3.487 \(\int \frac{\sqrt{a+b x} (A+B x)}{x^{11/2}} \, dx\)

Optimal. Leaf size=117 \[ \frac{16 b^2 (a+b x)^{3/2} (2 A b-3 a B)}{315 a^4 x^{3/2}}-\frac{8 b (a+b x)^{3/2} (2 A b-3 a B)}{105 a^3 x^{5/2}}+\frac{2 (a+b x)^{3/2} (2 A b-3 a B)}{21 a^2 x^{7/2}}-\frac{2 A (a+b x)^{3/2}}{9 a x^{9/2}} \]

[Out]

(-2*A*(a + b*x)^(3/2))/(9*a*x^(9/2)) + (2*(2*A*b - 3*a*B)*(a + b*x)^(3/2))/(21*a^2*x^(7/2)) - (8*b*(2*A*b - 3*
a*B)*(a + b*x)^(3/2))/(105*a^3*x^(5/2)) + (16*b^2*(2*A*b - 3*a*B)*(a + b*x)^(3/2))/(315*a^4*x^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0395779, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ \frac{16 b^2 (a+b x)^{3/2} (2 A b-3 a B)}{315 a^4 x^{3/2}}-\frac{8 b (a+b x)^{3/2} (2 A b-3 a B)}{105 a^3 x^{5/2}}+\frac{2 (a+b x)^{3/2} (2 A b-3 a B)}{21 a^2 x^{7/2}}-\frac{2 A (a+b x)^{3/2}}{9 a x^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(A + B*x))/x^(11/2),x]

[Out]

(-2*A*(a + b*x)^(3/2))/(9*a*x^(9/2)) + (2*(2*A*b - 3*a*B)*(a + b*x)^(3/2))/(21*a^2*x^(7/2)) - (8*b*(2*A*b - 3*
a*B)*(a + b*x)^(3/2))/(105*a^3*x^(5/2)) + (16*b^2*(2*A*b - 3*a*B)*(a + b*x)^(3/2))/(315*a^4*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^{11/2}} \, dx &=-\frac{2 A (a+b x)^{3/2}}{9 a x^{9/2}}+\frac{\left (2 \left (-3 A b+\frac{9 a B}{2}\right )\right ) \int \frac{\sqrt{a+b x}}{x^{9/2}} \, dx}{9 a}\\ &=-\frac{2 A (a+b x)^{3/2}}{9 a x^{9/2}}+\frac{2 (2 A b-3 a B) (a+b x)^{3/2}}{21 a^2 x^{7/2}}+\frac{(4 b (2 A b-3 a B)) \int \frac{\sqrt{a+b x}}{x^{7/2}} \, dx}{21 a^2}\\ &=-\frac{2 A (a+b x)^{3/2}}{9 a x^{9/2}}+\frac{2 (2 A b-3 a B) (a+b x)^{3/2}}{21 a^2 x^{7/2}}-\frac{8 b (2 A b-3 a B) (a+b x)^{3/2}}{105 a^3 x^{5/2}}-\frac{\left (8 b^2 (2 A b-3 a B)\right ) \int \frac{\sqrt{a+b x}}{x^{5/2}} \, dx}{105 a^3}\\ &=-\frac{2 A (a+b x)^{3/2}}{9 a x^{9/2}}+\frac{2 (2 A b-3 a B) (a+b x)^{3/2}}{21 a^2 x^{7/2}}-\frac{8 b (2 A b-3 a B) (a+b x)^{3/2}}{105 a^3 x^{5/2}}+\frac{16 b^2 (2 A b-3 a B) (a+b x)^{3/2}}{315 a^4 x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0267633, size = 73, normalized size = 0.62 \[ -\frac{2 (a+b x)^{3/2} \left (-6 a^2 b x (5 A+6 B x)+5 a^3 (7 A+9 B x)+24 a b^2 x^2 (A+B x)-16 A b^3 x^3\right )}{315 a^4 x^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*(A + B*x))/x^(11/2),x]

[Out]

(-2*(a + b*x)^(3/2)*(-16*A*b^3*x^3 + 24*a*b^2*x^2*(A + B*x) - 6*a^2*b*x*(5*A + 6*B*x) + 5*a^3*(7*A + 9*B*x)))/
(315*a^4*x^(9/2))

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 77, normalized size = 0.7 \begin{align*} -{\frac{-32\,A{b}^{3}{x}^{3}+48\,B{x}^{3}a{b}^{2}+48\,aA{b}^{2}{x}^{2}-72\,B{x}^{2}{a}^{2}b-60\,{a}^{2}Abx+90\,{a}^{3}Bx+70\,A{a}^{3}}{315\,{a}^{4}} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b*x+a)^(1/2)/x^(11/2),x)

[Out]

-2/315*(b*x+a)^(3/2)*(-16*A*b^3*x^3+24*B*a*b^2*x^3+24*A*a*b^2*x^2-36*B*a^2*b*x^2-30*A*a^2*b*x+45*B*a^3*x+35*A*
a^3)/x^(9/2)/a^4

________________________________________________________________________________________

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^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.6826, size = 231, normalized size = 1.97 \begin{align*} -\frac{2 \,{\left (35 \, A a^{4} + 8 \,{\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{4} - 4 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 3 \,{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 5 \,{\left (9 \, B a^{4} + A a^{3} b\right )} x\right )} \sqrt{b x + a}}{315 \, a^{4} x^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/x^(11/2),x, algorithm="fricas")

[Out]

-2/315*(35*A*a^4 + 8*(3*B*a*b^3 - 2*A*b^4)*x^4 - 4*(3*B*a^2*b^2 - 2*A*a*b^3)*x^3 + 3*(3*B*a^3*b - 2*A*a^2*b^2)
*x^2 + 5*(9*B*a^4 + A*a^3*b)*x)*sqrt(b*x + a)/(a^4*x^(9/2))

________________________________________________________________________________________

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**(11/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.35234, size = 201, normalized size = 1.72 \begin{align*} \frac{{\left ({\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (3 \, B a b^{8} - 2 \, A b^{9}\right )}{\left (b x + a\right )}}{a^{5} b^{15}} - \frac{9 \,{\left (3 \, B a^{2} b^{8} - 2 \, A a b^{9}\right )}}{a^{5} b^{15}}\right )} + \frac{63 \,{\left (3 \, B a^{3} b^{8} - 2 \, A a^{2} b^{9}\right )}}{a^{5} b^{15}}\right )} - \frac{105 \,{\left (B a^{4} b^{8} - A a^{3} b^{9}\right )}}{a^{5} b^{15}}\right )}{\left (b x + a\right )}^{\frac{3}{2}} b}{322560 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{9}{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^(11/2),x, algorithm="giac")

[Out]

1/322560*((b*x + a)*(4*(b*x + a)*(2*(3*B*a*b^8 - 2*A*b^9)*(b*x + a)/(a^5*b^15) - 9*(3*B*a^2*b^8 - 2*A*a*b^9)/(
a^5*b^15)) + 63*(3*B*a^3*b^8 - 2*A*a^2*b^9)/(a^5*b^15)) - 105*(B*a^4*b^8 - A*a^3*b^9)/(a^5*b^15))*(b*x + a)^(3
/2)*b/(((b*x + a)*b - a*b)^(9/2)*abs(b))