3.485 \(\int \frac{\sqrt{a+b x} (A+B x)}{x^{7/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{2 (a+b x)^{3/2} (2 A b-5 a B)}{15 a^2 x^{3/2}}-\frac{2 A (a+b x)^{3/2}}{5 a x^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0153268, 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} \[ \frac{2 (a+b x)^{3/2} (2 A b-5 a B)}{15 a^2 x^{3/2}}-\frac{2 A (a+b x)^{3/2}}{5 a x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(A + B*x))/x^(7/2),x]

[Out]

(-2*A*(a + b*x)^(3/2))/(5*a*x^(5/2)) + (2*(2*A*b - 5*a*B)*(a + b*x)^(3/2))/(15*a^2*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 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^{7/2}} \, dx &=-\frac{2 A (a+b x)^{3/2}}{5 a x^{5/2}}+\frac{\left (2 \left (-A b+\frac{5 a B}{2}\right )\right ) \int \frac{\sqrt{a+b x}}{x^{5/2}} \, dx}{5 a}\\ &=-\frac{2 A (a+b x)^{3/2}}{5 a x^{5/2}}+\frac{2 (2 A b-5 a B) (a+b x)^{3/2}}{15 a^2 x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0132321, size = 36, normalized size = 0.68 \[ -\frac{2 (a+b x)^{3/2} (3 a A+5 a B x-2 A b x)}{15 a^2 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*(A + B*x))/x^(7/2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 31, normalized size = 0.6 \begin{align*} -{\frac{-4\,Abx+10\,Bax+6\,Aa}{15\,{a}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b*x+a)^(1/2)/x^(7/2),x)

[Out]

-2/15*(b*x+a)^(3/2)*(-2*A*b*x+5*B*a*x+3*A*a)/x^(5/2)/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)*(b*x+a)^(1/2)/x^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.58863, size = 126, normalized size = 2.38 \begin{align*} -\frac{2 \,{\left (3 \, A a^{2} +{\left (5 \, B a b - 2 \, A b^{2}\right )} x^{2} +{\left (5 \, B a^{2} + A a b\right )} x\right )} \sqrt{b x + a}}{15 \, a^{2} x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/x^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 120.542, size = 110, normalized size = 2.08 \begin{align*} A \left (- \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{5 x^{2}} - \frac{2 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{15 a x} + \frac{4 b^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}}{15 a^{2}}\right ) + B \left (- \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 x} - \frac{2 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{3 a}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)**(1/2)/x**(7/2),x)

[Out]

A*(-2*sqrt(b)*sqrt(a/(b*x) + 1)/(5*x**2) - 2*b**(3/2)*sqrt(a/(b*x) + 1)/(15*a*x) + 4*b**(5/2)*sqrt(a/(b*x) + 1
)/(15*a**2)) + B*(-2*sqrt(b)*sqrt(a/(b*x) + 1)/(3*x) - 2*b**(3/2)*sqrt(a/(b*x) + 1)/(3*a))

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Giac [A]  time = 1.33986, size = 107, normalized size = 2.02 \begin{align*} \frac{{\left (b x + a\right )}^{\frac{3}{2}} b{\left (\frac{{\left (5 \, B a b^{4} - 2 \, A b^{5}\right )}{\left (b x + a\right )}}{a^{3} b^{9}} - \frac{5 \,{\left (B a^{2} b^{4} - A a b^{5}\right )}}{a^{3} b^{9}}\right )}}{960 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{5}{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^(7/2),x, algorithm="giac")

[Out]

1/960*(b*x + a)^(3/2)*b*((5*B*a*b^4 - 2*A*b^5)*(b*x + a)/(a^3*b^9) - 5*(B*a^2*b^4 - A*a*b^5)/(a^3*b^9))/(((b*x
+ a)*b - a*b)^(5/2)*abs(b))