3.19.12 \(\int \frac {(a+b x) (d+e x)}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=68 \[ -\frac {d+e x}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac {e}{6 b^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {768, 607} \begin {gather*} -\frac {d+e x}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac {e}{6 b^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(d + e*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

Rule 607

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*(a + b*x + c*x^2)^(p + 1))/((2*p + 1)*(b + 2
*c*x)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 768

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(2*c*(p + 1)), x] - Dist[(e*g*m)/(2*c*(p + 1)), Int[(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1]
&& GtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(a+b x) (d+e x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=-\frac {d+e x}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac {e \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac {d+e x}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac {e}{6 b^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 32, normalized size = 0.47 \begin {gather*} \frac {-a e-2 b d-3 b e x}{6 b^2 \left ((a+b x)^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(d + e*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

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IntegrateAlgebraic [B]  time = 0.84, size = 235, normalized size = 3.46 \begin {gather*} \frac {2 \left (-2 a^4 b e+2 a^3 b^2 d-a b^4 e x^3-2 b^5 d x^3-3 b^5 e x^4\right )+2 \sqrt {b^2} \sqrt {a^2+2 a b x+b^2 x^2} \left (-2 a^3 e+2 a^2 b d+2 a^2 b e x-2 a b^2 d x-2 a b^2 e x^2+2 b^3 d x^2+3 b^3 e x^3\right )}{3 x^3 \sqrt {a^2+2 a b x+b^2 x^2} \left (4 a^2 b^6+8 a b^7 x+4 b^8 x^2\right )+3 \sqrt {b^2} x^3 \left (-4 a^3 b^5-12 a^2 b^6 x-12 a b^7 x^2-4 b^8 x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)*(d + e*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(2*a^2*b*d - 2*a^3*e - 2*a*b^2*d*x + 2*a^2*b*e*x + 2*b^3*d*x^2 - 2*
a*b^2*e*x^2 + 3*b^3*e*x^3) + 2*(2*a^3*b^2*d - 2*a^4*b*e - 2*b^5*d*x^3 - a*b^4*e*x^3 - 3*b^5*e*x^4))/(3*x^3*Sqr
t[a^2 + 2*a*b*x + b^2*x^2]*(4*a^2*b^6 + 8*a*b^7*x + 4*b^8*x^2) + 3*Sqrt[b^2]*x^3*(-4*a^3*b^5 - 12*a^2*b^6*x -
12*a*b^7*x^2 - 4*b^8*x^3))

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fricas [A]  time = 0.44, size = 50, normalized size = 0.74 \begin {gather*} -\frac {3 \, b e x + 2 \, b d + a e}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x + a\right )} {\left (e x + d\right )}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x + a)*(e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), x)

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maple [A]  time = 0.06, size = 35, normalized size = 0.51 \begin {gather*} -\frac {\left (b x +a \right )^{2} \left (3 b e x +a e +2 b d \right )}{6 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

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

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maxima [B]  time = 0.60, size = 118, normalized size = 1.74 \begin {gather*} -\frac {b d + a e}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {e}{2 \, b^{4} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {2 \, a e}{3 \, b^{5} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {a d}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} - \frac {a^{2} e}{4 \, b^{6} {\left (x + \frac {a}{b}\right )}^{4}} + \frac {{\left (b d + a e\right )} a}{4 \, b^{6} {\left (x + \frac {a}{b}\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.16, size = 43, normalized size = 0.63 \begin {gather*} -\frac {\left (a\,e+2\,b\,d+3\,b\,e\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,b^2\,{\left (a+b\,x\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(d + e*x))/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right ) \left (d + e x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((a + b*x)*(d + e*x)/((a + b*x)**2)**(5/2), x)

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