3.1716 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=104 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2} (B d-A e)}{3 (d+e x)^3 (b d-a e)^2}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{2 (d+e x)^2 (b d-a e)^2} \]

[Out]

((A*b - a*B)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*(b*d - a*e)^2*(d + e*x)^2) + ((B*d - A*e)*(a^2 + 2*a*
b*x + b^2*x^2)^(3/2))/(3*(b*d - a*e)^2*(d + e*x)^3)

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Rubi [A]  time = 0.0591883, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {769, 646, 37} \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2} (B d-A e)}{3 (d+e x)^3 (b d-a e)^2}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{2 (d+e x)^2 (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^4,x]

[Out]

((A*b - a*B)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*(b*d - a*e)^2*(d + e*x)^2) + ((B*d - A*e)*(a^2 + 2*a*
b*x + b^2*x^2)^(3/2))/(3*(b*d - a*e)^2*(d + e*x)^3)

Rule 769

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(-2*c*(e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)^2), x] + Dist[(2*c*f -
b*g)/(2*c*d - b*e), 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]
&& EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && NeQ[2*c*f - b*g, 0] && NeQ[2*c*d - b*e, 0]

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx &=\frac{(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (b d-a e)^2 (d+e x)^3}+\frac{(A b-a B) \int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^3} \, dx}{b d-a e}\\ &=\frac{(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (b d-a e)^2 (d+e x)^3}+\frac{\left ((A b-a B) \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{a b+b^2 x}{(d+e x)^3} \, dx}{(b d-a e) \left (a b+b^2 x\right )}\\ &=\frac{(A b-a B) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 (b d-a e)^2 (d+e x)^2}+\frac{(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (b d-a e)^2 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0391157, size = 81, normalized size = 0.78 \[ -\frac{\sqrt{(a+b x)^2} \left (a e (2 A e+B (d+3 e x))+b \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )\right )}{6 e^3 (a+b x) (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^4,x]

[Out]

-(Sqrt[(a + b*x)^2]*(a*e*(2*A*e + B*(d + 3*e*x)) + b*(A*e*(d + 3*e*x) + 2*B*(d^2 + 3*d*e*x + 3*e^2*x^2))))/(6*
e^3*(a + b*x)*(d + e*x)^3)

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Maple [A]  time = 0.005, size = 87, normalized size = 0.8 \begin{align*} -{\frac{6\,B{x}^{2}b{e}^{2}+3\,Axb{e}^{2}+3\,aB{e}^{2}x+6\,Bxbde+2\,aA{e}^{2}+Abde+aBde+2\,Bb{d}^{2}}{6\, \left ( ex+d \right ) ^{3}{e}^{3} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^4,x)

[Out]

-1/6*(6*B*b*e^2*x^2+3*A*b*e^2*x+3*B*a*e^2*x+6*B*b*d*e*x+2*A*a*e^2+A*b*d*e+B*a*d*e+2*B*b*d^2)*((b*x+a)^2)^(1/2)
/(e*x+d)^3/e^3/(b*x+a)

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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)*((b*x+a)^2)^(1/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.52624, size = 200, normalized size = 1.92 \begin{align*} -\frac{6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} +{\left (B a + A b\right )} d e + 3 \,{\left (2 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

-1/6*(6*B*b*e^2*x^2 + 2*B*b*d^2 + 2*A*a*e^2 + (B*a + A*b)*d*e + 3*(2*B*b*d*e + (B*a + A*b)*e^2)*x)/(e^6*x^3 +
3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^3)

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Sympy [A]  time = 1.85542, size = 107, normalized size = 1.03 \begin{align*} - \frac{2 A a e^{2} + A b d e + B a d e + 2 B b d^{2} + 6 B b e^{2} x^{2} + x \left (3 A b e^{2} + 3 B a e^{2} + 6 B b d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)**2)**(1/2)/(e*x+d)**4,x)

[Out]

-(2*A*a*e**2 + A*b*d*e + B*a*d*e + 2*B*b*d**2 + 6*B*b*e**2*x**2 + x*(3*A*b*e**2 + 3*B*a*e**2 + 6*B*b*d*e))/(6*
d**3*e**3 + 18*d**2*e**4*x + 18*d*e**5*x**2 + 6*e**6*x**3)

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Giac [A]  time = 1.11973, size = 158, normalized size = 1.52 \begin{align*} -\frac{{\left (6 \, B b x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, B b d x e \mathrm{sgn}\left (b x + a\right ) + 2 \, B b d^{2} \mathrm{sgn}\left (b x + a\right ) + 3 \, B a x e^{2} \mathrm{sgn}\left (b x + a\right ) + 3 \, A b x e^{2} \mathrm{sgn}\left (b x + a\right ) + B a d e \mathrm{sgn}\left (b x + a\right ) + A b d e \mathrm{sgn}\left (b x + a\right ) + 2 \, A a e^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

-1/6*(6*B*b*x^2*e^2*sgn(b*x + a) + 6*B*b*d*x*e*sgn(b*x + a) + 2*B*b*d^2*sgn(b*x + a) + 3*B*a*x*e^2*sgn(b*x + a
) + 3*A*b*x*e^2*sgn(b*x + a) + B*a*d*e*sgn(b*x + a) + A*b*d*e*sgn(b*x + a) + 2*A*a*e^2*sgn(b*x + a))*e^(-3)/(x
*e + d)^3