3.1237 \(\int \frac{A+B x}{(d+e x)^{9/2} (b x+c x^2)} \, dx\)

Optimal. Leaf size=301 \[ \frac{2 \left (B c^3 d^4-A e \left (4 b^2 c d e^2-b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{d^4 \sqrt{d+e x} (c d-b e)^4}+\frac{2 \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{3 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac{2 c^{7/2} (b B-A c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{9/2}}+\frac{2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (d+e x)^{5/2} (c d-b e)^2}+\frac{2 (B d-A e)}{7 d (d+e x)^{7/2} (c d-b e)}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{9/2}} \]

[Out]

(2*(B*d - A*e))/(7*d*(c*d - b*e)*(d + e*x)^(7/2)) + (2*(B*c*d^2 - A*e*(2*c*d - b*e)))/(5*d^2*(c*d - b*e)^2*(d
+ e*x)^(5/2)) + (2*(B*c^2*d^3 - A*e*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2)))/(3*d^3*(c*d - b*e)^3*(d + e*x)^(3/2))
+ (2*(B*c^3*d^4 - A*e*(4*c^3*d^3 - 6*b*c^2*d^2*e + 4*b^2*c*d*e^2 - b^3*e^3)))/(d^4*(c*d - b*e)^4*Sqrt[d + e*x]
) - (2*A*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(9/2)) - (2*c^(7/2)*(b*B - A*c)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/
Sqrt[c*d - b*e]])/(b*(c*d - b*e)^(9/2))

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Rubi [A]  time = 0.551613, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {828, 826, 1166, 208} \[ \frac{2 \left (B c^3 d^4-A e \left (4 b^2 c d e^2-b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )\right )}{d^4 \sqrt{d+e x} (c d-b e)^4}+\frac{2 \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{3 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac{2 c^{7/2} (b B-A c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{9/2}}+\frac{2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (d+e x)^{5/2} (c d-b e)^2}+\frac{2 (B d-A e)}{7 d (d+e x)^{7/2} (c d-b e)}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^(9/2)*(b*x + c*x^2)),x]

[Out]

(2*(B*d - A*e))/(7*d*(c*d - b*e)*(d + e*x)^(7/2)) + (2*(B*c*d^2 - A*e*(2*c*d - b*e)))/(5*d^2*(c*d - b*e)^2*(d
+ e*x)^(5/2)) + (2*(B*c^2*d^3 - A*e*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2)))/(3*d^3*(c*d - b*e)^3*(d + e*x)^(3/2))
+ (2*(B*c^3*d^4 - A*e*(4*c^3*d^3 - 6*b*c^2*d^2*e + 4*b^2*c*d*e^2 - b^3*e^3)))/(d^4*(c*d - b*e)^4*Sqrt[d + e*x]
) - (2*A*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(9/2)) - (2*c^(7/2)*(b*B - A*c)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/
Sqrt[c*d - b*e]])/(b*(c*d - b*e)^(9/2))

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + 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)^{9/2} \left (b x+c x^2\right )} \, dx &=\frac{2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac{\int \frac{A (c d-b e)+c (B d-A e) x}{(d+e x)^{7/2} \left (b x+c x^2\right )} \, dx}{d (c d-b e)}\\ &=\frac{2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac{2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac{\int \frac{A (c d-b e)^2+c \left (B c d^2-A e (2 c d-b e)\right ) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{d^2 (c d-b e)^2}\\ &=\frac{2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac{2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac{2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{3 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac{\int \frac{A (c d-b e)^3+c \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{d^3 (c d-b e)^3}\\ &=\frac{2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac{2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac{2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{3 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac{2 \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )}{d^4 (c d-b e)^4 \sqrt{d+e x}}+\frac{\int \frac{A (c d-b e)^4+c \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{d^4 (c d-b e)^4}\\ &=\frac{2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac{2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac{2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{3 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac{2 \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )}{d^4 (c d-b e)^4 \sqrt{d+e x}}+\frac{2 \operatorname{Subst}\left (\int \frac{A e (c d-b e)^4-c d \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )+c \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{d^4 (c d-b e)^4}\\ &=\frac{2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac{2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac{2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{3 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac{2 \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )}{d^4 (c d-b e)^4 \sqrt{d+e x}}+\frac{(2 A c) \operatorname{Subst}\left (\int \frac{1}{-\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{b d^4}+\frac{\left (2 c^4 (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{b (c d-b e)^4}\\ &=\frac{2 (B d-A e)}{7 d (c d-b e) (d+e x)^{7/2}}+\frac{2 \left (B c d^2-A e (2 c d-b e)\right )}{5 d^2 (c d-b e)^2 (d+e x)^{5/2}}+\frac{2 \left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{3 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac{2 \left (B c^3 d^4-A e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )}{d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{9/2}}-\frac{2 c^{7/2} (b B-A c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.039185, size = 91, normalized size = 0.3 \[ \frac{2 \left (d (b B-A c) \, _2F_1\left (-\frac{7}{2},1;-\frac{5}{2};\frac{c (d+e x)}{c d-b e}\right )+A (c d-b e) \, _2F_1\left (-\frac{7}{2},1;-\frac{5}{2};\frac{e x}{d}+1\right )\right )}{7 b d (d+e x)^{7/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^(9/2)*(b*x + c*x^2)),x]

[Out]

(2*((b*B - A*c)*d*Hypergeometric2F1[-7/2, 1, -5/2, (c*(d + e*x))/(c*d - b*e)] + A*(c*d - b*e)*Hypergeometric2F
1[-7/2, 1, -5/2, 1 + (e*x)/d]))/(7*b*d*(c*d - b*e)*(d + e*x)^(7/2))

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Maple [A]  time = 0.028, size = 489, normalized size = 1.6 \begin{align*}{\frac{2\,Ae}{7\,d \left ( be-cd \right ) } \left ( ex+d \right ) ^{-{\frac{7}{2}}}}-{\frac{2\,B}{7\,be-7\,cd} \left ( ex+d \right ) ^{-{\frac{7}{2}}}}+{\frac{2\,Ab{e}^{2}}{5\,{d}^{2} \left ( be-cd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}-{\frac{4\,Ace}{5\,d \left ( be-cd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}+{\frac{2\,Bc}{5\, \left ( be-cd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}+{\frac{2\,A{b}^{2}{e}^{3}}{3\,{d}^{3} \left ( be-cd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-2\,{\frac{Abc{e}^{2}}{{d}^{2} \left ( be-cd \right ) ^{3} \left ( ex+d \right ) ^{3/2}}}+2\,{\frac{A{c}^{2}e}{d \left ( be-cd \right ) ^{3} \left ( ex+d \right ) ^{3/2}}}-{\frac{2\,B{c}^{2}}{3\, \left ( be-cd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{A{b}^{3}{e}^{4}}{{d}^{4} \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}-8\,{\frac{A{b}^{2}c{e}^{3}}{{d}^{3} \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}+12\,{\frac{Ab{c}^{2}{e}^{2}}{{d}^{2} \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}-8\,{\frac{A{c}^{3}e}{d \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}+2\,{\frac{B{c}^{3}}{ \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}-2\,{\frac{A}{b{d}^{9/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-2\,{\frac{{c}^{5}A}{ \left ( be-cd \right ) ^{4}b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+2\,{\frac{B{c}^{4}}{ \left ( be-cd \right ) ^{4}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^(9/2)/(c*x^2+b*x),x)

[Out]

2/7/d/(b*e-c*d)/(e*x+d)^(7/2)*A*e-2/7/(b*e-c*d)/(e*x+d)^(7/2)*B+2/5/d^2/(b*e-c*d)^2/(e*x+d)^(5/2)*A*b*e^2-4/5/
d/(b*e-c*d)^2/(e*x+d)^(5/2)*A*c*e+2/5/(b*e-c*d)^2/(e*x+d)^(5/2)*B*c+2/3/d^3/(b*e-c*d)^3/(e*x+d)^(3/2)*A*b^2*e^
3-2/d^2/(b*e-c*d)^3/(e*x+d)^(3/2)*A*b*c*e^2+2/d/(b*e-c*d)^3/(e*x+d)^(3/2)*A*c^2*e-2/3/(b*e-c*d)^3/(e*x+d)^(3/2
)*B*c^2+2/d^4/(b*e-c*d)^4/(e*x+d)^(1/2)*A*b^3*e^4-8/d^3/(b*e-c*d)^4/(e*x+d)^(1/2)*A*b^2*c*e^3+12/d^2/(b*e-c*d)
^4/(e*x+d)^(1/2)*A*b*c^2*e^2-8/d/(b*e-c*d)^4/(e*x+d)^(1/2)*A*c^3*e+2/(b*e-c*d)^4/(e*x+d)^(1/2)*B*c^3-2*A*arcta
nh((e*x+d)^(1/2)/d^(1/2))/b/d^(9/2)-2/(b*e-c*d)^4*c^5/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*
c)^(1/2))*A+2/(b*e-c*d)^4*c^4/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(9/2)/(c*x^2+b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 138.776, size = 9810, normalized size = 32.59 \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)^(9/2)/(c*x^2+b*x),x, algorithm="fricas")

[Out]

[-1/105*(105*((B*b*c^3 - A*c^4)*d^5*e^4*x^4 + 4*(B*b*c^3 - A*c^4)*d^6*e^3*x^3 + 6*(B*b*c^3 - A*c^4)*d^7*e^2*x^
2 + 4*(B*b*c^3 - A*c^4)*d^8*e*x + (B*b*c^3 - A*c^4)*d^9)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d
- b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - 105*(A*c^4*d^8 - 4*A*b*c^3*d^7*e + 6*A*b^2*c^2*d^6*e^2
- 4*A*b^3*c*d^5*e^3 + A*b^4*d^4*e^4 + (A*c^4*d^4*e^4 - 4*A*b*c^3*d^3*e^5 + 6*A*b^2*c^2*d^2*e^6 - 4*A*b^3*c*d*
e^7 + A*b^4*e^8)*x^4 + 4*(A*c^4*d^5*e^3 - 4*A*b*c^3*d^4*e^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4*
d*e^7)*x^3 + 6*(A*c^4*d^6*e^2 - 4*A*b*c^3*d^5*e^3 + 6*A*b^2*c^2*d^4*e^4 - 4*A*b^3*c*d^3*e^5 + A*b^4*d^2*e^6)*x
^2 + 4*(A*c^4*d^7*e - 4*A*b*c^3*d^6*e^2 + 6*A*b^2*c^2*d^5*e^3 - 4*A*b^3*c*d^4*e^4 + A*b^4*d^3*e^5)*x)*sqrt(d)*
log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(176*B*b*c^3*d^8 + 176*A*b^4*d^4*e^4 - 2*(61*B*b^2*c^2 + 291*
A*b*c^3)*d^7*e + 66*(B*b^3*c + 15*A*b^2*c^2)*d^6*e^2 - (15*B*b^4 + 689*A*b^3*c)*d^5*e^3 + 105*(B*b*c^3*d^5*e^3
- 4*A*b*c^3*d^4*e^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4*d*e^7)*x^3 + 35*(10*B*b*c^3*d^6*e^2 + 6
0*A*b^2*c^2*d^4*e^4 - 40*A*b^3*c*d^3*e^5 + 10*A*b^4*d^2*e^6 - (B*b^2*c^2 + 39*A*b*c^3)*d^5*e^3)*x^2 + 7*(58*B*
b*c^3*d^7*e - 232*A*b^3*c*d^4*e^4 + 58*A*b^4*d^3*e^5 - 8*(2*B*b^2*c^2 + 27*A*b*c^3)*d^6*e^2 + 3*(B*b^3*c + 115
*A*b^2*c^2)*d^5*e^3)*x)*sqrt(e*x + d))/(b*c^4*d^13 - 4*b^2*c^3*d^12*e + 6*b^3*c^2*d^11*e^2 - 4*b^4*c*d^10*e^3
+ b^5*d^9*e^4 + (b*c^4*d^9*e^4 - 4*b^2*c^3*d^8*e^5 + 6*b^3*c^2*d^7*e^6 - 4*b^4*c*d^6*e^7 + b^5*d^5*e^8)*x^4 +
4*(b*c^4*d^10*e^3 - 4*b^2*c^3*d^9*e^4 + 6*b^3*c^2*d^8*e^5 - 4*b^4*c*d^7*e^6 + b^5*d^6*e^7)*x^3 + 6*(b*c^4*d^11
*e^2 - 4*b^2*c^3*d^10*e^3 + 6*b^3*c^2*d^9*e^4 - 4*b^4*c*d^8*e^5 + b^5*d^7*e^6)*x^2 + 4*(b*c^4*d^12*e - 4*b^2*c
^3*d^11*e^2 + 6*b^3*c^2*d^10*e^3 - 4*b^4*c*d^9*e^4 + b^5*d^8*e^5)*x), -1/105*(210*((B*b*c^3 - A*c^4)*d^5*e^4*x
^4 + 4*(B*b*c^3 - A*c^4)*d^6*e^3*x^3 + 6*(B*b*c^3 - A*c^4)*d^7*e^2*x^2 + 4*(B*b*c^3 - A*c^4)*d^8*e*x + (B*b*c^
3 - A*c^4)*d^9)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) - 1
05*(A*c^4*d^8 - 4*A*b*c^3*d^7*e + 6*A*b^2*c^2*d^6*e^2 - 4*A*b^3*c*d^5*e^3 + A*b^4*d^4*e^4 + (A*c^4*d^4*e^4 - 4
*A*b*c^3*d^3*e^5 + 6*A*b^2*c^2*d^2*e^6 - 4*A*b^3*c*d*e^7 + A*b^4*e^8)*x^4 + 4*(A*c^4*d^5*e^3 - 4*A*b*c^3*d^4*e
^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4*d*e^7)*x^3 + 6*(A*c^4*d^6*e^2 - 4*A*b*c^3*d^5*e^3 + 6*A*b
^2*c^2*d^4*e^4 - 4*A*b^3*c*d^3*e^5 + A*b^4*d^2*e^6)*x^2 + 4*(A*c^4*d^7*e - 4*A*b*c^3*d^6*e^2 + 6*A*b^2*c^2*d^5
*e^3 - 4*A*b^3*c*d^4*e^4 + A*b^4*d^3*e^5)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(176*B*b
*c^3*d^8 + 176*A*b^4*d^4*e^4 - 2*(61*B*b^2*c^2 + 291*A*b*c^3)*d^7*e + 66*(B*b^3*c + 15*A*b^2*c^2)*d^6*e^2 - (1
5*B*b^4 + 689*A*b^3*c)*d^5*e^3 + 105*(B*b*c^3*d^5*e^3 - 4*A*b*c^3*d^4*e^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^
2*e^6 + A*b^4*d*e^7)*x^3 + 35*(10*B*b*c^3*d^6*e^2 + 60*A*b^2*c^2*d^4*e^4 - 40*A*b^3*c*d^3*e^5 + 10*A*b^4*d^2*e
^6 - (B*b^2*c^2 + 39*A*b*c^3)*d^5*e^3)*x^2 + 7*(58*B*b*c^3*d^7*e - 232*A*b^3*c*d^4*e^4 + 58*A*b^4*d^3*e^5 - 8*
(2*B*b^2*c^2 + 27*A*b*c^3)*d^6*e^2 + 3*(B*b^3*c + 115*A*b^2*c^2)*d^5*e^3)*x)*sqrt(e*x + d))/(b*c^4*d^13 - 4*b^
2*c^3*d^12*e + 6*b^3*c^2*d^11*e^2 - 4*b^4*c*d^10*e^3 + b^5*d^9*e^4 + (b*c^4*d^9*e^4 - 4*b^2*c^3*d^8*e^5 + 6*b^
3*c^2*d^7*e^6 - 4*b^4*c*d^6*e^7 + b^5*d^5*e^8)*x^4 + 4*(b*c^4*d^10*e^3 - 4*b^2*c^3*d^9*e^4 + 6*b^3*c^2*d^8*e^5
- 4*b^4*c*d^7*e^6 + b^5*d^6*e^7)*x^3 + 6*(b*c^4*d^11*e^2 - 4*b^2*c^3*d^10*e^3 + 6*b^3*c^2*d^9*e^4 - 4*b^4*c*d
^8*e^5 + b^5*d^7*e^6)*x^2 + 4*(b*c^4*d^12*e - 4*b^2*c^3*d^11*e^2 + 6*b^3*c^2*d^10*e^3 - 4*b^4*c*d^9*e^4 + b^5*
d^8*e^5)*x), 1/105*(210*(A*c^4*d^8 - 4*A*b*c^3*d^7*e + 6*A*b^2*c^2*d^6*e^2 - 4*A*b^3*c*d^5*e^3 + A*b^4*d^4*e^4
+ (A*c^4*d^4*e^4 - 4*A*b*c^3*d^3*e^5 + 6*A*b^2*c^2*d^2*e^6 - 4*A*b^3*c*d*e^7 + A*b^4*e^8)*x^4 + 4*(A*c^4*d^5*
e^3 - 4*A*b*c^3*d^4*e^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4*d*e^7)*x^3 + 6*(A*c^4*d^6*e^2 - 4*A*
b*c^3*d^5*e^3 + 6*A*b^2*c^2*d^4*e^4 - 4*A*b^3*c*d^3*e^5 + A*b^4*d^2*e^6)*x^2 + 4*(A*c^4*d^7*e - 4*A*b*c^3*d^6*
e^2 + 6*A*b^2*c^2*d^5*e^3 - 4*A*b^3*c*d^4*e^4 + A*b^4*d^3*e^5)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) -
105*((B*b*c^3 - A*c^4)*d^5*e^4*x^4 + 4*(B*b*c^3 - A*c^4)*d^6*e^3*x^3 + 6*(B*b*c^3 - A*c^4)*d^7*e^2*x^2 + 4*(B*
b*c^3 - A*c^4)*d^8*e*x + (B*b*c^3 - A*c^4)*d^9)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*s
qrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 2*(176*B*b*c^3*d^8 + 176*A*b^4*d^4*e^4 - 2*(61*B*b^2*c^2 + 291*
A*b*c^3)*d^7*e + 66*(B*b^3*c + 15*A*b^2*c^2)*d^6*e^2 - (15*B*b^4 + 689*A*b^3*c)*d^5*e^3 + 105*(B*b*c^3*d^5*e^3
- 4*A*b*c^3*d^4*e^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4*d*e^7)*x^3 + 35*(10*B*b*c^3*d^6*e^2 + 6
0*A*b^2*c^2*d^4*e^4 - 40*A*b^3*c*d^3*e^5 + 10*A*b^4*d^2*e^6 - (B*b^2*c^2 + 39*A*b*c^3)*d^5*e^3)*x^2 + 7*(58*B*
b*c^3*d^7*e - 232*A*b^3*c*d^4*e^4 + 58*A*b^4*d^3*e^5 - 8*(2*B*b^2*c^2 + 27*A*b*c^3)*d^6*e^2 + 3*(B*b^3*c + 115
*A*b^2*c^2)*d^5*e^3)*x)*sqrt(e*x + d))/(b*c^4*d^13 - 4*b^2*c^3*d^12*e + 6*b^3*c^2*d^11*e^2 - 4*b^4*c*d^10*e^3
+ b^5*d^9*e^4 + (b*c^4*d^9*e^4 - 4*b^2*c^3*d^8*e^5 + 6*b^3*c^2*d^7*e^6 - 4*b^4*c*d^6*e^7 + b^5*d^5*e^8)*x^4 +
4*(b*c^4*d^10*e^3 - 4*b^2*c^3*d^9*e^4 + 6*b^3*c^2*d^8*e^5 - 4*b^4*c*d^7*e^6 + b^5*d^6*e^7)*x^3 + 6*(b*c^4*d^11
*e^2 - 4*b^2*c^3*d^10*e^3 + 6*b^3*c^2*d^9*e^4 - 4*b^4*c*d^8*e^5 + b^5*d^7*e^6)*x^2 + 4*(b*c^4*d^12*e - 4*b^2*c
^3*d^11*e^2 + 6*b^3*c^2*d^10*e^3 - 4*b^4*c*d^9*e^4 + b^5*d^8*e^5)*x), -2/105*(105*((B*b*c^3 - A*c^4)*d^5*e^4*x
^4 + 4*(B*b*c^3 - A*c^4)*d^6*e^3*x^3 + 6*(B*b*c^3 - A*c^4)*d^7*e^2*x^2 + 4*(B*b*c^3 - A*c^4)*d^8*e*x + (B*b*c^
3 - A*c^4)*d^9)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) - 1
05*(A*c^4*d^8 - 4*A*b*c^3*d^7*e + 6*A*b^2*c^2*d^6*e^2 - 4*A*b^3*c*d^5*e^3 + A*b^4*d^4*e^4 + (A*c^4*d^4*e^4 - 4
*A*b*c^3*d^3*e^5 + 6*A*b^2*c^2*d^2*e^6 - 4*A*b^3*c*d*e^7 + A*b^4*e^8)*x^4 + 4*(A*c^4*d^5*e^3 - 4*A*b*c^3*d^4*e
^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4*d*e^7)*x^3 + 6*(A*c^4*d^6*e^2 - 4*A*b*c^3*d^5*e^3 + 6*A*b
^2*c^2*d^4*e^4 - 4*A*b^3*c*d^3*e^5 + A*b^4*d^2*e^6)*x^2 + 4*(A*c^4*d^7*e - 4*A*b*c^3*d^6*e^2 + 6*A*b^2*c^2*d^5
*e^3 - 4*A*b^3*c*d^4*e^4 + A*b^4*d^3*e^5)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (176*B*b*c^3*d^8 + 17
6*A*b^4*d^4*e^4 - 2*(61*B*b^2*c^2 + 291*A*b*c^3)*d^7*e + 66*(B*b^3*c + 15*A*b^2*c^2)*d^6*e^2 - (15*B*b^4 + 689
*A*b^3*c)*d^5*e^3 + 105*(B*b*c^3*d^5*e^3 - 4*A*b*c^3*d^4*e^4 + 6*A*b^2*c^2*d^3*e^5 - 4*A*b^3*c*d^2*e^6 + A*b^4
*d*e^7)*x^3 + 35*(10*B*b*c^3*d^6*e^2 + 60*A*b^2*c^2*d^4*e^4 - 40*A*b^3*c*d^3*e^5 + 10*A*b^4*d^2*e^6 - (B*b^2*c
^2 + 39*A*b*c^3)*d^5*e^3)*x^2 + 7*(58*B*b*c^3*d^7*e - 232*A*b^3*c*d^4*e^4 + 58*A*b^4*d^3*e^5 - 8*(2*B*b^2*c^2
+ 27*A*b*c^3)*d^6*e^2 + 3*(B*b^3*c + 115*A*b^2*c^2)*d^5*e^3)*x)*sqrt(e*x + d))/(b*c^4*d^13 - 4*b^2*c^3*d^12*e
+ 6*b^3*c^2*d^11*e^2 - 4*b^4*c*d^10*e^3 + b^5*d^9*e^4 + (b*c^4*d^9*e^4 - 4*b^2*c^3*d^8*e^5 + 6*b^3*c^2*d^7*e^6
- 4*b^4*c*d^6*e^7 + b^5*d^5*e^8)*x^4 + 4*(b*c^4*d^10*e^3 - 4*b^2*c^3*d^9*e^4 + 6*b^3*c^2*d^8*e^5 - 4*b^4*c*d^
7*e^6 + b^5*d^6*e^7)*x^3 + 6*(b*c^4*d^11*e^2 - 4*b^2*c^3*d^10*e^3 + 6*b^3*c^2*d^9*e^4 - 4*b^4*c*d^8*e^5 + b^5*
d^7*e^6)*x^2 + 4*(b*c^4*d^12*e - 4*b^2*c^3*d^11*e^2 + 6*b^3*c^2*d^10*e^3 - 4*b^4*c*d^9*e^4 + b^5*d^8*e^5)*x)]

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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)**(9/2)/(c*x**2+b*x),x)

[Out]

Timed out

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Giac [B]  time = 1.52986, size = 830, normalized size = 2.76 \begin{align*} \frac{2 \,{\left (B b c^{4} - A c^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b c^{4} d^{4} - 4 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} + b^{5} e^{4}\right )} \sqrt{-c^{2} d + b c e}} + \frac{2 \,{\left (105 \,{\left (x e + d\right )}^{3} B c^{3} d^{4} + 35 \,{\left (x e + d\right )}^{2} B c^{3} d^{5} + 21 \,{\left (x e + d\right )} B c^{3} d^{6} + 15 \, B c^{3} d^{7} - 420 \,{\left (x e + d\right )}^{3} A c^{3} d^{3} e - 35 \,{\left (x e + d\right )}^{2} B b c^{2} d^{4} e - 105 \,{\left (x e + d\right )}^{2} A c^{3} d^{4} e - 42 \,{\left (x e + d\right )} B b c^{2} d^{5} e - 42 \,{\left (x e + d\right )} A c^{3} d^{5} e - 45 \, B b c^{2} d^{6} e - 15 \, A c^{3} d^{6} e + 630 \,{\left (x e + d\right )}^{3} A b c^{2} d^{2} e^{2} + 210 \,{\left (x e + d\right )}^{2} A b c^{2} d^{3} e^{2} + 21 \,{\left (x e + d\right )} B b^{2} c d^{4} e^{2} + 105 \,{\left (x e + d\right )} A b c^{2} d^{4} e^{2} + 45 \, B b^{2} c d^{5} e^{2} + 45 \, A b c^{2} d^{5} e^{2} - 420 \,{\left (x e + d\right )}^{3} A b^{2} c d e^{3} - 140 \,{\left (x e + d\right )}^{2} A b^{2} c d^{2} e^{3} - 84 \,{\left (x e + d\right )} A b^{2} c d^{3} e^{3} - 15 \, B b^{3} d^{4} e^{3} - 45 \, A b^{2} c d^{4} e^{3} + 105 \,{\left (x e + d\right )}^{3} A b^{3} e^{4} + 35 \,{\left (x e + d\right )}^{2} A b^{3} d e^{4} + 21 \,{\left (x e + d\right )} A b^{3} d^{2} e^{4} + 15 \, A b^{3} d^{3} e^{4}\right )}}{105 \,{\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}\right )}{\left (x e + d\right )}^{\frac{7}{2}}} + \frac{2 \, A \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(9/2)/(c*x^2+b*x),x, algorithm="giac")

[Out]

2*(B*b*c^4 - A*c^5)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b*c^4*d^4 - 4*b^2*c^3*d^3*e + 6*b^3*c^2*d^2
*e^2 - 4*b^4*c*d*e^3 + b^5*e^4)*sqrt(-c^2*d + b*c*e)) + 2/105*(105*(x*e + d)^3*B*c^3*d^4 + 35*(x*e + d)^2*B*c^
3*d^5 + 21*(x*e + d)*B*c^3*d^6 + 15*B*c^3*d^7 - 420*(x*e + d)^3*A*c^3*d^3*e - 35*(x*e + d)^2*B*b*c^2*d^4*e - 1
05*(x*e + d)^2*A*c^3*d^4*e - 42*(x*e + d)*B*b*c^2*d^5*e - 42*(x*e + d)*A*c^3*d^5*e - 45*B*b*c^2*d^6*e - 15*A*c
^3*d^6*e + 630*(x*e + d)^3*A*b*c^2*d^2*e^2 + 210*(x*e + d)^2*A*b*c^2*d^3*e^2 + 21*(x*e + d)*B*b^2*c*d^4*e^2 +
105*(x*e + d)*A*b*c^2*d^4*e^2 + 45*B*b^2*c*d^5*e^2 + 45*A*b*c^2*d^5*e^2 - 420*(x*e + d)^3*A*b^2*c*d*e^3 - 140*
(x*e + d)^2*A*b^2*c*d^2*e^3 - 84*(x*e + d)*A*b^2*c*d^3*e^3 - 15*B*b^3*d^4*e^3 - 45*A*b^2*c*d^4*e^3 + 105*(x*e
+ d)^3*A*b^3*e^4 + 35*(x*e + d)^2*A*b^3*d*e^4 + 21*(x*e + d)*A*b^3*d^2*e^4 + 15*A*b^3*d^3*e^4)/((c^4*d^8 - 4*b
*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4)*(x*e + d)^(7/2)) + 2*A*arctan(sqrt(x*e + d)/sq
rt(-d))/(b*sqrt(-d)*d^4)