3.1451 \(\int \frac{A+B x}{(d+e x)^{3/2} (a-c x^2)} \, dx\)

Optimal. Leaf size=197 \[ -\frac{2 (B d-A e)}{\sqrt{d+e x} \left (c d^2-a e^2\right )}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \sqrt [4]{c} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.402406, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {829, 827, 1166, 208} \[ -\frac{2 (B d-A e)}{\sqrt{d+e x} \left (c d^2-a e^2\right )}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \sqrt [4]{c} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^(3/2)*(a - c*x^2)),x]

[Out]

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

Rule 829

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((e*f - d*g)*(d
+ e*x)^(m + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d*f + a*
e*g - c*(e*f - d*g)*x, x])/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&
FractionQ[m] && LtQ[m, -1]

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)^{3/2} \left (a-c x^2\right )} \, dx &=-\frac{2 (B d-A e)}{\left (c d^2-a e^2\right ) \sqrt{d+e x}}+\frac{\int \frac{-A c d+a B e-c (B d-A e) x}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx}{-c d^2+a e^2}\\ &=-\frac{2 (B d-A e)}{\left (c d^2-a e^2\right ) \sqrt{d+e x}}-\frac{2 \operatorname{Subst}\left (\int \frac{c d (B d-A e)+e (-A c d+a B e)-c (B d-A e) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt{d+e x}\right )}{c d^2-a e^2}\\ &=-\frac{2 (B d-A e)}{\left (c d^2-a e^2\right ) \sqrt{d+e x}}+\frac{\left (\left (B-\frac{A \sqrt{c}}{\sqrt{a}}\right ) \sqrt{c}\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{c} d-\sqrt{a} e}+\frac{\left (\left (B+\frac{A \sqrt{c}}{\sqrt{a}}\right ) \sqrt{c}\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{c} d+\sqrt{a} e}\\ &=-\frac{2 (B d-A e)}{\left (c d^2-a e^2\right ) \sqrt{d+e x}}+\frac{\left (B-\frac{A \sqrt{c}}{\sqrt{a}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}+\frac{\left (B+\frac{A \sqrt{c}}{\sqrt{a}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{a} e}}\right )}{\sqrt [4]{c} \left (\sqrt{c} d+\sqrt{a} e\right )^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.35821, size = 271, normalized size = 1.38 \[ \frac{\frac{B \left (\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt [4]{c}}-\frac{(B d-A e) \left (\left (\sqrt{a} e+\sqrt{c} d\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{a} e}\right )+\left (\sqrt{a} e-\sqrt{c} d\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{a} e}\right )\right )}{\sqrt{d+e x} \left (c d^2-a e^2\right )}}{\sqrt{a} e} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^(3/2)*(a - c*x^2)),x]

[Out]

((B*(-(ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]/Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ArcTanh[(c^
(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]/Sqrt[Sqrt[c]*d + Sqrt[a]*e]))/c^(1/4) - ((B*d - A*e)*((Sqrt[
c]*d + Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)] + (-(Sqrt[c]*d)
+ Sqrt[a]*e)*Hypergeometric2F1[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]))/((c*d^2 - a*e^2)*
Sqrt[d + e*x]))/(Sqrt[a]*e)

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Maple [B]  time = 0.026, size = 588, normalized size = 3. \begin{align*} -{\frac{A{c}^{2}de}{a{e}^{2}-c{d}^{2}}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{B{e}^{2}ac}{a{e}^{2}-c{d}^{2}}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{Ace}{a{e}^{2}-c{d}^{2}}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{Bcd}{a{e}^{2}-c{d}^{2}}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{A{c}^{2}de}{a{e}^{2}-c{d}^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{B{e}^{2}ac}{a{e}^{2}-c{d}^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{Ace}{a{e}^{2}-c{d}^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{Bcd}{a{e}^{2}-c{d}^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-2\,{\frac{Ae}{ \left ( a{e}^{2}-c{d}^{2} \right ) \sqrt{ex+d}}}+2\,{\frac{Bd}{ \left ( a{e}^{2}-c{d}^{2} \right ) \sqrt{ex+d}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a),x)

[Out]

-1/(a*e^2-c*d^2)*c^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))*A*d*e+1/(a*e^2-c*d^2)*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*B*e^2+1/(a*e^2-c*d^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))*A*e-1/(a*e^2-c*d^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-1/(a*e^2-c*d^2)*c^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))*A*d*e+1/(a*e^2-c*d^2)*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*B*e^2-1/(a*e^2-c*d^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))*A*e+1/(a*e^2-c*d^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/(a*e^2-c*d^2)/(
e*x+d)^(1/2)*A*e+2/(a*e^2-c*d^2)/(e*x+d)^(1/2)*B*d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{B x + A}{{\left (c x^{2} - a\right )}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a),x, algorithm="maxima")

[Out]

-integrate((B*x + A)/((c*x^2 - a)*(e*x + d)^(3/2)), x)

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Fricas [B]  time = 20.0065, size = 12672, normalized size = 64.32 \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)^(3/2)/(-c*x^2+a),x, algorithm="fricas")

[Out]

1/2*((c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3
- 3*(B^2*a^2 + A^2*a*c)*d*e^2 + (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((4*A^2*B^2*c^
4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*
B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c +
A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a
^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(a*c^3*d^6 - 3*a
^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((2*(A*B^3*a*c^2 - A^3*B*c^3)*d^3 - 3*(B^4*a^2*c - A^4*c^3)*d^
2*e + 6*(A*B^3*a^2*c - A^3*B*a*c^2)*d*e^2 - (B^4*a^3 - A^4*a*c^2)*e^3)*sqrt(e*x + d) + (2*A*B^2*a*c^3*d^5 - (3
*B^3*a^2*c^2 + 7*A^2*B*a*c^3)*d^4*e + 2*(7*A*B^2*a^2*c^2 + 3*A^3*a*c^3)*d^3*e^2 - 4*(B^3*a^3*c + 4*A^2*B*a^2*c
^2)*d^2*e^3 + 2*(4*A*B^2*a^3*c + A^3*a^2*c^2)*d*e^4 - (B^3*a^4 + A^2*B*a^3*c)*e^5 + (A*a*c^5*d^8 - 2*B*a^2*c^4
*d^7*e - 2*A*a^2*c^4*d^6*e^2 + 6*B*a^3*c^3*d^5*e^3 - 6*B*a^4*c^2*d^3*e^5 + 2*A*a^4*c^2*d^2*e^6 + 2*B*a^5*c*d*e
^7 - A*a^5*c*e^8)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2
*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^
4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*
c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e
^10 + a^7*c*e^12)))*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d
*e^2 + (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 +
A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3
)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (
B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c
^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*
d^2*e^4 - a^4*e^6))) - (c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a
*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 + (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*
sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4
)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 -
12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c
^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12))
)/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((2*(A*B^3*a*c^2 - A^3*B*c^3)*d^3 - 3*(B^4*a
^2*c - A^4*c^3)*d^2*e + 6*(A*B^3*a^2*c - A^3*B*a*c^2)*d*e^2 - (B^4*a^3 - A^4*a*c^2)*e^3)*sqrt(e*x + d) - (2*A*
B^2*a*c^3*d^5 - (3*B^3*a^2*c^2 + 7*A^2*B*a*c^3)*d^4*e + 2*(7*A*B^2*a^2*c^2 + 3*A^3*a*c^3)*d^3*e^2 - 4*(B^3*a^3
*c + 4*A^2*B*a^2*c^2)*d^2*e^3 + 2*(4*A*B^2*a^3*c + A^3*a^2*c^2)*d*e^4 - (B^3*a^4 + A^2*B*a^3*c)*e^5 + (A*a*c^5
*d^8 - 2*B*a^2*c^4*d^7*e - 2*A*a^2*c^4*d^6*e^2 + 6*B*a^3*c^3*d^5*e^3 - 6*B*a^4*c^2*d^3*e^5 + 2*A*a^4*c^2*d^2*e
^6 + 2*B*a^5*c*d*e^7 - A*a^5*c*e^8)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^
2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^
2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4
*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8
- 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^
2*a^2 + A^2*a*c)*d*e^2 + (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((4*A^2*B^2*c^4*d^6 -
12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2
*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*
a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*
d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*
d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) + (c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt(-(6*A*B*a*c*d^2*e + 2*A*
B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d
^2*e^4 - a^4*e^6)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2
*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^
4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*
c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e
^10 + a^7*c*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((2*(A*B^3*a*c^2 - A^3*B*c
^3)*d^3 - 3*(B^4*a^2*c - A^4*c^3)*d^2*e + 6*(A*B^3*a^2*c - A^3*B*a*c^2)*d*e^2 - (B^4*a^3 - A^4*a*c^2)*e^3)*sqr
t(e*x + d) + (2*A*B^2*a*c^3*d^5 - (3*B^3*a^2*c^2 + 7*A^2*B*a*c^3)*d^4*e + 2*(7*A*B^2*a^2*c^2 + 3*A^3*a*c^3)*d^
3*e^2 - 4*(B^3*a^3*c + 4*A^2*B*a^2*c^2)*d^2*e^3 + 2*(4*A*B^2*a^3*c + A^3*a^2*c^2)*d*e^4 - (B^3*a^4 + A^2*B*a^3
*c)*e^5 - (A*a*c^5*d^8 - 2*B*a^2*c^4*d^7*e - 2*A*a^2*c^4*d^6*e^2 + 6*B*a^3*c^3*d^5*e^3 - 6*B*a^4*c^2*d^3*e^5 +
2*A*a^4*c^2*d^2*e^6 + 2*B*a^5*c*d*e^7 - A*a^5*c*e^8)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d
^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6
*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A
^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 +
15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a*c + A^
2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((4
*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e
^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*
B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10
*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(a*c^
3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) - (c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt(-(6*A*
B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d
^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^
2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^
2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4
*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8
- 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((2*(A*B
^3*a*c^2 - A^3*B*c^3)*d^3 - 3*(B^4*a^2*c - A^4*c^3)*d^2*e + 6*(A*B^3*a^2*c - A^3*B*a*c^2)*d*e^2 - (B^4*a^3 - A
^4*a*c^2)*e^3)*sqrt(e*x + d) - (2*A*B^2*a*c^3*d^5 - (3*B^3*a^2*c^2 + 7*A^2*B*a*c^3)*d^4*e + 2*(7*A*B^2*a^2*c^2
+ 3*A^3*a*c^3)*d^3*e^2 - 4*(B^3*a^3*c + 4*A^2*B*a^2*c^2)*d^2*e^3 + 2*(4*A*B^2*a^3*c + A^3*a^2*c^2)*d*e^4 - (B
^3*a^4 + A^2*B*a^3*c)*e^5 - (A*a*c^5*d^8 - 2*B*a^2*c^4*d^7*e - 2*A*a^2*c^4*d^6*e^2 + 6*B*a^3*c^3*d^5*e^3 - 6*B
*a^4*c^2*d^3*e^5 + 2*A*a^4*c^2*d^2*e^6 + 2*B*a^5*c*d*e^7 - A*a^5*c*e^8)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*
c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*
a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^
5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*
a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e
^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4
- a^4*e^6)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3
+ 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3
)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^1
2 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a
^7*c*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) - 4*(B*d - A*e)*sqrt(e*x + d))/(c*d
^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{A}{- a d \sqrt{d + e x} - a e x \sqrt{d + e x} + c d x^{2} \sqrt{d + e x} + c e x^{3} \sqrt{d + e x}}\, dx - \int \frac{B x}{- a d \sqrt{d + e x} - a e x \sqrt{d + e x} + c d x^{2} \sqrt{d + e x} + c e x^{3} \sqrt{d + e x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**(3/2)/(-c*x**2+a),x)

[Out]

-Integral(A/(-a*d*sqrt(d + e*x) - a*e*x*sqrt(d + e*x) + c*d*x**2*sqrt(d + e*x) + c*e*x**3*sqrt(d + e*x)), x) -
Integral(B*x/(-a*d*sqrt(d + e*x) - a*e*x*sqrt(d + e*x) + c*d*x**2*sqrt(d + e*x) + c*e*x**3*sqrt(d + e*x)), x)

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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)^(3/2)/(-c*x^2+a),x, algorithm="giac")

[Out]

Timed out