3.1204 \(\int \frac {A+B x}{(d+e x)^2 (b x+c x^2)^{3/2}} \, dx\)
Optimal. Leaf size=245 \[ -\frac {e \sqrt {b x+c x^2} \left (b^2 (-e) (B d-3 A e)-2 b c d (2 A e+B d)+4 A c^2 d^2\right )}{b^2 d^2 (d+e x) (c d-b e)^2}-\frac {2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}+\frac {e (3 A e (2 c d-b e)-B d (4 c d-b e)) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{5/2} (c d-b e)^{5/2}} \]
[Out]
1/2*e*(3*A*e*(-b*e+2*c*d)-B*d*(-b*e+4*c*d))*arctanh(1/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2+b
*x)^(1/2))/d^(5/2)/(-b*e+c*d)^(5/2)-2*(A*b*(-b*e+c*d)+c*(2*A*c*d-b*(A*e+B*d))*x)/b^2/d/(-b*e+c*d)/(e*x+d)/(c*x
^2+b*x)^(1/2)-e*(4*A*c^2*d^2-b^2*e*(-3*A*e+B*d)-2*b*c*d*(2*A*e+B*d))*(c*x^2+b*x)^(1/2)/b^2/d^2/(-b*e+c*d)^2/(e
*x+d)
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Rubi [A] time = 0.31, antiderivative size = 245, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used =
{822, 806, 724, 206} \[ -\frac {e \sqrt {b x+c x^2} \left (b^2 (-e) (B d-3 A e)-2 b c d (2 A e+B d)+4 A c^2 d^2\right )}{b^2 d^2 (d+e x) (c d-b e)^2}-\frac {2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)}+\frac {e (3 A e (2 c d-b e)-B d (4 c d-b e)) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{5/2} (c d-b e)^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^2*(b*x + c*x^2)^(3/2)),x]
[Out]
(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*(d + e*x)*Sqrt[b*x + c*x^2]) - (e*(4
*A*c^2*d^2 - b^2*e*(B*d - 3*A*e) - 2*b*c*d*(B*d + 2*A*e))*Sqrt[b*x + c*x^2])/(b^2*d^2*(c*d - b*e)^2*(d + e*x))
+ (e*(3*A*e*(2*c*d - b*e) - B*d*(4*c*d - b*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqr
t[b*x + c*x^2])])/(2*d^(5/2)*(c*d - b*e)^(5/2))
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 724
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]
Rule 806
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]
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])
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x) \sqrt {b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} b e (b B d+2 A c d-3 A b e)-c e (b B d-2 A c d+A b e) x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx}{b^2 d (c d-b e)}\\ &=-\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x) \sqrt {b x+c x^2}}-\frac {e \left (4 A c^2 d^2-b^2 e (B d-3 A e)-2 b c d (B d+2 A e)\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 (d+e x)}+\frac {(e (3 A e (2 c d-b e)-B d (4 c d-b e))) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{2 d^2 (c d-b e)^2}\\ &=-\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x) \sqrt {b x+c x^2}}-\frac {e \left (4 A c^2 d^2-b^2 e (B d-3 A e)-2 b c d (B d+2 A e)\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 (d+e x)}-\frac {(e (3 A e (2 c d-b e)-B d (4 c d-b e))) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{d^2 (c d-b e)^2}\\ &=-\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x) \sqrt {b x+c x^2}}-\frac {e \left (4 A c^2 d^2-b^2 e (B d-3 A e)-2 b c d (B d+2 A e)\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 (d+e x)}+\frac {e (3 A e (2 c d-b e)-B d (4 c d-b e)) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 d^{5/2} (c d-b e)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 226, normalized size = 0.92 \[ \frac {x \left (\frac {c x (b+c x) \left (b^2 e (B d-3 A e)+2 b c d (2 A e+B d)-4 A c^2 d^2\right )}{b^2 d (b e-c d)}+\frac {e \sqrt {x} (b+c x)^{3/2} (3 A e (b e-2 c d)+B d (4 c d-b e)) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )}{d^{3/2} (c d-b e)^{3/2}}+\frac {(b+c x) (-3 A b e+2 A c d+b B d)}{b d}+\frac {(b+c x) (A e-B d)}{d+e x}\right )}{d (x (b+c x))^{3/2} (b e-c d)} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^2*(b*x + c*x^2)^(3/2)),x]
[Out]
(x*(((b*B*d + 2*A*c*d - 3*A*b*e)*(b + c*x))/(b*d) + (c*(-4*A*c^2*d^2 + b^2*e*(B*d - 3*A*e) + 2*b*c*d*(B*d + 2*
A*e))*x*(b + c*x))/(b^2*d*(-(c*d) + b*e)) + ((-(B*d) + A*e)*(b + c*x))/(d + e*x) + (e*(B*d*(4*c*d - b*e) + 3*A
*e*(-2*c*d + b*e))*Sqrt[x]*(b + c*x)^(3/2)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(d^(3/2
)*(c*d - b*e)^(3/2))))/(d*(-(c*d) + b*e)*(x*(b + c*x))^(3/2))
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fricas [B] time = 1.23, size = 1240, normalized size = 5.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^2/(c*x^2+b*x)^(3/2),x, algorithm="fricas")
[Out]
[1/2*(((4*B*b^2*c^2*d^2*e^2 + 3*A*b^3*c*e^4 - (B*b^3*c + 6*A*b^2*c^2)*d*e^3)*x^3 + (4*B*b^2*c^2*d^3*e + 3*A*b^
4*e^4 + 3*(B*b^3*c - 2*A*b^2*c^2)*d^2*e^2 - (B*b^4 + 3*A*b^3*c)*d*e^3)*x^2 + (4*B*b^3*c*d^3*e + 3*A*b^4*d*e^3
- (B*b^4 + 6*A*b^3*c)*d^2*e^2)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x - 2*sqrt(c*d^2 - b*d*e)*sqrt(
c*x^2 + b*x))/(e*x + d)) - 2*(2*A*b*c^3*d^5 - 6*A*b^2*c^2*d^4*e + 6*A*b^3*c*d^3*e^2 - 2*A*b^4*d^2*e^3 - (3*A*b
^3*c*d*e^4 + 2*(B*b*c^3 - 2*A*c^4)*d^4*e - (B*b^2*c^2 - 8*A*b*c^3)*d^3*e^2 - (B*b^3*c + 7*A*b^2*c^2)*d^2*e^3)*
x^2 - (B*b^3*c*d^3*e^2 + 3*A*b^4*d*e^4 + 2*(B*b*c^3 - 2*A*c^4)*d^5 - 2*(B*b^2*c^2 - 3*A*b*c^3)*d^4*e - (B*b^4
+ 5*A*b^3*c)*d^2*e^3)*x)*sqrt(c*x^2 + b*x))/((b^2*c^4*d^6*e - 3*b^3*c^3*d^5*e^2 + 3*b^4*c^2*d^4*e^3 - b^5*c*d^
3*e^4)*x^3 + (b^2*c^4*d^7 - 2*b^3*c^3*d^6*e + 2*b^5*c*d^4*e^3 - b^6*d^3*e^4)*x^2 + (b^3*c^3*d^7 - 3*b^4*c^2*d^
6*e + 3*b^5*c*d^5*e^2 - b^6*d^4*e^3)*x), -(((4*B*b^2*c^2*d^2*e^2 + 3*A*b^3*c*e^4 - (B*b^3*c + 6*A*b^2*c^2)*d*e
^3)*x^3 + (4*B*b^2*c^2*d^3*e + 3*A*b^4*e^4 + 3*(B*b^3*c - 2*A*b^2*c^2)*d^2*e^2 - (B*b^4 + 3*A*b^3*c)*d*e^3)*x^
2 + (4*B*b^3*c*d^3*e + 3*A*b^4*d*e^3 - (B*b^4 + 6*A*b^3*c)*d^2*e^2)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^
2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + (2*A*b*c^3*d^5 - 6*A*b^2*c^2*d^4*e + 6*A*b^3*c*d^3*e^2 - 2*A*b
^4*d^2*e^3 - (3*A*b^3*c*d*e^4 + 2*(B*b*c^3 - 2*A*c^4)*d^4*e - (B*b^2*c^2 - 8*A*b*c^3)*d^3*e^2 - (B*b^3*c + 7*A
*b^2*c^2)*d^2*e^3)*x^2 - (B*b^3*c*d^3*e^2 + 3*A*b^4*d*e^4 + 2*(B*b*c^3 - 2*A*c^4)*d^5 - 2*(B*b^2*c^2 - 3*A*b*c
^3)*d^4*e - (B*b^4 + 5*A*b^3*c)*d^2*e^3)*x)*sqrt(c*x^2 + b*x))/((b^2*c^4*d^6*e - 3*b^3*c^3*d^5*e^2 + 3*b^4*c^2
*d^4*e^3 - b^5*c*d^3*e^4)*x^3 + (b^2*c^4*d^7 - 2*b^3*c^3*d^6*e + 2*b^5*c*d^4*e^3 - b^6*d^3*e^4)*x^2 + (b^3*c^3
*d^7 - 3*b^4*c^2*d^6*e + 3*b^5*c*d^5*e^2 - b^6*d^4*e^3)*x)]
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giac [B] time = 1.62, size = 1181, normalized size = 4.82 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^2/(c*x^2+b*x)^(3/2),x, algorithm="giac")
[Out]
-1/2*((4*B*b^2*c*d^2*e^3*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) + 4*sqrt(c*d^2 - b*d*e)*B*b*c^(
3/2)*d^2*e^2 - 8*sqrt(c*d^2 - b*d*e)*A*c^(5/2)*d^2*e^2 - B*b^3*d*e^4*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*
e)*sqrt(c))) - 6*A*b^2*c*d*e^4*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) + 2*sqrt(c*d^2 - b*d*e)*B
*b^2*sqrt(c)*d*e^3 + 8*sqrt(c*d^2 - b*d*e)*A*b*c^(3/2)*d*e^3 + 3*A*b^3*e^5*log(abs(2*c*d - b*e - 2*sqrt(c*d^2
- b*d*e)*sqrt(c))) - 6*sqrt(c*d^2 - b*d*e)*A*b^2*sqrt(c)*e^4)*sgn(1/(x*e + d))/(sqrt(c*d^2 - b*d*e)*b^2*c^2*d^
4 - 2*sqrt(c*d^2 - b*d*e)*b^3*c*d^3*e + sqrt(c*d^2 - b*d*e)*b^4*d^2*e^2) + 2*(((2*B*b*c^2*d^3*e^8*sgn(1/(x*e +
d)) - 4*A*c^3*d^3*e^8*sgn(1/(x*e + d)) + 2*B*b^2*c*d^2*e^9*sgn(1/(x*e + d)) + 6*A*b*c^2*d^2*e^9*sgn(1/(x*e +
d)) - B*b^3*d*e^10*sgn(1/(x*e + d)) - 8*A*b^2*c*d*e^10*sgn(1/(x*e + d)) + 3*A*b^3*e^11*sgn(1/(x*e + d)))/(b^2*
c^2*d^4*e^5*sgn(1/(x*e + d))^2 - 2*b^3*c*d^3*e^6*sgn(1/(x*e + d))^2 + b^4*d^2*e^7*sgn(1/(x*e + d))^2) - (B*b^2
*c*d^3*e^10*sgn(1/(x*e + d)) - B*b^3*d^2*e^11*sgn(1/(x*e + d)) - A*b^2*c*d^2*e^11*sgn(1/(x*e + d)) + A*b^3*d*e
^12*sgn(1/(x*e + d)))*e^(-1)/((b^2*c^2*d^4*e^5*sgn(1/(x*e + d))^2 - 2*b^3*c*d^3*e^6*sgn(1/(x*e + d))^2 + b^4*d
^2*e^7*sgn(1/(x*e + d))^2)*(x*e + d)))*e^(-1)/(x*e + d) - (2*B*b*c^2*d^2*e^7*sgn(1/(x*e + d)) - 4*A*c^3*d^2*e^
7*sgn(1/(x*e + d)) + B*b^2*c*d*e^8*sgn(1/(x*e + d)) + 4*A*b*c^2*d*e^8*sgn(1/(x*e + d)) - 3*A*b^2*c*e^9*sgn(1/(
x*e + d)))/(b^2*c^2*d^4*e^5*sgn(1/(x*e + d))^2 - 2*b^3*c*d^3*e^6*sgn(1/(x*e + d))^2 + b^4*d^2*e^7*sgn(1/(x*e +
d))^2))/sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2) - (4*B*c*d^2*e^4 -
B*b*d*e^5 - 6*A*c*d*e^5 + 3*A*b*e^6)*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*(sqrt(c - 2*c*d/(x*e + d) + c
*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2) + sqrt(c*d^2*e^2 - b*d*e^3)*e^(-1)/(x*e + d))))/((c^2*d^
4*e - 2*b*c*d^3*e^2 + b^2*d^2*e^3)*sqrt(c*d^2 - b*d*e)*sgn(1/(x*e + d))))*e^(-2)
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maple [B] time = 0.06, size = 2069, normalized size = 8.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(e*x+d)^2/(c*x^2+b*x)^(3/2),x)
[Out]
-1/e/(b*e-c*d)/(x+d/e)/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*B-6/(b*e-c*d)^2/b/((x+d/e)^2*
c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c^2*A+3/(b*e-c*d)^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d
/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2
))/(x+d/e))*c*B-3*e^2/(b*e-c*d)^2/d^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c*A+3*e/(b*e
-c*d)^2/d/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c*B+6/e/(b*e-c*d)^2/b/((x+d/e)^2*c-(b*e-
c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c^2*B*d+3/2*e^2/(b*e-c*d)^2/d^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c
*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)
^(1/2))/(x+d/e))*b*A-3/2*e/(b*e-c*d)^2/d/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e
+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b*B-8*c^2/(b*e
-c*d)/d/b^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*A+12/e*c^2/(b*e-c*d)/b^2/((x+d/e)^2*c-
(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*B-2*B/(b*e-c*d)/d/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x
+d/e)/e)^(1/2)*x*c+9*e/(b*e-c*d)^2/d/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c*A-12/(b*e-c*d
)^2/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c^2*B-2*B/(b*e-c*d)/d/((x+d/e)^2*c-(b*e-c*d)
*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)-9/(b*e-c*d)^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c*
B+12/e/(b*e-c*d)^2/b^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c^3*B*d+12*e/(b*e-c*d)^2/d/
b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c^2*A+B/(b*e-c*d)/d/(-(b*e-c*d)*d/e^2)^(1/2)*ln(
(-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*
(x+d/e)/e)^(1/2))/(x+d/e))+1/(b*e-c*d)/d/(x+d/e)/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*A-3
*e/(b*e-c*d)^2/d/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1
/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*c*A+6/e*c/(b*e-c*d)/b/((x+d/e)^2*c-(b*
e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*B-4*c/(b*e-c*d)/d/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)
/e)^(1/2)*A-12/(b*e-c*d)^2/b^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c^3*A-3*e^2/(b*e-c*
d)^2/d^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*A+3*e/(b*e-c*d)^2/d/((x+d/e)^2*c-(b*e-c*d
)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*B
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^2/(c*x^2+b*x)^(3/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive, negative or zero?
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x}{{\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)/((b*x + c*x^2)^(3/2)*(d + e*x)^2),x)
[Out]
int((A + B*x)/((b*x + c*x^2)^(3/2)*(d + e*x)^2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {A + B x}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)**2/(c*x**2+b*x)**(3/2),x)
[Out]
Integral((A + B*x)/((x*(b + c*x))**(3/2)*(d + e*x)**2), x)
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