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

Optimal. Leaf size=170 \[ \frac{128 a^3 \sqrt{a x+b \sqrt{x}}}{231 b^4 x^{3/2}}-\frac{320 a^2 \sqrt{a x+b \sqrt{x}}}{693 b^3 x^2}+\frac{1024 a^5 \sqrt{a x+b \sqrt{x}}}{693 b^6 \sqrt{x}}-\frac{512 a^4 \sqrt{a x+b \sqrt{x}}}{693 b^5 x}+\frac{40 a \sqrt{a x+b \sqrt{x}}}{99 b^2 x^{5/2}}-\frac{4 \sqrt{a x+b \sqrt{x}}}{11 b x^3} \]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(11*b*x^3) + (40*a*Sqrt[b*Sqrt[x] + a*x])/(99*b^2*x^(5/2)) - (320*a^2*Sqrt[b*Sqrt[x
] + a*x])/(693*b^3*x^2) + (128*a^3*Sqrt[b*Sqrt[x] + a*x])/(231*b^4*x^(3/2)) - (512*a^4*Sqrt[b*Sqrt[x] + a*x])/
(693*b^5*x) + (1024*a^5*Sqrt[b*Sqrt[x] + a*x])/(693*b^6*Sqrt[x])

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Rubi [A]  time = 0.244644, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2016, 2014} \[ \frac{128 a^3 \sqrt{a x+b \sqrt{x}}}{231 b^4 x^{3/2}}-\frac{320 a^2 \sqrt{a x+b \sqrt{x}}}{693 b^3 x^2}+\frac{1024 a^5 \sqrt{a x+b \sqrt{x}}}{693 b^6 \sqrt{x}}-\frac{512 a^4 \sqrt{a x+b \sqrt{x}}}{693 b^5 x}+\frac{40 a \sqrt{a x+b \sqrt{x}}}{99 b^2 x^{5/2}}-\frac{4 \sqrt{a x+b \sqrt{x}}}{11 b x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(11*b*x^3) + (40*a*Sqrt[b*Sqrt[x] + a*x])/(99*b^2*x^(5/2)) - (320*a^2*Sqrt[b*Sqrt[x
] + a*x])/(693*b^3*x^2) + (128*a^3*Sqrt[b*Sqrt[x] + a*x])/(231*b^4*x^(3/2)) - (512*a^4*Sqrt[b*Sqrt[x] + a*x])/
(693*b^5*x) + (1024*a^5*Sqrt[b*Sqrt[x] + a*x])/(693*b^6*Sqrt[x])

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])

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])

Rubi steps

\begin{align*} \int \frac{1}{x^{7/2} \sqrt{b \sqrt{x}+a x}} \, dx &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{11 b x^3}-\frac{(10 a) \int \frac{1}{x^3 \sqrt{b \sqrt{x}+a x}} \, dx}{11 b}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{11 b x^3}+\frac{40 a \sqrt{b \sqrt{x}+a x}}{99 b^2 x^{5/2}}+\frac{\left (80 a^2\right ) \int \frac{1}{x^{5/2} \sqrt{b \sqrt{x}+a x}} \, dx}{99 b^2}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{11 b x^3}+\frac{40 a \sqrt{b \sqrt{x}+a x}}{99 b^2 x^{5/2}}-\frac{320 a^2 \sqrt{b \sqrt{x}+a x}}{693 b^3 x^2}-\frac{\left (160 a^3\right ) \int \frac{1}{x^2 \sqrt{b \sqrt{x}+a x}} \, dx}{231 b^3}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{11 b x^3}+\frac{40 a \sqrt{b \sqrt{x}+a x}}{99 b^2 x^{5/2}}-\frac{320 a^2 \sqrt{b \sqrt{x}+a x}}{693 b^3 x^2}+\frac{128 a^3 \sqrt{b \sqrt{x}+a x}}{231 b^4 x^{3/2}}+\frac{\left (128 a^4\right ) \int \frac{1}{x^{3/2} \sqrt{b \sqrt{x}+a x}} \, dx}{231 b^4}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{11 b x^3}+\frac{40 a \sqrt{b \sqrt{x}+a x}}{99 b^2 x^{5/2}}-\frac{320 a^2 \sqrt{b \sqrt{x}+a x}}{693 b^3 x^2}+\frac{128 a^3 \sqrt{b \sqrt{x}+a x}}{231 b^4 x^{3/2}}-\frac{512 a^4 \sqrt{b \sqrt{x}+a x}}{693 b^5 x}-\frac{\left (256 a^5\right ) \int \frac{1}{x \sqrt{b \sqrt{x}+a x}} \, dx}{693 b^5}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{11 b x^3}+\frac{40 a \sqrt{b \sqrt{x}+a x}}{99 b^2 x^{5/2}}-\frac{320 a^2 \sqrt{b \sqrt{x}+a x}}{693 b^3 x^2}+\frac{128 a^3 \sqrt{b \sqrt{x}+a x}}{231 b^4 x^{3/2}}-\frac{512 a^4 \sqrt{b \sqrt{x}+a x}}{693 b^5 x}+\frac{1024 a^5 \sqrt{b \sqrt{x}+a x}}{693 b^6 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0548261, size = 83, normalized size = 0.49 \[ \frac{4 \sqrt{a x+b \sqrt{x}} \left (96 a^3 b^2 x^{3/2}-80 a^2 b^3 x-128 a^4 b x^2+256 a^5 x^{5/2}+70 a b^4 \sqrt{x}-63 b^5\right )}{693 b^6 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[b*Sqrt[x] + a*x]*(-63*b^5 + 70*a*b^4*Sqrt[x] - 80*a^2*b^3*x + 96*a^3*b^2*x^(3/2) - 128*a^4*b*x^2 + 256
*a^5*x^(5/2)))/(693*b^6*x^3)

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Maple [C]  time = 0.01, size = 284, normalized size = 1.7 \begin{align*}{\frac{1}{693\,{b}^{7}}\sqrt{b\sqrt{x}+ax} \left ( 2772\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{11/2}{x}^{11/2}-1386\,\sqrt{b\sqrt{x}+ax}{a}^{13/2}{x}^{13/2}-693\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ){x}^{13/2}{a}^{6}b-1386\,{a}^{13/2}{x}^{13/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }+693\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ){x}^{13/2}{a}^{6}b+1236\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{7/2}{x}^{9/2}{b}^{2}+532\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{3/2}{x}^{7/2}{b}^{4}-1748\,{a}^{9/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}b{x}^{5}-852\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{5/2}{x}^{4}{b}^{3}-252\, \left ( b\sqrt{x}+ax \right ) ^{3/2}\sqrt{a}{x}^{3}{b}^{5} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{\frac{1}{\sqrt{a}}}{x}^{-{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/693*(b*x^(1/2)+a*x)^(1/2)*(2772*(b*x^(1/2)+a*x)^(3/2)*a^(11/2)*x^(11/2)-1386*(b*x^(1/2)+a*x)^(1/2)*a^(13/2)*
x^(13/2)-693*ln(1/2*(2*a*x^(1/2)+2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+b)/a^(1/2))*x^(13/2)*a^6*b-1386*a^(13/2)*x^(1
3/2)*(x^(1/2)*(b+a*x^(1/2)))^(1/2)+693*ln(1/2*(2*(x^(1/2)*(b+a*x^(1/2)))^(1/2)*a^(1/2)+2*a*x^(1/2)+b)/a^(1/2))
*x^(13/2)*a^6*b+1236*(b*x^(1/2)+a*x)^(3/2)*a^(7/2)*x^(9/2)*b^2+532*(b*x^(1/2)+a*x)^(3/2)*a^(3/2)*x^(7/2)*b^4-1
748*a^(9/2)*(b*x^(1/2)+a*x)^(3/2)*b*x^5-852*(b*x^(1/2)+a*x)^(3/2)*a^(5/2)*x^4*b^3-252*(b*x^(1/2)+a*x)^(3/2)*a^
(1/2)*x^3*b^5)/(x^(1/2)*(b+a*x^(1/2)))^(1/2)/b^7/a^(1/2)/x^(13/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a x + b \sqrt{x}} x^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.13488, size = 178, normalized size = 1.05 \begin{align*} -\frac{4 \,{\left (128 \, a^{4} b x^{2} + 80 \, a^{2} b^{3} x + 63 \, b^{5} - 2 \,{\left (128 \, a^{5} x^{2} + 48 \, a^{3} b^{2} x + 35 \, a b^{4}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{693 \, b^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/693*(128*a^4*b*x^2 + 80*a^2*b^3*x + 63*b^5 - 2*(128*a^5*x^2 + 48*a^3*b^2*x + 35*a*b^4)*sqrt(x))*sqrt(a*x +
b*sqrt(x))/(b^6*x^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.18865, size = 239, normalized size = 1.41 \begin{align*} \frac{4 \,{\left (3696 \, a^{\frac{5}{2}}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{5} + 7920 \, a^{2} b{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{4} + 6930 \, a^{\frac{3}{2}} b^{2}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{3} + 3080 \, a b^{3}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{2} + 693 \, \sqrt{a} b^{4}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} + 63 \, b^{5}\right )}}{693 \,{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/693*(3696*a^(5/2)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^5 + 7920*a^2*b*(sqrt(a)*sqrt(x) - sqrt(a*x + b*s
qrt(x)))^4 + 6930*a^(3/2)*b^2*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^3 + 3080*a*b^3*(sqrt(a)*sqrt(x) - sqrt
(a*x + b*sqrt(x)))^2 + 693*sqrt(a)*b^4*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) + 63*b^5)/(sqrt(a)*sqrt(x) -
sqrt(a*x + b*sqrt(x)))^11