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

Optimal. Leaf size=84 \[ -\frac{4 b (a+b x)^{3/2} (4 A b-7 a B)}{105 a^3 x^{3/2}}+\frac{2 (a+b x)^{3/2} (4 A b-7 a B)}{35 a^2 x^{5/2}}-\frac{2 A (a+b x)^{3/2}}{7 a x^{7/2}} \]

[Out]

(-2*A*(a + b*x)^(3/2))/(7*a*x^(7/2)) + (2*(4*A*b - 7*a*B)*(a + b*x)^(3/2))/(35*a^2*x^(5/2)) - (4*b*(4*A*b - 7*
a*B)*(a + b*x)^(3/2))/(105*a^3*x^(3/2))

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Rubi [A]  time = 0.0273294, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ -\frac{4 b (a+b x)^{3/2} (4 A b-7 a B)}{105 a^3 x^{3/2}}+\frac{2 (a+b x)^{3/2} (4 A b-7 a B)}{35 a^2 x^{5/2}}-\frac{2 A (a+b x)^{3/2}}{7 a x^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*(a + b*x)^(3/2))/(7*a*x^(7/2)) + (2*(4*A*b - 7*a*B)*(a + b*x)^(3/2))/(35*a^2*x^(5/2)) - (4*b*(4*A*b - 7*
a*B)*(a + b*x)^(3/2))/(105*a^3*x^(3/2))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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{\sqrt{a+b x} (A+B x)}{x^{9/2}} \, dx &=-\frac{2 A (a+b x)^{3/2}}{7 a x^{7/2}}+\frac{\left (2 \left (-2 A b+\frac{7 a B}{2}\right )\right ) \int \frac{\sqrt{a+b x}}{x^{7/2}} \, dx}{7 a}\\ &=-\frac{2 A (a+b x)^{3/2}}{7 a x^{7/2}}+\frac{2 (4 A b-7 a B) (a+b x)^{3/2}}{35 a^2 x^{5/2}}+\frac{(2 b (4 A b-7 a B)) \int \frac{\sqrt{a+b x}}{x^{5/2}} \, dx}{35 a^2}\\ &=-\frac{2 A (a+b x)^{3/2}}{7 a x^{7/2}}+\frac{2 (4 A b-7 a B) (a+b x)^{3/2}}{35 a^2 x^{5/2}}-\frac{4 b (4 A b-7 a B) (a+b x)^{3/2}}{105 a^3 x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0206333, size = 57, normalized size = 0.68 \[ -\frac{2 (a+b x)^{3/2} \left (3 a^2 (5 A+7 B x)-2 a b x (6 A+7 B x)+8 A b^2 x^2\right )}{105 a^3 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*x)^(3/2)*(8*A*b^2*x^2 + 3*a^2*(5*A + 7*B*x) - 2*a*b*x*(6*A + 7*B*x)))/(105*a^3*x^(7/2))

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Maple [A]  time = 0.005, size = 53, normalized size = 0.6 \begin{align*} -{\frac{16\,A{b}^{2}{x}^{2}-28\,B{x}^{2}ab-24\,aAbx+42\,{a}^{2}Bx+30\,A{a}^{2}}{105\,{a}^{3}} \left ( bx+a \right ) ^{{\frac{3}{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)^(1/2)/x^(9/2),x)

[Out]

-2/105*(b*x+a)^(3/2)*(8*A*b^2*x^2-14*B*a*b*x^2-12*A*a*b*x+21*B*a^2*x+15*A*a^2)/x^(7/2)/a^3

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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)^(1/2)/x^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.59901, size = 180, normalized size = 2.14 \begin{align*} -\frac{2 \,{\left (15 \, A a^{3} - 2 \,{\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} +{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2} + 3 \,{\left (7 \, B a^{3} + A a^{2} b\right )} x\right )} \sqrt{b x + a}}{105 \, a^{3} x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(15*A*a^3 - 2*(7*B*a*b^2 - 4*A*b^3)*x^3 + (7*B*a^2*b - 4*A*a*b^2)*x^2 + 3*(7*B*a^3 + A*a^2*b)*x)*sqrt(b
*x + a)/(a^3*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)**(1/2)/x**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.3256, size = 154, normalized size = 1.83 \begin{align*} -\frac{{\left (b x + a\right )}^{\frac{3}{2}}{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (7 \, B a b^{6} - 4 \, A b^{7}\right )}{\left (b x + a\right )}}{a^{4} b^{12}} - \frac{7 \,{\left (7 \, B a^{2} b^{6} - 4 \, A a b^{7}\right )}}{a^{4} b^{12}}\right )} + \frac{35 \,{\left (B a^{3} b^{6} - A a^{2} b^{7}\right )}}{a^{4} b^{12}}\right )} b}{80640 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{7}{2}}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/80640*(b*x + a)^(3/2)*((b*x + a)*(2*(7*B*a*b^6 - 4*A*b^7)*(b*x + a)/(a^4*b^12) - 7*(7*B*a^2*b^6 - 4*A*a*b^7
)/(a^4*b^12)) + 35*(B*a^3*b^6 - A*a^2*b^7)/(a^4*b^12))*b/(((b*x + a)*b - a*b)^(7/2)*abs(b))