7 + (5*B^3*a^2*c^5 + 9*A^2*B*a*c^6)*d^6*e + 2*(16*A*B^2*a^2*c^5 + 5*A^3*a*c^6)*d^5*e^2 + 5*
(3*B^3*a^3*c^4 + 11*A^2*B*a^2*c^5)*d^4*e^3 + 10*(5*A*B^2*a^3*c^4 + 2*A^3*a^2*c^5)*d^3*e^4 + (11*B^3*a^4*c^3 +
31*A^2*B*a^3*c^4)*d^2*e^5 + 2*(6*A*B^2*a^4*c^3 + A^3*a^3*c^4)*d*e^6 + (B^3*a^5*c^2 + A^2*B*a^4*c^3)*e^7 + (A*a
*c^8*d^2 + 2*B*a^2*c^7*d*e + A*a^2*c^7*e^2)*sqrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*
(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4
*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^
4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B
^4*a^5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e^9 + (B^4*a^6 + 2*A^
2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))*sqrt((10*A*B*a*c^2*d^4*e + 20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (
B^2*a*c^2 + A^2*c^3)*d^5 + 10*(B^2*a^2*c + A^2*a*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^2*a^2*c)*d*e^4 - a*c^4*sqrt((4*
A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e
^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4
+ 504*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4
*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8
+ 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c^4)))
- 15*c^2*sqrt((10*A*B*a*c^2*d^4*e + 20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (B^2*a*c^2 + A^2*c^3)*d^5 + 10*(B^
2*a^2*c + A^2*a*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^2*a^2*c)*d*e^4 - a*c^4*sqrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^
5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a
*c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*a^2
*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B
*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2
)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c^4))*log(-(2*(A*B^3*a*c^5 - A^3*B*c^6)
*d^9 + 5*(B^4*a^2*c^4 - A^4*c^6)*d^8*e + 16*(A*B^3*a^2*c^4 - A^3*B*a*c^5)*d^7*e^2 - 28*(A*B^3*a^3*c^3 - A^3*B*
a^2*c^4)*d^5*e^4 - 14*(B^4*a^4*c^2 - A^4*a^2*c^4)*d^4*e^5 + 8*(B^4*a^5*c - A^4*a^3*c^3)*d^2*e^7 + 10*(A*B^3*a^
5*c - A^3*B*a^4*c^2)*d*e^8 + (B^4*a^6 - A^4*a^4*c^2)*e^9)*sqrt(e*x + d) - (2*A*B^2*a*c^6*d^7 + (5*B^3*a^2*c^5
+ 9*A^2*B*a*c^6)*d^6*e + 2*(16*A*B^2*a^2*c^5 + 5*A^3*a*c^6)*d^5*e^2 + 5*(3*B^3*a^3*c^4 + 11*A^2*B*a^2*c^5)*d^4
*e^3 + 10*(5*A*B^2*a^3*c^4 + 2*A^3*a^2*c^5)*d^3*e^4 + (11*B^3*a^4*c^3 + 31*A^2*B*a^3*c^4)*d^2*e^5 + 2*(6*A*B^2
*a^4*c^3 + A^3*a^3*c^4)*d*e^6 + (B^3*a^5*c^2 + A^2*B*a^4*c^3)*e^7 + (A*a*c^8*d^2 + 2*B*a^2*c^7*d*e + A*a^2*c^7
*e^2)*sqrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*
A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*
a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^
4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3
*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^
9)))*sqrt((10*A*B*a*c^2*d^4*e + 20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (B^2*a*c^2 + A^2*c^3)*d^5 + 10*(B^2*a^2
*c + A^2*a*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^2*a^2*c)*d*e^4 - a*c^4*sqrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A
^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)
*d^7*e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)
*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*
c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e
^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c^4))) + 4*(3*B*c*e^2*x^2 + 23*B*c*d^2 + 35*
A*c*d*e + 15*B*a*e^2 + (11*B*c*d*e + 5*A*c*e^2)*x)*sqrt(e*x + d))/c^2

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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),x)

[Out]

Timed out

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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),x, algorithm="giac")

[Out]

Timed out