3.1774 \(\int \frac{A+B x}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\)
Optimal. Leaf size=332 \[ -\frac{b (A b-a B)}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b (2 a B e-3 A b e+b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{e (a+b x) (a B e-3 A b e+2 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}-\frac{e (a+b x) (B d-A e)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}-\frac{3 b e (a+b x) \log (a+b x) (a B e-2 A b e+b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{3 b e (a+b x) \log (d+e x) (a B e-2 A b e+b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5} \]
[Out]
-((b*(b*B*d - 3*A*b*e + 2*a*B*e))/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) - (b*(A*b - a*B))/(2*(b*d - a
*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*(B*d - A*e)*(a + b*x))/(2*(b*d - a*e)^3*(d + e*x)^2*Sqrt[a
^2 + 2*a*b*x + b^2*x^2]) - (e*(2*b*B*d - 3*A*b*e + a*B*e)*(a + b*x))/((b*d - a*e)^4*(d + e*x)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2]) - (3*b*e*(b*B*d - 2*A*b*e + a*B*e)*(a + b*x)*Log[a + b*x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (3*b*e*(b*B*d - 2*A*b*e + a*B*e)*(a + b*x)*Log[d + e*x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])
________________________________________________________________________________________
Rubi [A] time = 0.319145, antiderivative size = 332, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used =
{770, 77} \[ -\frac{b (A b-a B)}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b (2 a B e-3 A b e+b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{e (a+b x) (a B e-3 A b e+2 b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}-\frac{e (a+b x) (B d-A e)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}-\frac{3 b e (a+b x) \log (a+b x) (a B e-2 A b e+b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{3 b e (a+b x) \log (d+e x) (a B e-2 A b e+b B d)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
[Out]
-((b*(b*B*d - 3*A*b*e + 2*a*B*e))/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) - (b*(A*b - a*B))/(2*(b*d - a
*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*(B*d - A*e)*(a + b*x))/(2*(b*d - a*e)^3*(d + e*x)^2*Sqrt[a
^2 + 2*a*b*x + b^2*x^2]) - (e*(2*b*B*d - 3*A*b*e + a*B*e)*(a + b*x))/((b*d - a*e)^4*(d + e*x)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2]) - (3*b*e*(b*B*d - 2*A*b*e + a*B*e)*(a + b*x)*Log[a + b*x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (3*b*e*(b*B*d - 2*A*b*e + a*B*e)*(a + b*x)*Log[d + e*x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])
Rule 770
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac{A+B x}{\left (a b+b^2 x\right )^3 (d+e x)^3} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac{A b-a B}{b (b d-a e)^3 (a+b x)^3}+\frac{b B d-3 A b e+2 a B e}{b (b d-a e)^4 (a+b x)^2}+\frac{3 e (-b B d+2 A b e-a B e)}{b (b d-a e)^5 (a+b x)}-\frac{e^2 (-B d+A e)}{b^3 (b d-a e)^3 (d+e x)^3}-\frac{e^2 (-2 b B d+3 A b e-a B e)}{b^3 (b d-a e)^4 (d+e x)^2}-\frac{3 e^2 (-b B d+2 A b e-a B e)}{b^2 (b d-a e)^5 (d+e x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{b (b B d-3 A b e+2 a B e)}{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b (A b-a B)}{2 (b d-a e)^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (B d-A e) (a+b x)}{2 (b d-a e)^3 (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (2 b B d-3 A b e+a B e) (a+b x)}{(b d-a e)^4 (d+e x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 b e (b B d-2 A b e+a B e) (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b e (b B d-2 A b e+a B e) (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.245234, size = 220, normalized size = 0.66 \[ \frac{(a+b x) \left (\frac{e (a+b x)^2 (b d-a e)^2 (A e-B d)}{(d+e x)^2}-2 b (a+b x) (b d-a e) (2 a B e-3 A b e+b B d)+\frac{2 e (a+b x)^2 (b d-a e) (-a B e+3 A b e-2 b B d)}{d+e x}-6 b e (a+b x)^2 \log (a+b x) (a B e-2 A b e+b B d)+6 b e (a+b x)^2 \log (d+e x) (a B e-2 A b e+b B d)-b (A b-a B) (b d-a e)^2\right )}{2 \left ((a+b x)^2\right )^{3/2} (b d-a e)^5} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
[Out]
((a + b*x)*(-(b*(A*b - a*B)*(b*d - a*e)^2) - 2*b*(b*d - a*e)*(b*B*d - 3*A*b*e + 2*a*B*e)*(a + b*x) + (e*(b*d -
a*e)^2*(-(B*d) + A*e)*(a + b*x)^2)/(d + e*x)^2 + (2*e*(b*d - a*e)*(-2*b*B*d + 3*A*b*e - a*B*e)*(a + b*x)^2)/(
d + e*x) - 6*b*e*(b*B*d - 2*A*b*e + a*B*e)*(a + b*x)^2*Log[a + b*x] + 6*b*e*(b*B*d - 2*A*b*e + a*B*e)*(a + b*x
)^2*Log[d + e*x]))/(2*(b*d - a*e)^5*((a + b*x)^2)^(3/2))
________________________________________________________________________________________
Maple [B] time = 0.023, size = 1271, normalized size = 3.8 \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)^3/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
[Out]
1/2*(A*d^4*b^4+14*B*a*b^3*d^3*e*x-14*B*a^3*b*d*e^3*x-24*A*a*b^3*d^2*e^2*x+12*A*ln(e*x+d)*x^2*a^2*b^2*e^4+12*A*
ln(e*x+d)*x^2*b^4*d^2*e^2-12*A*ln(b*x+a)*x^2*a^2*b^2*e^4-12*A*ln(b*x+a)*x^2*b^4*d^2*e^2-6*B*ln(e*x+d)*x^2*a^3*
b*e^4+6*B*ln(b*x+a)*x^2*a^3*b*e^4+6*B*ln(b*x+a)*x^2*b^4*d^3*e+12*A*ln(e*x+d)*a^2*b^2*d^2*e^2-12*A*ln(b*x+a)*a^
2*b^2*d^2*e^2-6*B*ln(e*x+d)*a^3*b*d^2*e^2-6*B*ln(e*x+d)*a^2*b^2*d^3*e+6*B*ln(b*x+a)*a^3*b*d^2*e^2+6*B*ln(b*x+a
)*a^2*b^2*d^3*e-6*B*ln(e*x+d)*x^4*a*b^3*e^4-6*B*ln(e*x+d)*x^4*b^4*d*e^3+6*B*ln(b*x+a)*x^4*a*b^3*e^4+6*B*ln(b*x
+a)*x^4*b^4*d*e^3+24*A*ln(e*x+d)*x^3*a*b^3*e^4+24*A*ln(e*x+d)*x^3*b^4*d*e^3-24*A*ln(b*x+a)*x^3*a*b^3*e^4-24*A*
ln(b*x+a)*x^3*b^4*d*e^3-12*B*ln(e*x+d)*x^3*a^2*b^2*e^4-12*B*ln(e*x+d)*x^3*b^4*d^2*e^2+12*B*ln(b*x+a)*x^3*a^2*b
^2*e^4+12*B*ln(b*x+a)*x^3*b^4*d^2*e^2-9*B*x^2*a^2*b^2*d*e^3+9*B*x^2*a*b^3*d^2*e^2+24*A*a^2*b^2*d*e^3*x-6*B*ln(
e*x+d)*x^2*b^4*d^3*e+2*B*b^4*d^4*x-2*B*e^4*a^4*x-B*a^4*d*e^3+B*a*b^3*d^4+18*A*x^2*a^2*b^2*e^4-18*A*x^2*b^4*d^2
*e^2+6*B*x^3*b^4*d^2*e^2-6*B*x^3*a^2*b^2*e^4+12*A*x^3*a*b^3*e^4-12*A*x^3*b^4*d*e^3+12*A*ln(e*x+d)*x^4*b^4*e^4-
12*A*ln(b*x+a)*x^4*b^4*e^4-9*B*x^2*a^3*b*e^4+9*B*x^2*b^4*d^3*e+4*A*a^3*b*e^4*x-4*A*b^4*d^3*e*x+8*A*a^3*b*d*e^3
-9*B*a^3*b*d^2*e^2+9*B*a^2*b^2*d^3*e-8*A*a*b^3*d^3*e-12*B*ln(e*x+d)*x*a^3*b*d*e^3-24*B*ln(e*x+d)*x*a^2*b^2*d^2
*e^2-12*B*ln(e*x+d)*x*a*b^3*d^3*e-48*A*ln(b*x+a)*x^2*a*b^3*d*e^3-30*B*ln(e*x+d)*x^2*a^2*b^2*d*e^3-24*B*ln(e*x+
d)*x^3*a*b^3*d*e^3+24*B*ln(b*x+a)*x^3*a*b^3*d*e^3+48*A*ln(e*x+d)*x^2*a*b^3*d*e^3-30*B*ln(e*x+d)*x^2*a*b^3*d^2*
e^2+30*B*ln(b*x+a)*x^2*a^2*b^2*d*e^3+30*B*ln(b*x+a)*x^2*a*b^3*d^2*e^2+24*A*ln(e*x+d)*x*a^2*b^2*d*e^3+12*B*ln(b
*x+a)*x*a^3*b*d*e^3+24*B*ln(b*x+a)*x*a^2*b^2*d^2*e^2+12*B*ln(b*x+a)*x*a*b^3*d^3*e-A*a^4*e^4+24*A*ln(e*x+d)*x*a
*b^3*d^2*e^2-24*A*ln(b*x+a)*x*a^2*b^2*d*e^3-24*A*ln(b*x+a)*x*a*b^3*d^2*e^2)*(b*x+a)/(e*x+d)^2/(a*e-b*d)^5/((b*
x+a)^2)^(3/2)
________________________________________________________________________________________
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)^3/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 1.55218, size = 2421, normalized size = 7.29 \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/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")
[Out]
1/2*(9*B*a^3*b*d^2*e^2 + A*a^4*e^4 - (B*a*b^3 + A*b^4)*d^4 - (9*B*a^2*b^2 - 8*A*a*b^3)*d^3*e + (B*a^4 - 8*A*a^
3*b)*d*e^3 - 6*(B*b^4*d^2*e^2 - 2*A*b^4*d*e^3 - (B*a^2*b^2 - 2*A*a*b^3)*e^4)*x^3 - 9*(B*b^4*d^3*e - B*a^2*b^2*
d*e^3 + (B*a*b^3 - 2*A*b^4)*d^2*e^2 - (B*a^3*b - 2*A*a^2*b^2)*e^4)*x^2 - 2*(B*b^4*d^4 - 12*A*a*b^3*d^2*e^2 + (
7*B*a*b^3 - 2*A*b^4)*d^3*e - (7*B*a^3*b - 12*A*a^2*b^2)*d*e^3 - (B*a^4 - 2*A*a^3*b)*e^4)*x - 6*(B*a^2*b^2*d^3*
e + (B*a^3*b - 2*A*a^2*b^2)*d^2*e^2 + (B*b^4*d*e^3 + (B*a*b^3 - 2*A*b^4)*e^4)*x^4 + 2*(B*b^4*d^2*e^2 + 2*(B*a*
b^3 - A*b^4)*d*e^3 + (B*a^2*b^2 - 2*A*a*b^3)*e^4)*x^3 + (B*b^4*d^3*e + (5*B*a*b^3 - 2*A*b^4)*d^2*e^2 + (5*B*a^
2*b^2 - 8*A*a*b^3)*d*e^3 + (B*a^3*b - 2*A*a^2*b^2)*e^4)*x^2 + 2*(B*a*b^3*d^3*e + 2*(B*a^2*b^2 - A*a*b^3)*d^2*e
^2 + (B*a^3*b - 2*A*a^2*b^2)*d*e^3)*x)*log(b*x + a) + 6*(B*a^2*b^2*d^3*e + (B*a^3*b - 2*A*a^2*b^2)*d^2*e^2 + (
B*b^4*d*e^3 + (B*a*b^3 - 2*A*b^4)*e^4)*x^4 + 2*(B*b^4*d^2*e^2 + 2*(B*a*b^3 - A*b^4)*d*e^3 + (B*a^2*b^2 - 2*A*a
*b^3)*e^4)*x^3 + (B*b^4*d^3*e + (5*B*a*b^3 - 2*A*b^4)*d^2*e^2 + (5*B*a^2*b^2 - 8*A*a*b^3)*d*e^3 + (B*a^3*b - 2
*A*a^2*b^2)*e^4)*x^2 + 2*(B*a*b^3*d^3*e + 2*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + (B*a^3*b - 2*A*a^2*b^2)*d*e^3)*x)*
log(e*x + d))/(a^2*b^5*d^7 - 5*a^3*b^4*d^6*e + 10*a^4*b^3*d^5*e^2 - 10*a^5*b^2*d^4*e^3 + 5*a^6*b*d^3*e^4 - a^7
*d^2*e^5 + (b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^
2*e^7)*x^4 + 2*(b^7*d^6*e - 4*a*b^6*d^5*e^2 + 5*a^2*b^5*d^4*e^3 - 5*a^4*b^3*d^2*e^5 + 4*a^5*b^2*d*e^6 - a^6*b*
e^7)*x^3 + (b^7*d^7 - a*b^6*d^6*e - 9*a^2*b^5*d^5*e^2 + 25*a^3*b^4*d^4*e^3 - 25*a^4*b^3*d^3*e^4 + 9*a^5*b^2*d^
2*e^5 + a^6*b*d*e^6 - a^7*e^7)*x^2 + 2*(a*b^6*d^7 - 4*a^2*b^5*d^6*e + 5*a^3*b^4*d^5*e^2 - 5*a^5*b^2*d^3*e^4 +
4*a^6*b*d^2*e^5 - a^7*d*e^6)*x)
________________________________________________________________________________________
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)**3/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")
[Out]
sage0*x