3.1.79 \(\int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx\)

Optimal. Leaf size=112 \[ \frac {64 a^3 \sqrt {a x+b \sqrt {x}}}{35 b^4 \sqrt {x}}-\frac {32 a^2 \sqrt {a x+b \sqrt {x}}}{35 b^3 x}+\frac {24 a \sqrt {a x+b \sqrt {x}}}{35 b^2 x^{3/2}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2} \]

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Rubi [A]  time = 0.15, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2016, 2014} \begin {gather*} \frac {64 a^3 \sqrt {a x+b \sqrt {x}}}{35 b^4 \sqrt {x}}-\frac {32 a^2 \sqrt {a x+b \sqrt {x}}}{35 b^3 x}+\frac {24 a \sqrt {a x+b \sqrt {x}}}{35 b^2 x^{3/2}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*Sqrt[b*Sqrt[x] + a*x]),x]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(7*b*x^2) + (24*a*Sqrt[b*Sqrt[x] + a*x])/(35*b^2*x^(3/2)) - (32*a^2*Sqrt[b*Sqrt[x]
+ a*x])/(35*b^3*x) + (64*a^3*Sqrt[b*Sqrt[x] + a*x])/(35*b^4*Sqrt[x])

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps

\begin {align*} \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{7 b x^2}-\frac {(6 a) \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx}{7 b}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{7 b x^2}+\frac {24 a \sqrt {b \sqrt {x}+a x}}{35 b^2 x^{3/2}}+\frac {\left (24 a^2\right ) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{35 b^2}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{7 b x^2}+\frac {24 a \sqrt {b \sqrt {x}+a x}}{35 b^2 x^{3/2}}-\frac {32 a^2 \sqrt {b \sqrt {x}+a x}}{35 b^3 x}-\frac {\left (16 a^3\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{35 b^3}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{7 b x^2}+\frac {24 a \sqrt {b \sqrt {x}+a x}}{35 b^2 x^{3/2}}-\frac {32 a^2 \sqrt {b \sqrt {x}+a x}}{35 b^3 x}+\frac {64 a^3 \sqrt {b \sqrt {x}+a x}}{35 b^4 \sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 59, normalized size = 0.53 \begin {gather*} \frac {4 \sqrt {a x+b \sqrt {x}} \left (16 a^3 x^{3/2}-8 a^2 b x+6 a b^2 \sqrt {x}-5 b^3\right )}{35 b^4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*Sqrt[b*Sqrt[x] + a*x]),x]

[Out]

(4*Sqrt[b*Sqrt[x] + a*x]*(-5*b^3 + 6*a*b^2*Sqrt[x] - 8*a^2*b*x + 16*a^3*x^(3/2)))/(35*b^4*x^2)

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IntegrateAlgebraic [A]  time = 0.18, size = 59, normalized size = 0.53 \begin {gather*} \frac {4 \sqrt {a x+b \sqrt {x}} \left (16 a^3 x^{3/2}-8 a^2 b x+6 a b^2 \sqrt {x}-5 b^3\right )}{35 b^4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^(5/2)*Sqrt[b*Sqrt[x] + a*x]),x]

[Out]

(4*Sqrt[b*Sqrt[x] + a*x]*(-5*b^3 + 6*a*b^2*Sqrt[x] - 8*a^2*b*x + 16*a^3*x^(3/2)))/(35*b^4*x^2)

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fricas [A]  time = 0.85, size = 50, normalized size = 0.45 \begin {gather*} -\frac {4 \, {\left (8 \, a^{2} b x + 5 \, b^{3} - 2 \, {\left (8 \, a^{3} x + 3 \, a b^{2}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{35 \, b^{4} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.21, size = 115, normalized size = 1.03 \begin {gather*} \frac {4 \, {\left (70 \, a^{\frac {3}{2}} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{3} + 84 \, a b {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{2} + 35 \, \sqrt {a} b^{2} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + 5 \, b^{3}\right )}}{35 \, {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="giac")

[Out]

4/35*(70*a^(3/2)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^3 + 84*a*b*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))
)^2 + 35*sqrt(a)*b^2*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) + 5*b^3)/(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x
)))^7

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maple [C]  time = 0.06, size = 240, normalized size = 2.14 \begin {gather*} \frac {\sqrt {a x +b \sqrt {x}}\, \left (35 a^{4} b \,x^{\frac {9}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-35 a^{4} b \,x^{\frac {9}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-70 \sqrt {a x +b \sqrt {x}}\, a^{\frac {9}{2}} x^{\frac {9}{2}}-70 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {9}{2}} x^{\frac {9}{2}}+140 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} x^{\frac {7}{2}}-76 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b \,x^{3}+44 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{2} x^{\frac {5}{2}}-20 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{3} x^{2}\right )}{35 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\, b^{5} x^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(a*x+b*x^(1/2))^(1/2),x)

[Out]

1/35*(a*x+b*x^(1/2))^(1/2)*(140*(a*x+b*x^(1/2))^(3/2)*a^(7/2)*x^(7/2)-70*(a*x+b*x^(1/2))^(1/2)*a^(9/2)*x^(9/2)
+35*ln(1/2*(2*a*x^(1/2)+b+2*((a*x^(1/2)+b)*x^(1/2))^(1/2)*a^(1/2))/a^(1/2))*x^(9/2)*a^4*b-70*a^(9/2)*x^(9/2)*(
(a*x^(1/2)+b)*x^(1/2))^(1/2)-35*ln(1/2*(2*a*x^(1/2)+b+2*(a*x+b*x^(1/2))^(1/2)*a^(1/2))/a^(1/2))*x^(9/2)*a^4*b+
44*(a*x+b*x^(1/2))^(3/2)*a^(3/2)*x^(5/2)*b^2-76*a^(5/2)*(a*x+b*x^(1/2))^(3/2)*b*x^3-20*(a*x+b*x^(1/2))^(3/2)*a
^(1/2)*x^2*b^3)/((a*x^(1/2)+b)*x^(1/2))^(1/2)/b^5/x^(9/2)/a^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + b \sqrt {x}} x^{\frac {5}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(a*x + b*sqrt(x))*x^(5/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^{5/2}\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a*x + b*x^(1/2))^(1/2)),x)

[Out]

int(1/(x^(5/2)*(a*x + b*x^(1/2))^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{\frac {5}{2}} \sqrt {a x + b \sqrt {x}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(5/2)/(b*x**(1/2)+a*x)**(1/2),x)

[Out]

Integral(1/(x**(5/2)*sqrt(a*x + b*sqrt(x))), x)

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