3.11.60 \(\int \frac {A+B x}{(d+e x)^3 (b x+c x^2)^{3/2}} \, dx\)
Optimal. Leaf size=374 \[ -\frac {e \sqrt {b x+c x^2} \left (b^2 (-e) (B d-5 A e)-4 b c d (2 A e+B d)+8 A c^2 d^2\right )}{2 b^2 d^2 (d+e x)^2 (c d-b e)^2}-\frac {3 e \left (B d \left (b^2 e^2-4 b c d e+8 c^2 d^2\right )-A e \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 d^{7/2} (c d-b e)^{7/2}}-\frac {2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x)^2 (c d-b e)}-\frac {e \sqrt {b x+c x^2} \left (3 b^3 e^2 (B d-5 A e)-2 b^2 c d e (5 B d-19 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )}{4 b^2 d^3 (d+e x) (c d-b e)^3} \]
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Rubi [A] time = 0.59, antiderivative size = 374, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used =
{822, 834, 806, 724, 206} \begin {gather*} -\frac {e \sqrt {b x+c x^2} \left (-2 b^2 c d e (5 B d-19 A e)+3 b^3 e^2 (B d-5 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )}{4 b^2 d^3 (d+e x) (c d-b e)^3}-\frac {3 e \left (B d \left (b^2 e^2-4 b c d e+8 c^2 d^2\right )-A e \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right ) \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 )}{8 d^{7/2} (c d-b e)^{7/2}}-\frac {e \sqrt {b x+c x^2} \left (b^2 (-e) (B d-5 A e)-4 b c d (2 A e+B d)+8 A c^2 d^2\right )}{2 b^2 d^2 (d+e x)^2 (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)^2 (c d-b e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^3*(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)^2*Sqrt[b*x + c*x^2]) - (e*
(8*A*c^2*d^2 - b^2*e*(B*d - 5*A*e) - 4*b*c*d*(B*d + 2*A*e))*Sqrt[b*x + c*x^2])/(2*b^2*d^2*(c*d - b*e)^2*(d + e
*x)^2) - (e*(16*A*c^3*d^3 - 2*b^2*c*d*e*(5*B*d - 19*A*e) + 3*b^3*e^2*(B*d - 5*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e)
)*Sqrt[b*x + c*x^2])/(4*b^2*d^3*(c*d - b*e)^3*(d + e*x)) - (3*e*(B*d*(8*c^2*d^2 - 4*b*c*d*e + b^2*e^2) - A*e*(
16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*
x^2])])/(8*d^(7/2)*(c*d - b*e)^(7/2))
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])
Rule 834
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x)^3 \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)^2 \sqrt {b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} b e (b B d+4 A c d-5 A b e)-2 c e (b B d-2 A c d+A b e) x}{(d+e x)^3 \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)^2 \sqrt {b x+c x^2}}-\frac {e \left (8 A c^2 d^2-b^2 e (B d-5 A e)-4 b c d (B d+2 A e)\right ) \sqrt {b x+c x^2}}{2 b^2 d^2 (c d-b e)^2 (d+e x)^2}+\frac {\int \frac {-\frac {1}{4} b e \left (8 A c^2 d^2+4 b c d (2 B d-7 A e)-3 b^2 e (B d-5 A e)\right )-\frac {1}{2} c e \left (8 A c^2 d^2-b^2 e (B d-5 A e)-4 b c d (B d+2 A e)\right ) x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx}{b^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)^2 \sqrt {b x+c x^2}}-\frac {e \left (8 A c^2 d^2-b^2 e (B d-5 A e)-4 b c d (B d+2 A e)\right ) \sqrt {b x+c x^2}}{2 b^2 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (16 A c^3 d^3-2 b^2 c d e (5 B d-19 A e)+3 b^3 e^2 (B d-5 A e)-8 b c^2 d^2 (B d+3 A e)\right ) \sqrt {b x+c x^2}}{4 b^2 d^3 (c d-b e)^3 (d+e x)}-\frac {\left (3 e \left (B d \left (8 c^2 d^2-4 b c d e+b^2 e^2\right )-A e \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 d^3 (c d-b e)^3}\\ &=-\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)^2 \sqrt {b x+c x^2}}-\frac {e \left (8 A c^2 d^2-b^2 e (B d-5 A e)-4 b c d (B d+2 A e)\right ) \sqrt {b x+c x^2}}{2 b^2 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (16 A c^3 d^3-2 b^2 c d e (5 B d-19 A e)+3 b^3 e^2 (B d-5 A e)-8 b c^2 d^2 (B d+3 A e)\right ) \sqrt {b x+c x^2}}{4 b^2 d^3 (c d-b e)^3 (d+e x)}+\frac {\left (3 e \left (B d \left (8 c^2 d^2-4 b c d e+b^2 e^2\right )-A e \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\right )\right ) \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 )}{4 d^3 (c d-b e)^3}\\ &=-\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)^2 \sqrt {b x+c x^2}}-\frac {e \left (8 A c^2 d^2-b^2 e (B d-5 A e)-4 b c d (B d+2 A e)\right ) \sqrt {b x+c x^2}}{2 b^2 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (16 A c^3 d^3-2 b^2 c d e (5 B d-19 A e)+3 b^3 e^2 (B d-5 A e)-8 b c^2 d^2 (B d+3 A e)\right ) \sqrt {b x+c x^2}}{4 b^2 d^3 (c d-b e)^3 (d+e x)}-\frac {3 e \left (B d \left (8 c^2 d^2-4 b c d e+b^2 e^2\right )-A e \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\right ) \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 )}{8 d^{7/2} (c d-b e)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 1.50, size = 377, normalized size = 1.01 \begin {gather*} \frac {2 b^2 d^{5/2} (B d-A e) (c d-b e)^{5/2}+(d+e x) \left (b^2 d^{3/2} (c d-b e)^{3/2} (5 A e (b e-2 c d)+B d (6 c d-b e))-(d+e x) \left (3 b^2 e \sqrt {x} \sqrt {b+c x} \left (A e \left (-5 b^2 e^2+16 b c d e-16 c^2 d^2\right )+B d \left (b^2 e^2-4 b c d e+8 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )+b \sqrt {d} (c d-b e)^{3/2} \left (3 b^2 e (5 A e-B d)+4 b c d (2 B d-7 A e)+8 A c^2 d^2\right )+c \sqrt {d} x \sqrt {c d-b e} \left (3 b^3 e^2 (B d-5 A e)+2 b^2 c d e (19 A e-5 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )\right )}{4 b^2 d^{7/2} \sqrt {x (b+c x)} (d+e x)^2 (c d-b e)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^3*(b*x + c*x^2)^(3/2)),x]
[Out]
(2*b^2*d^(5/2)*(B*d - A*e)*(c*d - b*e)^(5/2) + (d + e*x)*(b^2*d^(3/2)*(c*d - b*e)^(3/2)*(B*d*(6*c*d - b*e) + 5
*A*e*(-2*c*d + b*e)) - (d + e*x)*(b*Sqrt[d]*(c*d - b*e)^(3/2)*(8*A*c^2*d^2 + 4*b*c*d*(2*B*d - 7*A*e) + 3*b^2*e
*(-(B*d) + 5*A*e)) + c*Sqrt[d]*Sqrt[c*d - b*e]*(16*A*c^3*d^3 + 3*b^3*e^2*(B*d - 5*A*e) - 8*b*c^2*d^2*(B*d + 3*
A*e) + 2*b^2*c*d*e*(-5*B*d + 19*A*e))*x + 3*b^2*e*(A*e*(-16*c^2*d^2 + 16*b*c*d*e - 5*b^2*e^2) + B*d*(8*c^2*d^2
- 4*b*c*d*e + b^2*e^2))*Sqrt[x]*Sqrt[b + c*x]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])))/(
4*b^2*d^(7/2)*(c*d - b*e)^(7/2)*Sqrt[x*(b + c*x)]*(d + e*x)^2)
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IntegrateAlgebraic [B] time = 58.96, size = 10583, normalized size = 28.30 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/((d + e*x)^3*(b*x + c*x^2)^(3/2)),x]
[Out]
Result too large to show
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fricas [B] time = 0.53, size = 2282, normalized size = 6.10
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^3/(c*x^2+b*x)^(3/2),x, algorithm="fricas")
[Out]
[-1/8*(3*((8*B*b^2*c^3*d^3*e^3 - 5*A*b^4*c*e^6 - 4*(B*b^3*c^2 + 4*A*b^2*c^3)*d^2*e^4 + (B*b^4*c + 16*A*b^3*c^2
)*d*e^5)*x^4 + (16*B*b^2*c^3*d^4*e^2 - 32*A*b^2*c^3*d^3*e^3 - 5*A*b^5*e^6 - 2*(B*b^4*c - 8*A*b^3*c^2)*d^2*e^4
+ (B*b^5 + 6*A*b^4*c)*d*e^5)*x^3 + (8*B*b^2*c^3*d^5*e - 10*A*b^5*d*e^5 + 4*(3*B*b^3*c^2 - 4*A*b^2*c^3)*d^4*e^2
- (7*B*b^4*c + 16*A*b^3*c^2)*d^3*e^3 + (2*B*b^5 + 27*A*b^4*c)*d^2*e^4)*x^2 + (8*B*b^3*c^2*d^5*e - 5*A*b^5*d^2
*e^4 - 4*(B*b^4*c + 4*A*b^3*c^2)*d^4*e^2 + (B*b^5 + 16*A*b^4*c)*d^3*e^3)*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*(8*A*b*c^4*d^7 - 32*A*b^2*c^3*d^6*e + 4
8*A*b^3*c^2*d^5*e^2 - 32*A*b^4*c*d^4*e^3 + 8*A*b^5*d^3*e^4 + (15*A*b^4*c*d*e^6 - 8*(B*b*c^4 - 2*A*c^5)*d^5*e^2
- 2*(B*b^2*c^3 + 20*A*b*c^4)*d^4*e^3 + (13*B*b^3*c^2 + 62*A*b^2*c^3)*d^3*e^4 - (3*B*b^4*c + 53*A*b^3*c^2)*d^2
*e^5)*x^3 + (15*A*b^5*d*e^6 - 16*(B*b*c^4 - 2*A*c^5)*d^6*e + 4*(B*b^2*c^3 - 18*A*b*c^4)*d^5*e^2 + (7*B*b^3*c^2
+ 80*A*b^2*c^3)*d^4*e^3 + (8*B*b^4*c - 27*A*b^3*c^2)*d^3*e^4 - (3*B*b^5 + 28*A*b^4*c)*d^2*e^5)*x^2 + (25*A*b^
5*d^2*e^5 - 8*(B*b*c^4 - 2*A*c^5)*d^7 + 8*(B*b^2*c^3 - 3*A*b*c^4)*d^6*e - 4*(3*B*b^3*c^2 + 4*A*b^2*c^3)*d^5*e^
2 + (17*B*b^4*c + 80*A*b^3*c^2)*d^4*e^3 - (5*B*b^5 + 81*A*b^4*c)*d^3*e^4)*x)*sqrt(c*x^2 + b*x))/((b^2*c^5*d^8*
e^2 - 4*b^3*c^4*d^7*e^3 + 6*b^4*c^3*d^6*e^4 - 4*b^5*c^2*d^5*e^5 + b^6*c*d^4*e^6)*x^4 + (2*b^2*c^5*d^9*e - 7*b^
3*c^4*d^8*e^2 + 8*b^4*c^3*d^7*e^3 - 2*b^5*c^2*d^6*e^4 - 2*b^6*c*d^5*e^5 + b^7*d^4*e^6)*x^3 + (b^2*c^5*d^10 - 2
*b^3*c^4*d^9*e - 2*b^4*c^3*d^8*e^2 + 8*b^5*c^2*d^7*e^3 - 7*b^6*c*d^6*e^4 + 2*b^7*d^5*e^5)*x^2 + (b^3*c^4*d^10
- 4*b^4*c^3*d^9*e + 6*b^5*c^2*d^8*e^2 - 4*b^6*c*d^7*e^3 + b^7*d^6*e^4)*x), -1/4*(3*((8*B*b^2*c^3*d^3*e^3 - 5*A
*b^4*c*e^6 - 4*(B*b^3*c^2 + 4*A*b^2*c^3)*d^2*e^4 + (B*b^4*c + 16*A*b^3*c^2)*d*e^5)*x^4 + (16*B*b^2*c^3*d^4*e^2
- 32*A*b^2*c^3*d^3*e^3 - 5*A*b^5*e^6 - 2*(B*b^4*c - 8*A*b^3*c^2)*d^2*e^4 + (B*b^5 + 6*A*b^4*c)*d*e^5)*x^3 + (
8*B*b^2*c^3*d^5*e - 10*A*b^5*d*e^5 + 4*(3*B*b^3*c^2 - 4*A*b^2*c^3)*d^4*e^2 - (7*B*b^4*c + 16*A*b^3*c^2)*d^3*e^
3 + (2*B*b^5 + 27*A*b^4*c)*d^2*e^4)*x^2 + (8*B*b^3*c^2*d^5*e - 5*A*b^5*d^2*e^4 - 4*(B*b^4*c + 4*A*b^3*c^2)*d^4
*e^2 + (B*b^5 + 16*A*b^4*c)*d^3*e^3)*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)) + (8*A*b*c^4*d^7 - 32*A*b^2*c^3*d^6*e + 48*A*b^3*c^2*d^5*e^2 - 32*A*b^4*c*d^4*e^3 + 8*A*b^5*d^3
*e^4 + (15*A*b^4*c*d*e^6 - 8*(B*b*c^4 - 2*A*c^5)*d^5*e^2 - 2*(B*b^2*c^3 + 20*A*b*c^4)*d^4*e^3 + (13*B*b^3*c^2
+ 62*A*b^2*c^3)*d^3*e^4 - (3*B*b^4*c + 53*A*b^3*c^2)*d^2*e^5)*x^3 + (15*A*b^5*d*e^6 - 16*(B*b*c^4 - 2*A*c^5)*d
^6*e + 4*(B*b^2*c^3 - 18*A*b*c^4)*d^5*e^2 + (7*B*b^3*c^2 + 80*A*b^2*c^3)*d^4*e^3 + (8*B*b^4*c - 27*A*b^3*c^2)*
d^3*e^4 - (3*B*b^5 + 28*A*b^4*c)*d^2*e^5)*x^2 + (25*A*b^5*d^2*e^5 - 8*(B*b*c^4 - 2*A*c^5)*d^7 + 8*(B*b^2*c^3 -
3*A*b*c^4)*d^6*e - 4*(3*B*b^3*c^2 + 4*A*b^2*c^3)*d^5*e^2 + (17*B*b^4*c + 80*A*b^3*c^2)*d^4*e^3 - (5*B*b^5 + 8
1*A*b^4*c)*d^3*e^4)*x)*sqrt(c*x^2 + b*x))/((b^2*c^5*d^8*e^2 - 4*b^3*c^4*d^7*e^3 + 6*b^4*c^3*d^6*e^4 - 4*b^5*c^
2*d^5*e^5 + b^6*c*d^4*e^6)*x^4 + (2*b^2*c^5*d^9*e - 7*b^3*c^4*d^8*e^2 + 8*b^4*c^3*d^7*e^3 - 2*b^5*c^2*d^6*e^4
- 2*b^6*c*d^5*e^5 + b^7*d^4*e^6)*x^3 + (b^2*c^5*d^10 - 2*b^3*c^4*d^9*e - 2*b^4*c^3*d^8*e^2 + 8*b^5*c^2*d^7*e^3
- 7*b^6*c*d^6*e^4 + 2*b^7*d^5*e^5)*x^2 + (b^3*c^4*d^10 - 4*b^4*c^3*d^9*e + 6*b^5*c^2*d^8*e^2 - 4*b^6*c*d^7*e^
3 + b^7*d^6*e^4)*x)]
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giac [B] time = 0.39, size = 1095, normalized size = 2.93 \begin {gather*} \frac {2 \, {\left (\frac {{\left (B b c^{3} d^{6} - 2 \, A c^{4} d^{6} + 3 \, A b c^{3} d^{5} e - 3 \, A b^{2} c^{2} d^{4} e^{2} + A b^{3} c d^{3} e^{3}\right )} x}{b^{2} c^{3} d^{9} - 3 \, b^{3} c^{2} d^{8} e + 3 \, b^{4} c d^{7} e^{2} - b^{5} d^{6} e^{3}} - \frac {A b c^{3} d^{6} - 3 \, A b^{2} c^{2} d^{5} e + 3 \, A b^{3} c d^{4} e^{2} - A b^{4} d^{3} e^{3}}{b^{2} c^{3} d^{9} - 3 \, b^{3} c^{2} d^{8} e + 3 \, b^{4} c d^{7} e^{2} - b^{5} d^{6} e^{3}}\right )}}{\sqrt {c x^{2} + b x}} - \frac {3 \, {\left (8 \, B c^{2} d^{3} e - 4 \, B b c d^{2} e^{2} - 16 \, A c^{2} d^{2} e^{2} + B b^{2} d e^{3} + 16 \, A b c d e^{3} - 5 \, A b^{2} e^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right )}{4 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} \sqrt {-c d^{2} + b d e}} + \frac {40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B c^{\frac {5}{2}} d^{4} e + 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B c^{2} d^{3} e^{2} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b c^{2} d^{4} e - 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b c^{\frac {3}{2}} d^{3} e^{2} - 56 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A c^{\frac {5}{2}} d^{3} e^{2} + 10 \, B b^{2} c^{\frac {3}{2}} d^{4} e - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b c d^{2} e^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A c^{2} d^{2} e^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{2} c d^{3} e^{2} - 56 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b c^{2} d^{3} e^{2} + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{2} \sqrt {c} d^{2} e^{3} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b c^{\frac {3}{2}} d^{2} e^{3} - 3 \, B b^{3} \sqrt {c} d^{3} e^{2} - 14 \, A b^{2} c^{\frac {3}{2}} d^{3} e^{2} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{2} d e^{4} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b c d e^{4} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{3} d^{2} e^{3} + 44 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{2} c d^{2} e^{3} - 13 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{2} \sqrt {c} d e^{4} + 7 \, A b^{3} \sqrt {c} d^{2} e^{3} - 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{2} e^{5} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{3} d e^{4}}{4 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^3/(c*x^2+b*x)^(3/2),x, algorithm="giac")
[Out]
2*((B*b*c^3*d^6 - 2*A*c^4*d^6 + 3*A*b*c^3*d^5*e - 3*A*b^2*c^2*d^4*e^2 + A*b^3*c*d^3*e^3)*x/(b^2*c^3*d^9 - 3*b^
3*c^2*d^8*e + 3*b^4*c*d^7*e^2 - b^5*d^6*e^3) - (A*b*c^3*d^6 - 3*A*b^2*c^2*d^5*e + 3*A*b^3*c*d^4*e^2 - A*b^4*d^
3*e^3)/(b^2*c^3*d^9 - 3*b^3*c^2*d^8*e + 3*b^4*c*d^7*e^2 - b^5*d^6*e^3))/sqrt(c*x^2 + b*x) - 3/4*(8*B*c^2*d^3*e
- 4*B*b*c*d^2*e^2 - 16*A*c^2*d^2*e^2 + B*b^2*d*e^3 + 16*A*b*c*d*e^3 - 5*A*b^2*e^4)*arctan(-((sqrt(c)*x - sqrt
(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)
*sqrt(-c*d^2 + b*d*e)) + 1/4*(40*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*c^(5/2)*d^4*e + 16*(sqrt(c)*x - sqrt(c*x^
2 + b*x))^3*B*c^2*d^3*e^2 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b*c^2*d^4*e - 28*(sqrt(c)*x - sqrt(c*x^2 + b*
x))^2*B*b*c^(3/2)*d^3*e^2 - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*c^(5/2)*d^3*e^2 + 10*B*b^2*c^(3/2)*d^4*e -
12*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b*c*d^2*e^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^2*d^2*e^3 - 24*(
sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^2*c*d^3*e^2 - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b*c^2*d^3*e^2 + 9*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^2*B*b^2*sqrt(c)*d^2*e^3 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b*c^(3/2)*d^2*e^3 -
3*B*b^3*sqrt(c)*d^3*e^2 - 14*A*b^2*c^(3/2)*d^3*e^2 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*d*e^4 + 24*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^3*A*b*c*d*e^4 + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^3*d^2*e^3 + 44*(sqrt(c)*x -
sqrt(c*x^2 + b*x))*A*b^2*c*d^2*e^3 - 13*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^2*sqrt(c)*d*e^4 + 7*A*b^3*sqrt(
c)*d^2*e^3 - 7*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*e^5 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*d*e^4)/((
c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x -
sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^2)
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maple [B] time = 0.07, size = 3735, normalized size = 9.99 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(e*x+d)^3/(c*x^2+b*x)^(3/2),x)
[Out]
-15/2/(b*e-c*d)^3/(-(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^2*B+17*B/(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+30/(b*e-c*d)^3/((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-45*e/(b*e-c*d)^3/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^3*A-13*e/(
b*e-c*d)^2/d^2*c^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*A-15/4*e^3/(b*e-c*d)^3/d^3/((
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*A-30/e/(b*e-c*d)^3/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^4*B*d-15/2*e^2/(b*e-c*d)^3/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*c*A+15/4*e^2/(b*e-c*d)^3/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*b*B+
15/2*e/(b*e-c*d)^3/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*c*B+B/e/(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)-3*B*e/(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-30*e/(b*e-c*d)^3/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^2*A-8
*e/(b*e-c*d)^2/d^2*c/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*A+1/2/e/(b*e-c*d)/d/(x+d/e)^2/(
(x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*A-15/4*e^3/(b*e-c*d)^3/d^3/((x+d/e)^2*c-(b*e-c*d)*d/e
^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b^2*A+15/4*e^2/(b*e-c*d)^3/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^2*B+5/2/e/(b*e-c*d)^2/(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)*c*B-8*B/e
*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-5/4/(b*e-c*d)^2/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*B-5/2/(b*e-c*d)^2/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)*c*A+30/(b*e-c*d)^3/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^4*A+45/(b*e-c*d)^3/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^3*B+13/(b*e-c*d)^2/
d*c^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)*A-19*B/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^2-9/2*B/(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-1/2/e^2/(b*e-c*d)/(x+d/e)^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+15/(b*e-c*d)^3
/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^3*A-45/2*e/(b*e-c*d)^3/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^2*B+26/(b*e-c*d)^2/d*c^3/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-75/4*e/(b*e-c*d)^3/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*c*B-3*B*
e/(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+25*B/(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^2-38*B/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+3/2*B*e/(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-4*B/e*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)+5/4*e/(b*e-c*d)^2/d^2/(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*A+75/4*e^2/(b*e-c*d)^3/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*c*A-15/8*e^2/(b*e-c*d)^3/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^2*B+3/2*e*c/(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))*A+15/2*e/(b
*e-c*d)^3/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^2*A-15/e/(b*e-c*d)^3/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^3*B*d+15/8*e^3/(b*e-c*d)^3/d^3/(-(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^2*A+45/2*e^2/(b*e-c*d)^3/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
^2*A
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^3/(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 \begin {gather*} \int \frac {A+B\,x}{{\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^3} \,d x \end {gather*}
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)^3),x)
[Out]
int((A + B*x)/((b*x + c*x^2)^(3/2)*(d + e*x)^3), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)**3/(c*x**2+b*x)**(3/2),x)
[Out]
Integral((A + B*x)/((x*(b + c*x))**(3/2)*(d + e*x)**3), x)
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