3.575 \(\int \frac{1}{x^{7/2} \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{16 b^2 \sqrt{a+b x}}{15 a^3 \sqrt{x}}+\frac{8 b \sqrt{a+b x}}{15 a^2 x^{3/2}}-\frac{2 \sqrt{a+b x}}{5 a x^{5/2}} \]

[Out]

(-2*Sqrt[a + b*x])/(5*a*x^(5/2)) + (8*b*Sqrt[a + b*x])/(15*a^2*x^(3/2)) - (16*b^2*Sqrt[a + b*x])/(15*a^3*Sqrt[
x])

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Rubi [A]  time = 0.0102229, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac{16 b^2 \sqrt{a+b x}}{15 a^3 \sqrt{x}}+\frac{8 b \sqrt{a+b x}}{15 a^2 x^{3/2}}-\frac{2 \sqrt{a+b x}}{5 a x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(7/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x])/(5*a*x^(5/2)) + (8*b*Sqrt[a + b*x])/(15*a^2*x^(3/2)) - (16*b^2*Sqrt[a + b*x])/(15*a^3*Sqrt[
x])

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{1}{x^{7/2} \sqrt{a+b x}} \, dx &=-\frac{2 \sqrt{a+b x}}{5 a x^{5/2}}-\frac{(4 b) \int \frac{1}{x^{5/2} \sqrt{a+b x}} \, dx}{5 a}\\ &=-\frac{2 \sqrt{a+b x}}{5 a x^{5/2}}+\frac{8 b \sqrt{a+b x}}{15 a^2 x^{3/2}}+\frac{\left (8 b^2\right ) \int \frac{1}{x^{3/2} \sqrt{a+b x}} \, dx}{15 a^2}\\ &=-\frac{2 \sqrt{a+b x}}{5 a x^{5/2}}+\frac{8 b \sqrt{a+b x}}{15 a^2 x^{3/2}}-\frac{16 b^2 \sqrt{a+b x}}{15 a^3 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.008939, size = 40, normalized size = 0.59 \[ -\frac{2 \sqrt{a+b x} \left (3 a^2-4 a b x+8 b^2 x^2\right )}{15 a^3 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(7/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x]*(3*a^2 - 4*a*b*x + 8*b^2*x^2))/(15*a^3*x^(5/2))

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Maple [A]  time = 0.004, size = 35, normalized size = 0.5 \begin{align*} -{\frac{16\,{b}^{2}{x}^{2}-8\,abx+6\,{a}^{2}}{15\,{a}^{3}}\sqrt{bx+a}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(b*x+a)^(1/2)*(8*b^2*x^2-4*a*b*x+3*a^2)/x^(5/2)/a^3

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Maxima [A]  time = 1.14235, size = 62, normalized size = 0.91 \begin{align*} -\frac{2 \,{\left (\frac{15 \, \sqrt{b x + a} b^{2}}{\sqrt{x}} - \frac{10 \,{\left (b x + a\right )}^{\frac{3}{2}} b}{x^{\frac{3}{2}}} + \frac{3 \,{\left (b x + a\right )}^{\frac{5}{2}}}{x^{\frac{5}{2}}}\right )}}{15 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(15*sqrt(b*x + a)*b^2/sqrt(x) - 10*(b*x + a)^(3/2)*b/x^(3/2) + 3*(b*x + a)^(5/2)/x^(5/2))/a^3

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Fricas [A]  time = 1.81237, size = 88, normalized size = 1.29 \begin{align*} -\frac{2 \,{\left (8 \, b^{2} x^{2} - 4 \, a b x + 3 \, a^{2}\right )} \sqrt{b x + a}}{15 \, a^{3} x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(8*b^2*x^2 - 4*a*b*x + 3*a^2)*sqrt(b*x + a)/(a^3*x^(5/2))

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Sympy [B]  time = 30.2744, size = 287, normalized size = 4.22 \begin{align*} - \frac{6 a^{4} b^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac{4 a^{3} b^{\frac{11}{2}} x \sqrt{\frac{a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac{6 a^{2} b^{\frac{13}{2}} x^{2} \sqrt{\frac{a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac{24 a b^{\frac{15}{2}} x^{3} \sqrt{\frac{a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac{16 b^{\frac{17}{2}} x^{4} \sqrt{\frac{a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*a**4*b**(9/2)*sqrt(a/(b*x) + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 4*a**3*b**(11
/2)*x*sqrt(a/(b*x) + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 6*a**2*b**(13/2)*x**2*sq
rt(a/(b*x) + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 24*a*b**(15/2)*x**3*sqrt(a/(b*x)
+ 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 16*b**(17/2)*x**4*sqrt(a/(b*x) + 1)/(15*a*
*5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4)

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Giac [A]  time = 1.08107, size = 89, normalized size = 1.31 \begin{align*} \frac{\sqrt{b x + a}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (b x + a\right )}}{a^{3} b^{4}} - \frac{5}{a^{2} b^{4}}\right )} + \frac{15}{a b^{4}}\right )} b}{480 \,{\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(1/x^(7/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/480*sqrt(b*x + a)*(4*(b*x + a)*(2*(b*x + a)/(a^3*b^4) - 5/(a^2*b^4)) + 15/(a*b^4))*b/(((b*x + a)*b - a*b)^(5
/2)*abs(b))