3.1.65 \(\int \frac {1}{x^3 \sqrt {b \sqrt {x}+a x}} \, dx\)

Optimal. Leaf size=142 \[ -\frac {512 a^4 \sqrt {a x+b \sqrt {x}}}{315 b^5 \sqrt {x}}+\frac {256 a^3 \sqrt {a x+b \sqrt {x}}}{315 b^4 x}-\frac {64 a^2 \sqrt {a x+b \sqrt {x}}}{105 b^3 x^{3/2}}+\frac {32 a \sqrt {a x+b \sqrt {x}}}{63 b^2 x^2}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}} \]

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Rubi [A]  time = 0.20, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2016, 2014} \begin {gather*} -\frac {64 a^2 \sqrt {a x+b \sqrt {x}}}{105 b^3 x^{3/2}}-\frac {512 a^4 \sqrt {a x+b \sqrt {x}}}{315 b^5 \sqrt {x}}+\frac {256 a^3 \sqrt {a x+b \sqrt {x}}}{315 b^4 x}+\frac {32 a \sqrt {a x+b \sqrt {x}}}{63 b^2 x^2}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[b*Sqrt[x] + a*x]),x]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(9*b*x^(5/2)) + (32*a*Sqrt[b*Sqrt[x] + a*x])/(63*b^2*x^2) - (64*a^2*Sqrt[b*Sqrt[x]
+ a*x])/(105*b^3*x^(3/2)) + (256*a^3*Sqrt[b*Sqrt[x] + a*x])/(315*b^4*x) - (512*a^4*Sqrt[b*Sqrt[x] + a*x])/(315
*b^5*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^3 \sqrt {b \sqrt {x}+a x}} \, dx &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{9 b x^{5/2}}-\frac {(8 a) \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx}{9 b}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{9 b x^{5/2}}+\frac {32 a \sqrt {b \sqrt {x}+a x}}{63 b^2 x^2}+\frac {\left (16 a^2\right ) \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx}{21 b^2}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{9 b x^{5/2}}+\frac {32 a \sqrt {b \sqrt {x}+a x}}{63 b^2 x^2}-\frac {64 a^2 \sqrt {b \sqrt {x}+a x}}{105 b^3 x^{3/2}}-\frac {\left (64 a^3\right ) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{105 b^3}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{9 b x^{5/2}}+\frac {32 a \sqrt {b \sqrt {x}+a x}}{63 b^2 x^2}-\frac {64 a^2 \sqrt {b \sqrt {x}+a x}}{105 b^3 x^{3/2}}+\frac {256 a^3 \sqrt {b \sqrt {x}+a x}}{315 b^4 x}+\frac {\left (128 a^4\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{315 b^4}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{9 b x^{5/2}}+\frac {32 a \sqrt {b \sqrt {x}+a x}}{63 b^2 x^2}-\frac {64 a^2 \sqrt {b \sqrt {x}+a x}}{105 b^3 x^{3/2}}+\frac {256 a^3 \sqrt {b \sqrt {x}+a x}}{315 b^4 x}-\frac {512 a^4 \sqrt {b \sqrt {x}+a x}}{315 b^5 \sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 72, normalized size = 0.51 \begin {gather*} -\frac {4 \sqrt {a x+b \sqrt {x}} \left (128 a^4 x^2-64 a^3 b x^{3/2}+48 a^2 b^2 x-40 a b^3 \sqrt {x}+35 b^4\right )}{315 b^5 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x]*(35*b^4 - 40*a*b^3*Sqrt[x] + 48*a^2*b^2*x - 64*a^3*b*x^(3/2) + 128*a^4*x^2))/(315*b^
5*x^(5/2))

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IntegrateAlgebraic [A]  time = 0.20, size = 72, normalized size = 0.51 \begin {gather*} -\frac {4 \sqrt {a x+b \sqrt {x}} \left (128 a^4 x^2-64 a^3 b x^{3/2}+48 a^2 b^2 x-40 a b^3 \sqrt {x}+35 b^4\right )}{315 b^5 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x]*(35*b^4 - 40*a*b^3*Sqrt[x] + 48*a^2*b^2*x - 64*a^3*b*x^(3/2) + 128*a^4*x^2))/(315*b^
5*x^(5/2))

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fricas [A]  time = 0.70, size = 64, normalized size = 0.45 \begin {gather*} \frac {4 \, {\left (64 \, a^{3} b x^{2} + 40 \, a b^{3} x - {\left (128 \, a^{4} x^{2} + 48 \, a^{2} b^{2} x + 35 \, b^{4}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{315 \, b^{5} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/315*(64*a^3*b*x^2 + 40*a*b^3*x - (128*a^4*x^2 + 48*a^2*b^2*x + 35*b^4)*sqrt(x))*sqrt(a*x + b*sqrt(x))/(b^5*x
^3)

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giac [A]  time = 0.19, size = 146, normalized size = 1.03 \begin {gather*} \frac {4 \, {\left (1008 \, a^{2} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{4} + 1680 \, a^{\frac {3}{2}} b {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{3} + 1080 \, a b^{2} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{2} + 315 \, \sqrt {a} b^{3} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + 35 \, b^{4}\right )}}{315 \, {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/315*(1008*a^2*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^4 + 1680*a^(3/2)*b*(sqrt(a)*sqrt(x) - sqrt(a*x + b*s
qrt(x)))^3 + 1080*a*b^2*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^2 + 315*sqrt(a)*b^3*(sqrt(a)*sqrt(x) - sqrt(
a*x + b*sqrt(x))) + 35*b^4)/(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^9

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maple [C]  time = 0.06, size = 262, normalized size = 1.85 \begin {gather*} -\frac {\sqrt {a x +b \sqrt {x}}\, \left (315 a^{5} b \,x^{\frac {11}{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 )-315 a^{5} b \,x^{\frac {11}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-630 \sqrt {a x +b \sqrt {x}}\, a^{\frac {11}{2}} x^{\frac {11}{2}}-630 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {11}{2}} x^{\frac {11}{2}}+1260 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{\frac {9}{2}}-748 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b \,x^{4}+492 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2} x^{\frac {7}{2}}-300 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{3} x^{3}+140 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{4} x^{\frac {5}{2}}\right )}{315 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\, b^{6} x^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(a*x+b*x^(1/2))^(1/2)*(1260*(a*x+b*x^(1/2))^(3/2)*a^(9/2)*x^(9/2)-630*(a*x+b*x^(1/2))^(1/2)*a^(11/2)*x^
(11/2)+315*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^(11/2)*a^5*b-630*a^(11/2)
*x^(11/2)*((a*x^(1/2)+b)*x^(1/2))^(1/2)-315*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^
(11/2)*a^5*b+492*(a*x+b*x^(1/2))^(3/2)*a^(5/2)*x^(7/2)*b^2+140*(a*x+b*x^(1/2))^(3/2)*a^(1/2)*x^(5/2)*b^4-748*a
^(7/2)*(a*x+b*x^(1/2))^(3/2)*b*x^4-300*(a*x+b*x^(1/2))^(3/2)*a^(3/2)*x^3*b^3)/((a*x^(1/2)+b)*x^(1/2))^(1/2)/b^
6/x^(11/2)/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^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^3\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(x^3*(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^{3} \sqrt {a x + b \sqrt {x}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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