3.790 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{x^{9/2}} \, dx\)

Optimal. Leaf size=120 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^{5/2} (a+b x)}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)} \]

[Out]

(-2*a*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^(7/2)*(a + b*x)) - (2*(A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
5*x^(5/2)*(a + b*x)) - (2*b*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^(3/2)*(a + b*x))

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Rubi [A]  time = 0.0427632, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {770, 76} \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^{5/2} (a+b x)}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^(9/2),x]

[Out]

(-2*a*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^(7/2)*(a + b*x)) - (2*(A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
5*x^(5/2)*(a + b*x)) - (2*b*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^(3/2)*(a + b*x))

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 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{x^{9/2}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right ) (A+B x)}{x^{9/2}} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a A b}{x^{9/2}}+\frac{b (A b+a B)}{x^{7/2}}+\frac{b^2 B}{x^{5/2}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac{2 (A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0284987, size = 51, normalized size = 0.42 \[ -\frac{2 \sqrt{(a+b x)^2} (3 a (5 A+7 B x)+7 b x (3 A+5 B x))}{105 x^{7/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^(9/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(7*b*x*(3*A + 5*B*x) + 3*a*(5*A + 7*B*x)))/(105*x^(7/2)*(a + b*x))

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Maple [A]  time = 0.004, size = 44, normalized size = 0.4 \begin{align*} -{\frac{70\,Bb{x}^{2}+42\,Abx+42\,aBx+30\,aA}{105\,bx+105\,a}\sqrt{ \left ( bx+a \right ) ^{2}}{x}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*((b*x+a)^2)^(1/2)/x^(9/2),x)

[Out]

-2/105*(35*B*b*x^2+21*A*b*x+21*B*a*x+15*A*a)*((b*x+a)^2)^(1/2)/x^(7/2)/(b*x+a)

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Maxima [A]  time = 1.05445, size = 47, normalized size = 0.39 \begin{align*} -\frac{2 \,{\left (5 \, b x^{2} + 3 \, a x\right )} B}{15 \, x^{\frac{7}{2}}} - \frac{2 \,{\left (7 \, b x^{2} + 5 \, a x\right )} A}{35 \, x^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/x^(9/2),x, algorithm="maxima")

[Out]

-2/15*(5*b*x^2 + 3*a*x)*B/x^(7/2) - 2/35*(7*b*x^2 + 5*a*x)*A/x^(9/2)

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Fricas [A]  time = 1.49881, size = 77, normalized size = 0.64 \begin{align*} -\frac{2 \,{\left (35 \, B b x^{2} + 15 \, A a + 21 \,{\left (B a + A b\right )} x\right )}}{105 \, x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/x^(9/2),x, algorithm="fricas")

[Out]

-2/105*(35*B*b*x^2 + 15*A*a + 21*(B*a + A*b)*x)/x^(7/2)

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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)*((b*x+a)**2)**(1/2)/x**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.13674, size = 69, normalized size = 0.57 \begin{align*} -\frac{2 \,{\left (35 \, B b x^{2} \mathrm{sgn}\left (b x + a\right ) + 21 \, B a x \mathrm{sgn}\left (b x + a\right ) + 21 \, A b x \mathrm{sgn}\left (b x + a\right ) + 15 \, A a \mathrm{sgn}\left (b x + a\right )\right )}}{105 \, x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/x^(9/2),x, algorithm="giac")

[Out]

-2/105*(35*B*b*x^2*sgn(b*x + a) + 21*B*a*x*sgn(b*x + a) + 21*A*b*x*sgn(b*x + a) + 15*A*a*sgn(b*x + a))/x^(7/2)