3.509 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^{11/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{2 (a+b x)^{7/2} (2 A b-9 a B)}{63 a^2 x^{7/2}}-\frac{2 A (a+b x)^{7/2}}{9 a x^{9/2}} \]

[Out]

(-2*A*(a + b*x)^(7/2))/(9*a*x^(9/2)) + (2*(2*A*b - 9*a*B)*(a + b*x)^(7/2))/(63*a^2*x^(7/2))

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Rubi [A]  time = 0.0142352, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {78, 37} \[ \frac{2 (a+b x)^{7/2} (2 A b-9 a B)}{63 a^2 x^{7/2}}-\frac{2 A (a+b x)^{7/2}}{9 a x^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/x^(11/2),x]

[Out]

(-2*A*(a + b*x)^(7/2))/(9*a*x^(9/2)) + (2*(2*A*b - 9*a*B)*(a + b*x)^(7/2))/(63*a^2*x^(7/2))

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 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)^{5/2} (A+B x)}{x^{11/2}} \, dx &=-\frac{2 A (a+b x)^{7/2}}{9 a x^{9/2}}+\frac{\left (2 \left (-A b+\frac{9 a B}{2}\right )\right ) \int \frac{(a+b x)^{5/2}}{x^{9/2}} \, dx}{9 a}\\ &=-\frac{2 A (a+b x)^{7/2}}{9 a x^{9/2}}+\frac{2 (2 A b-9 a B) (a+b x)^{7/2}}{63 a^2 x^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0203764, size = 36, normalized size = 0.68 \[ -\frac{2 (a+b x)^{7/2} (7 a A+9 a B x-2 A b x)}{63 a^2 x^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/x^(11/2),x]

[Out]

(-2*(a + b*x)^(7/2)*(7*a*A - 2*A*b*x + 9*a*B*x))/(63*a^2*x^(9/2))

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Maple [A]  time = 0.003, size = 31, normalized size = 0.6 \begin{align*} -{\frac{-4\,Abx+18\,Bax+14\,Aa}{63\,{a}^{2}} \left ( bx+a \right ) ^{{\frac{7}{2}}}{x}^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)/x^(11/2),x)

[Out]

-2/63*(b*x+a)^(7/2)*(-2*A*b*x+9*B*a*x+7*A*a)/x^(9/2)/a^2

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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)^(5/2)*(B*x+A)/x^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.6027, size = 223, normalized size = 4.21 \begin{align*} -\frac{2 \,{\left (7 \, A a^{4} +{\left (9 \, B a b^{3} - 2 \, A b^{4}\right )} x^{4} +{\left (27 \, B a^{2} b^{2} + A a b^{3}\right )} x^{3} + 3 \,{\left (9 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} x^{2} +{\left (9 \, B a^{4} + 19 \, A a^{3} b\right )} x\right )} \sqrt{b x + a}}{63 \, a^{2} x^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^(11/2),x, algorithm="fricas")

[Out]

-2/63*(7*A*a^4 + (9*B*a*b^3 - 2*A*b^4)*x^4 + (27*B*a^2*b^2 + A*a*b^3)*x^3 + 3*(9*B*a^3*b + 5*A*a^2*b^2)*x^2 +
(9*B*a^4 + 19*A*a^3*b)*x)*sqrt(b*x + a)/(a^2*x^(9/2))

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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)**(5/2)*(B*x+A)/x**(11/2),x)

[Out]

Timed out

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Giac [B]  time = 1.3019, size = 116, normalized size = 2.19 \begin{align*} \frac{{\left (b x + a\right )}^{\frac{7}{2}} b{\left (\frac{{\left (9 \, B a^{3} b^{8} - 2 \, A a^{2} b^{9}\right )}{\left (b x + a\right )}}{a^{5} b^{15}} - \frac{9 \,{\left (B a^{4} b^{8} - A a^{3} b^{9}\right )}}{a^{5} b^{15}}\right )}}{64512 \,{\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)^(5/2)*(B*x+A)/x^(11/2),x, algorithm="giac")

[Out]

1/64512*(b*x + a)^(7/2)*b*((9*B*a^3*b^8 - 2*A*a^2*b^9)*(b*x + a)/(a^5*b^15) - 9*(B*a^4*b^8 - A*a^3*b^9)/(a^5*b
^15))/(((b*x + a)*b - a*b)^(9/2)*abs(b))