3.1251 \(\int \frac{(A+B x) \sqrt{d+e x}}{(b x+c x^2)^3} \, dx\)
Optimal. Leaf size=317 \[ \frac{\sqrt{c} \left (5 b^2 c e (7 A e+8 B d)-12 b c^2 d (7 A e+2 B d)+48 A c^3 d^2-15 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}}-\frac{\sqrt{d+e x} \left (b (c d-b e) (A b e-12 A c d+6 b B d)-c x \left (b^2 e (A e+11 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )\right )}{4 b^4 d \left (b x+c x^2\right ) (c d-b e)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 e (4 B d-A e)-12 b c d (A e+2 B d)+48 A c^2 d^2\right )}{4 b^5 d^{3/2}}-\frac{\sqrt{d+e x} (A b-x (b B-2 A c))}{2 b^2 \left (b x+c x^2\right )^2} \]
[Out]
-((A*b - (b*B - 2*A*c)*x)*Sqrt[d + e*x])/(2*b^2*(b*x + c*x^2)^2) - (Sqrt[d + e*x]*(b*(c*d - b*e)*(6*b*B*d - 12
*A*c*d + A*b*e) - c*(24*A*c^2*d^2 + b^2*e*(11*B*d + A*e) - 12*b*c*d*(B*d + 2*A*e))*x))/(4*b^4*d*(c*d - b*e)*(b
*x + c*x^2)) - ((48*A*c^2*d^2 + b^2*e*(4*B*d - A*e) - 12*b*c*d*(2*B*d + A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/
(4*b^5*d^(3/2)) + (Sqrt[c]*(48*A*c^3*d^2 - 15*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 7*A*e) + 5*b^2*c*e*(8*B*d + 7*A*
e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(3/2))
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Rubi [A] time = 0.699656, antiderivative size = 317, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used =
{820, 822, 826, 1166, 208} \[ \frac{\sqrt{c} \left (5 b^2 c e (7 A e+8 B d)-12 b c^2 d (7 A e+2 B d)+48 A c^3 d^2-15 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}}-\frac{\sqrt{d+e x} \left (b (c d-b e) (A b e-12 A c d+6 b B d)-c x \left (b^2 e (A e+11 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )\right )}{4 b^4 d \left (b x+c x^2\right ) (c d-b e)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 e (4 B d-A e)-12 b c d (A e+2 B d)+48 A c^2 d^2\right )}{4 b^5 d^{3/2}}-\frac{\sqrt{d+e x} (A b-x (b B-2 A c))}{2 b^2 \left (b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*Sqrt[d + e*x])/(b*x + c*x^2)^3,x]
[Out]
-((A*b - (b*B - 2*A*c)*x)*Sqrt[d + e*x])/(2*b^2*(b*x + c*x^2)^2) - (Sqrt[d + e*x]*(b*(c*d - b*e)*(6*b*B*d - 12
*A*c*d + A*b*e) - c*(24*A*c^2*d^2 + b^2*e*(11*B*d + A*e) - 12*b*c*d*(B*d + 2*A*e))*x))/(4*b^4*d*(c*d - b*e)*(b
*x + c*x^2)) - ((48*A*c^2*d^2 + b^2*e*(4*B*d - A*e) - 12*b*c*d*(2*B*d + A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/
(4*b^5*d^(3/2)) + (Sqrt[c]*(48*A*c^3*d^2 - 15*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 7*A*e) + 5*b^2*c*e*(8*B*d + 7*A*
e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(3/2))
Rule 820
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], 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] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p]
|| IntegersQ[2*m, 2*p])
Rule 822
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], 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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])
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) \sqrt{d+e x}}{\left (b x+c x^2\right )^3} \, dx &=-\frac{(A b-(b B-2 A c) x) \sqrt{d+e x}}{2 b^2 \left (b x+c x^2\right )^2}-\frac{\int \frac{\frac{1}{2} (12 A c d-b (6 B d+A e))-\frac{5}{2} (b B-2 A c) e x}{\sqrt{d+e x} \left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac{(A b-(b B-2 A c) x) \sqrt{d+e x}}{2 b^2 \left (b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (b (c d-b e) (6 b B d-12 A c d+A b e)-c \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}+\frac{\int \frac{\frac{1}{4} (c d-b e) \left (48 A c^2 d^2+b^2 e (4 B d-A e)-12 b c d (2 B d+A e)\right )+\frac{1}{4} c e \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d (c d-b e)}\\ &=-\frac{(A b-(b B-2 A c) x) \sqrt{d+e x}}{2 b^2 \left (b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (b (c d-b e) (6 b B d-12 A c d+A b e)-c \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} e (c d-b e) \left (48 A c^2 d^2+b^2 e (4 B d-A e)-12 b c d (2 B d+A e)\right )-\frac{1}{4} c d e \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\right )+\frac{1}{4} c e \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\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 )}{b^4 d (c d-b e)}\\ &=-\frac{(A b-(b B-2 A c) x) \sqrt{d+e x}}{2 b^2 \left (b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (b (c d-b e) (6 b B d-12 A c d+A b e)-c \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}+\frac{\left (c \left (48 A c^2 d^2+b^2 e (4 B d-A e)-12 b c d (2 B d+A e)\right )\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 )}{4 b^5 d}-\frac{\left (c \left (48 A c^3 d^2-15 b^3 B e^2-12 b c^2 d (2 B d+7 A e)+5 b^2 c e (8 B d+7 A e)\right )\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 )}{4 b^5 (c d-b e)}\\ &=-\frac{(A b-(b B-2 A c) x) \sqrt{d+e x}}{2 b^2 \left (b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (b (c d-b e) (6 b B d-12 A c d+A b e)-c \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}-\frac{\left (48 A c^2 d^2+b^2 e (4 B d-A e)-12 b c d (2 B d+A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{3/2}}+\frac{\sqrt{c} \left (48 A c^3 d^2-15 b^3 B e^2-12 b c^2 d (2 B d+7 A e)+5 b^2 c e (8 B d+7 A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}}\\ \end{align*}
Mathematica [A] time = 2.41356, size = 449, normalized size = 1.42 \[ \frac{\frac{(b+c x) \left (2 b c^{5/2} (d+e x)^{3/2} \left (b^2 c d e (10 A e+17 B d)+b^3 e^2 (A e-4 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )+(b+c x) \left (2 c^2 d^2 \left (-5 b^2 c e (7 A e+8 B d)+12 b c^2 d (7 A e+2 B d)-48 A c^3 d^2+15 b^3 B e^2\right ) \left (\sqrt{c} \sqrt{d+e x}-\sqrt{c d-b e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )\right )+2 c^{3/2} (c d-b e)^2 \left (\sqrt{d+e x}-\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right ) \left (b^2 e (4 B d-A e)-12 b c d (A e+2 B d)+48 A c^2 d^2\right )\right )\right )-2 b^2 c^{5/2} (d+e x)^{3/2} (c d-b e) \left (b^2 e (A e-4 B d)+3 b c d (3 A e+2 B d)-12 A c^2 d^2\right )}{b^4 c^{3/2} d (c d-b e)^2}-\frac{2 (d+e x)^{3/2} (-A b e-8 A c d+4 b B d)}{b d x}-\frac{4 A (d+e x)^{3/2}}{x^2}}{8 b d (b+c x)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*Sqrt[d + e*x])/(b*x + c*x^2)^3,x]
[Out]
((-4*A*(d + e*x)^(3/2))/x^2 - (2*(4*b*B*d - 8*A*c*d - A*b*e)*(d + e*x)^(3/2))/(b*d*x) + (-2*b^2*c^(5/2)*(c*d -
b*e)*(-12*A*c^2*d^2 + b^2*e*(-4*B*d + A*e) + 3*b*c*d*(2*B*d + 3*A*e))*(d + e*x)^(3/2) + (b + c*x)*(2*b*c^(5/2
)*(24*A*c^3*d^3 + b^3*e^2*(-4*B*d + A*e) - 12*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(17*B*d + 10*A*e))*(d + e*x)
^(3/2) + (b + c*x)*(2*c^(3/2)*(c*d - b*e)^2*(48*A*c^2*d^2 + b^2*e*(4*B*d - A*e) - 12*b*c*d*(2*B*d + A*e))*(Sqr
t[d + e*x] - Sqrt[d]*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]) + 2*c^2*d^2*(-48*A*c^3*d^2 + 15*b^3*B*e^2 + 12*b*c^2*d*(2
*B*d + 7*A*e) - 5*b^2*c*e*(8*B*d + 7*A*e))*(Sqrt[c]*Sqrt[d + e*x] - Sqrt[c*d - b*e]*ArcTanh[(Sqrt[c]*Sqrt[d +
e*x])/Sqrt[c*d - b*e]]))))/(b^4*c^(3/2)*d*(c*d - b*e)^2))/(8*b*d*(b + c*x)^2)
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Maple [B] time = 0.022, size = 829, normalized size = 2.6 \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)^(1/2)/(c*x^2+b*x)^3,x)
[Out]
13/4*e^2*c^2/b^3/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)-9/4*e^2*c/b^2/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)+3/e/b^4/x^2*(e*x+d)
^(3/2)*A*c+1/e/b^3/x^2*(e*x+d)^(1/2)*B*d+3*e/b^4/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c-1/4/b^3/x^2*(e*x+d
)^(1/2)*A+6/b^4*d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B*c-1/e/b^3/x^2*(e*x+d)^(3/2)*B-e/b^3/d^(1/2)*arctanh((
e*x+d)^(1/2)/d^(1/2))*B-1/4/b^3/x^2/d*(e*x+d)^(3/2)*A-12/b^5*d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c^2+1/4*
e^2/b^3/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A-3*e*c^3/b^4/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d+2*e*c^2/b^3/(c*e*
x+b*e)^2*B*(e*x+d)^(1/2)*d+35/4*e^2*c^2/b^3/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)
^(1/2))*A-15/4*e^2*c/b^2/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B-3/e/b^4/x
^2*(e*x+d)^(1/2)*A*c*d+11/4*e^2*c^3/b^3/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(3/2)*A-7/4*e^2*c^2/b^2/(c*e*x+b*e)^2/
(b*e-c*d)*(e*x+d)^(3/2)*B+12*c^4/b^5/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))
*A*d^2-6*c^3/b^4/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d^2-21*e*c^3/b^4/
(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d+10*e*c^2/b^3/(b*e-c*d)/((b*e-c*d
)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d-3*e*c^4/b^4/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(3/2)*A
*d+2*e*c^3/b^3/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(3/2)*B*d
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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)^(1/2)/(c*x^2+b*x)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 48.8015, size = 7023, normalized size = 22.15 \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)^(1/2)/(c*x^2+b*x)^3,x, algorithm="fricas")
[Out]
[1/8*(((24*(B*b*c^4 - 2*A*c^5)*d^4 - 4*(10*B*b^2*c^3 - 21*A*b*c^4)*d^3*e + 5*(3*B*b^3*c^2 - 7*A*b^2*c^3)*d^2*e
^2)*x^4 + 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^4 - 4*(10*B*b^3*c^2 - 21*A*b^2*c^3)*d^3*e + 5*(3*B*b^4*c - 7*A*b^3*c
^2)*d^2*e^2)*x^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^4 - 4*(10*B*b^4*c - 21*A*b^3*c^2)*d^3*e + 5*(3*B*b^5 - 7*A*
b^4*c)*d^2*e^2)*x^2)*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)) - ((A*b^3*c^2*e^3 - 24*(B*b*c^4 - 2*A*c^5)*d^3 + 4*(7*B*b^2*c^3 - 15*A*b*c^4)*d^2*e - (4*B*b
^3*c^2 - 11*A*b^2*c^3)*d*e^2)*x^4 + 2*(A*b^4*c*e^3 - 24*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + 4*(7*B*b^3*c^2 - 15*A*b^
2*c^3)*d^2*e - (4*B*b^4*c - 11*A*b^3*c^2)*d*e^2)*x^3 + (A*b^5*e^3 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + 4*(7*B*
b^4*c - 15*A*b^3*c^2)*d^2*e - (4*B*b^5 - 11*A*b^4*c)*d*e^2)*x^2)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) +
2*d)/x) - 2*(2*A*b^4*c*d^3 - 2*A*b^5*d^2*e - (A*b^3*c^2*d*e^2 - 12*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + (11*B*b^3*c^2
- 24*A*b^2*c^3)*d^2*e)*x^3 - (2*A*b^4*c*d*e^2 - 18*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + (17*B*b^4*c - 37*A*b^3*c^2
)*d^2*e)*x^2 - (A*b^5*d*e^2 - 4*(B*b^4*c - 2*A*b^3*c^2)*d^3 + (4*B*b^5 - 9*A*b^4*c)*d^2*e)*x)*sqrt(e*x + d))/(
(b^5*c^3*d^3 - b^6*c^2*d^2*e)*x^4 + 2*(b^6*c^2*d^3 - b^7*c*d^2*e)*x^3 + (b^7*c*d^3 - b^8*d^2*e)*x^2), -1/8*(2*
((24*(B*b*c^4 - 2*A*c^5)*d^4 - 4*(10*B*b^2*c^3 - 21*A*b*c^4)*d^3*e + 5*(3*B*b^3*c^2 - 7*A*b^2*c^3)*d^2*e^2)*x^
4 + 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^4 - 4*(10*B*b^3*c^2 - 21*A*b^2*c^3)*d^3*e + 5*(3*B*b^4*c - 7*A*b^3*c^2)*d^
2*e^2)*x^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^4 - 4*(10*B*b^4*c - 21*A*b^3*c^2)*d^3*e + 5*(3*B*b^5 - 7*A*b^4*c)
*d^2*e^2)*x^2)*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)) + ((
A*b^3*c^2*e^3 - 24*(B*b*c^4 - 2*A*c^5)*d^3 + 4*(7*B*b^2*c^3 - 15*A*b*c^4)*d^2*e - (4*B*b^3*c^2 - 11*A*b^2*c^3)
*d*e^2)*x^4 + 2*(A*b^4*c*e^3 - 24*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + 4*(7*B*b^3*c^2 - 15*A*b^2*c^3)*d^2*e - (4*B*b^
4*c - 11*A*b^3*c^2)*d*e^2)*x^3 + (A*b^5*e^3 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + 4*(7*B*b^4*c - 15*A*b^3*c^2)*
d^2*e - (4*B*b^5 - 11*A*b^4*c)*d*e^2)*x^2)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c
*d^3 - 2*A*b^5*d^2*e - (A*b^3*c^2*d*e^2 - 12*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + (11*B*b^3*c^2 - 24*A*b^2*c^3)*d^2*e
)*x^3 - (2*A*b^4*c*d*e^2 - 18*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + (17*B*b^4*c - 37*A*b^3*c^2)*d^2*e)*x^2 - (A*b^5*
d*e^2 - 4*(B*b^4*c - 2*A*b^3*c^2)*d^3 + (4*B*b^5 - 9*A*b^4*c)*d^2*e)*x)*sqrt(e*x + d))/((b^5*c^3*d^3 - b^6*c^2
*d^2*e)*x^4 + 2*(b^6*c^2*d^3 - b^7*c*d^2*e)*x^3 + (b^7*c*d^3 - b^8*d^2*e)*x^2), 1/8*(2*((A*b^3*c^2*e^3 - 24*(B
*b*c^4 - 2*A*c^5)*d^3 + 4*(7*B*b^2*c^3 - 15*A*b*c^4)*d^2*e - (4*B*b^3*c^2 - 11*A*b^2*c^3)*d*e^2)*x^4 + 2*(A*b^
4*c*e^3 - 24*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + 4*(7*B*b^3*c^2 - 15*A*b^2*c^3)*d^2*e - (4*B*b^4*c - 11*A*b^3*c^2)*d
*e^2)*x^3 + (A*b^5*e^3 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + 4*(7*B*b^4*c - 15*A*b^3*c^2)*d^2*e - (4*B*b^5 - 11
*A*b^4*c)*d*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + ((24*(B*b*c^4 - 2*A*c^5)*d^4 - 4*(10*B*b^2*c
^3 - 21*A*b*c^4)*d^3*e + 5*(3*B*b^3*c^2 - 7*A*b^2*c^3)*d^2*e^2)*x^4 + 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^4 - 4*(1
0*B*b^3*c^2 - 21*A*b^2*c^3)*d^3*e + 5*(3*B*b^4*c - 7*A*b^3*c^2)*d^2*e^2)*x^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^3)*d
^4 - 4*(10*B*b^4*c - 21*A*b^3*c^2)*d^3*e + 5*(3*B*b^5 - 7*A*b^4*c)*d^2*e^2)*x^2)*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)) - 2*(2*A*b^4*c*d^3 - 2*A*b^5*d^2
*e - (A*b^3*c^2*d*e^2 - 12*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + (11*B*b^3*c^2 - 24*A*b^2*c^3)*d^2*e)*x^3 - (2*A*b^4*c
*d*e^2 - 18*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + (17*B*b^4*c - 37*A*b^3*c^2)*d^2*e)*x^2 - (A*b^5*d*e^2 - 4*(B*b^4*c
- 2*A*b^3*c^2)*d^3 + (4*B*b^5 - 9*A*b^4*c)*d^2*e)*x)*sqrt(e*x + d))/((b^5*c^3*d^3 - b^6*c^2*d^2*e)*x^4 + 2*(b
^6*c^2*d^3 - b^7*c*d^2*e)*x^3 + (b^7*c*d^3 - b^8*d^2*e)*x^2), -1/4*(((24*(B*b*c^4 - 2*A*c^5)*d^4 - 4*(10*B*b^2
*c^3 - 21*A*b*c^4)*d^3*e + 5*(3*B*b^3*c^2 - 7*A*b^2*c^3)*d^2*e^2)*x^4 + 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^4 - 4*
(10*B*b^3*c^2 - 21*A*b^2*c^3)*d^3*e + 5*(3*B*b^4*c - 7*A*b^3*c^2)*d^2*e^2)*x^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^3)
*d^4 - 4*(10*B*b^4*c - 21*A*b^3*c^2)*d^3*e + 5*(3*B*b^5 - 7*A*b^4*c)*d^2*e^2)*x^2)*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)) - ((A*b^3*c^2*e^3 - 24*(B*b*c^4 - 2*A*c^5)*d^3
+ 4*(7*B*b^2*c^3 - 15*A*b*c^4)*d^2*e - (4*B*b^3*c^2 - 11*A*b^2*c^3)*d*e^2)*x^4 + 2*(A*b^4*c*e^3 - 24*(B*b^2*c
^3 - 2*A*b*c^4)*d^3 + 4*(7*B*b^3*c^2 - 15*A*b^2*c^3)*d^2*e - (4*B*b^4*c - 11*A*b^3*c^2)*d*e^2)*x^3 + (A*b^5*e^
3 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + 4*(7*B*b^4*c - 15*A*b^3*c^2)*d^2*e - (4*B*b^5 - 11*A*b^4*c)*d*e^2)*x^2)
*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (2*A*b^4*c*d^3 - 2*A*b^5*d^2*e - (A*b^3*c^2*d*e^2 - 12*(B*b^2*c^3
- 2*A*b*c^4)*d^3 + (11*B*b^3*c^2 - 24*A*b^2*c^3)*d^2*e)*x^3 - (2*A*b^4*c*d*e^2 - 18*(B*b^3*c^2 - 2*A*b^2*c^3)
*d^3 + (17*B*b^4*c - 37*A*b^3*c^2)*d^2*e)*x^2 - (A*b^5*d*e^2 - 4*(B*b^4*c - 2*A*b^3*c^2)*d^3 + (4*B*b^5 - 9*A*
b^4*c)*d^2*e)*x)*sqrt(e*x + d))/((b^5*c^3*d^3 - b^6*c^2*d^2*e)*x^4 + 2*(b^6*c^2*d^3 - b^7*c*d^2*e)*x^3 + (b^7*
c*d^3 - b^8*d^2*e)*x^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)**(1/2)/(c*x**2+b*x)**3,x)
[Out]
Timed out
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Giac [B] time = 1.37065, size = 1135, normalized size = 3.58 \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)^(1/2)/(c*x^2+b*x)^3,x, algorithm="giac")
[Out]
1/4*(24*B*b*c^3*d^2 - 48*A*c^4*d^2 - 40*B*b^2*c^2*d*e + 84*A*b*c^3*d*e + 15*B*b^3*c*e^2 - 35*A*b^2*c^2*e^2)*ar
ctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c*d - b^6*e)*sqrt(-c^2*d + b*c*e)) - 1/4*(12*(x*e + d)^(7/2)*
B*b*c^3*d^2*e - 24*(x*e + d)^(7/2)*A*c^4*d^2*e - 36*(x*e + d)^(5/2)*B*b*c^3*d^3*e + 72*(x*e + d)^(5/2)*A*c^4*d
^3*e + 36*(x*e + d)^(3/2)*B*b*c^3*d^4*e - 72*(x*e + d)^(3/2)*A*c^4*d^4*e - 12*sqrt(x*e + d)*B*b*c^3*d^5*e + 24
*sqrt(x*e + d)*A*c^4*d^5*e - 11*(x*e + d)^(7/2)*B*b^2*c^2*d*e^2 + 24*(x*e + d)^(7/2)*A*b*c^3*d*e^2 + 51*(x*e +
d)^(5/2)*B*b^2*c^2*d^2*e^2 - 108*(x*e + d)^(5/2)*A*b*c^3*d^2*e^2 - 69*(x*e + d)^(3/2)*B*b^2*c^2*d^3*e^2 + 144
*(x*e + d)^(3/2)*A*b*c^3*d^3*e^2 + 29*sqrt(x*e + d)*B*b^2*c^2*d^4*e^2 - 60*sqrt(x*e + d)*A*b*c^3*d^4*e^2 - (x*
e + d)^(7/2)*A*b^2*c^2*e^3 - 17*(x*e + d)^(5/2)*B*b^3*c*d*e^3 + 40*(x*e + d)^(5/2)*A*b^2*c^2*d*e^3 + 38*(x*e +
d)^(3/2)*B*b^3*c*d^2*e^3 - 85*(x*e + d)^(3/2)*A*b^2*c^2*d^2*e^3 - 21*sqrt(x*e + d)*B*b^3*c*d^3*e^3 + 46*sqrt(
x*e + d)*A*b^2*c^2*d^3*e^3 - 2*(x*e + d)^(5/2)*A*b^3*c*e^4 - 4*(x*e + d)^(3/2)*B*b^4*d*e^4 + 13*(x*e + d)^(3/2
)*A*b^3*c*d*e^4 + 4*sqrt(x*e + d)*B*b^4*d^2*e^4 - 9*sqrt(x*e + d)*A*b^3*c*d^2*e^4 - (x*e + d)^(3/2)*A*b^4*e^5
- sqrt(x*e + d)*A*b^4*d*e^5)/((b^4*c*d^2 - b^5*d*e)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e -
b*d*e)^2) - 1/4*(24*B*b*c*d^2 - 48*A*c^2*d^2 - 4*B*b^2*d*e + 12*A*b*c*d*e + A*b^2*e^2)*arctan(sqrt(x*e + d)/s
qrt(-d))/(b^5*sqrt(-d)*d)