3.2.28 \(\int \frac {1}{x^{3/2} (b \sqrt {x}+a x)^{3/2}} \, dx\) [128]

Optimal. Leaf size=107 \[ \frac {4}{b x \sqrt {b \sqrt {x}+a x}}-\frac {24 \sqrt {b \sqrt {x}+a x}}{5 b^2 x^{3/2}}+\frac {32 a \sqrt {b \sqrt {x}+a x}}{5 b^3 x}-\frac {64 a^2 \sqrt {b \sqrt {x}+a x}}{5 b^4 \sqrt {x}} \]

[Out]

4/b/x/(b*x^(1/2)+a*x)^(1/2)-24/5*(b*x^(1/2)+a*x)^(1/2)/b^2/x^(3/2)+32/5*a*(b*x^(1/2)+a*x)^(1/2)/b^3/x-64/5*a^2
*(b*x^(1/2)+a*x)^(1/2)/b^4/x^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2040, 2041, 2039} \begin {gather*} -\frac {64 a^2 \sqrt {a x+b \sqrt {x}}}{5 b^4 \sqrt {x}}+\frac {32 a \sqrt {a x+b \sqrt {x}}}{5 b^3 x}-\frac {24 \sqrt {a x+b \sqrt {x}}}{5 b^2 x^{3/2}}+\frac {4}{b x \sqrt {a x+b \sqrt {x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(b*Sqrt[x] + a*x)^(3/2)),x]

[Out]

4/(b*x*Sqrt[b*Sqrt[x] + a*x]) - (24*Sqrt[b*Sqrt[x] + a*x])/(5*b^2*x^(3/2)) + (32*a*Sqrt[b*Sqrt[x] + a*x])/(5*b
^3*x) - (64*a^2*Sqrt[b*Sqrt[x] + a*x])/(5*b^4*Sqrt[x])

Rule 2039

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] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2040

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] + Dist[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1)))
, Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] &&  !IntegerQ[p] && NeQ[n,
j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && LtQ[p, -1] && (IntegerQ[j] || GtQ[c, 0])

Rule 2041

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

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Mathematica [A]
time = 0.17, size = 70, normalized size = 0.65 \begin {gather*} -\frac {4 \sqrt {b \sqrt {x}+a x} \left (b^3-2 a b^2 \sqrt {x}+8 a^2 b x+16 a^3 x^{3/2}\right )}{5 b^4 \left (b+a \sqrt {x}\right ) x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*(b*Sqrt[x] + a*x)^(3/2)),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.39, size = 548, normalized size = 5.12

method	result	size
derivativedivides	\(-\frac {4}{5 b x \sqrt {b \sqrt {x}+a x}}-\frac {12 a \left (-\frac {2}{3 b \sqrt {x}\, \sqrt {b \sqrt {x}+a x}}+\frac {8 a \left (b +2 a \sqrt {x}\right )}{3 b^{3} \sqrt {b \sqrt {x}+a x}}\right )}{5 b}\)	\(72\)
default	\(\frac {2 \sqrt {b \sqrt {x}+a x}\, \left (10 x^{\frac {9}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {11}{2}}+10 x^{\frac {9}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {11}{2}}-30 x^{\frac {7}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {9}{2}}+10 x^{\frac {7}{2}} \left (\sqrt {x}\, \left (a \sqrt {x}+b \right )\right )^{\frac {3}{2}} a^{\frac {9}{2}}+10 x^{\frac {7}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {7}{2}} b^{2}+10 x^{\frac {7}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {7}{2}} b^{2}-5 x^{\frac {9}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{5} b +5 x^{\frac {9}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{5} b -16 x^{\frac {5}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2}+20 x^{4} \sqrt {b \sqrt {x}+a x}\, a^{\frac {9}{2}} b +20 x^{4} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {9}{2}} b -5 x^{\frac {7}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{3} b^{3}+5 x^{\frac {7}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{3} b^{3}-52 x^{3} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b -2 x^{\frac {3}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} \sqrt {a}\, b^{4}+4 x^{2} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{3}-10 x^{4} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{4} b^{2}+10 x^{4} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{4} b^{2}\right )}{5 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{5} \left (a \sqrt {x}+b \right )^{2} \sqrt {a}\, x^{\frac {7}{2}}}\)	\(548\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(3/2)/(b*x^(1/2)+a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/5*(b*x^(1/2)+a*x)^(1/2)*(10*x^(9/2)*(b*x^(1/2)+a*x)^(1/2)*a^(11/2)+10*x^(9/2)*(x^(1/2)*(a*x^(1/2)+b))^(1/2)*
a^(11/2)-30*x^(7/2)*(b*x^(1/2)+a*x)^(3/2)*a^(9/2)+10*x^(7/2)*(x^(1/2)*(a*x^(1/2)+b))^(3/2)*a^(9/2)+10*x^(7/2)*
(b*x^(1/2)+a*x)^(1/2)*a^(7/2)*b^2+10*x^(7/2)*(x^(1/2)*(a*x^(1/2)+b))^(1/2)*a^(7/2)*b^2-5*x^(9/2)*ln(1/2*(2*a*x
^(1/2)+2*(x^(1/2)*(a*x^(1/2)+b))^(1/2)*a^(1/2)+b)/a^(1/2))*a^5*b+5*x^(9/2)*ln(1/2*(2*a*x^(1/2)+2*(b*x^(1/2)+a*
x)^(1/2)*a^(1/2)+b)/a^(1/2))*a^5*b-16*x^(5/2)*(b*x^(1/2)+a*x)^(3/2)*a^(5/2)*b^2+20*x^4*(b*x^(1/2)+a*x)^(1/2)*a
^(9/2)*b+20*x^4*(x^(1/2)*(a*x^(1/2)+b))^(1/2)*a^(9/2)*b-5*x^(7/2)*ln(1/2*(2*a*x^(1/2)+2*(x^(1/2)*(a*x^(1/2)+b)
)^(1/2)*a^(1/2)+b)/a^(1/2))*a^3*b^3+5*x^(7/2)*ln(1/2*(2*a*x^(1/2)+2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+b)/a^(1/2))*
a^3*b^3-52*x^3*(b*x^(1/2)+a*x)^(3/2)*a^(7/2)*b-2*x^(3/2)*(b*x^(1/2)+a*x)^(3/2)*a^(1/2)*b^4+4*x^2*(b*x^(1/2)+a*
x)^(3/2)*a^(3/2)*b^3-10*x^4*ln(1/2*(2*a*x^(1/2)+2*(x^(1/2)*(a*x^(1/2)+b))^(1/2)*a^(1/2)+b)/a^(1/2))*a^4*b^2+10
*x^4*ln(1/2*(2*a*x^(1/2)+2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+b)/a^(1/2))*a^4*b^2)/(x^(1/2)*(a*x^(1/2)+b))^(1/2)/b^
5/(a*x^(1/2)+b)^2/a^(1/2)/x^(7/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(3/2)), x)

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Fricas [A]
time = 2.80, size = 79, normalized size = 0.74 \begin {gather*} \frac {4 \, {\left (8 \, a^{3} b x^{2} - 3 \, a b^{3} x - {\left (16 \, a^{4} x^{2} - 10 \, a^{2} b^{2} x - b^{4}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{5 \, {\left (a^{2} b^{4} x^{3} - b^{6} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="fricas")

[Out]

4/5*(8*a^3*b*x^2 - 3*a*b^3*x - (16*a^4*x^2 - 10*a^2*b^2*x - b^4)*sqrt(x))*sqrt(a*x + b*sqrt(x))/(a^2*b^4*x^3 -
b^6*x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{\frac {3}{2}} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(3/2)/(b*x**(1/2)+a*x)**(3/2),x)

[Out]

Integral(1/(x**(3/2)*(a*x + b*sqrt(x))**(3/2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(3/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^{3/2}\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(a*x + b*x^(1/2))^(3/2)),x)

[Out]

int(1/(x^(3/2)*(a*x + b*x^(1/2))^(3/2)), x)

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