3.116 \(\int \frac{1}{x^3 (b \sqrt{x}+a x)^{3/2}} \, dx\)

Optimal. Leaf size=195 \[ \frac{512 a^3 \sqrt{a x+b \sqrt{x}}}{77 b^5 x^{3/2}}-\frac{1280 a^2 \sqrt{a x+b \sqrt{x}}}{231 b^4 x^2}+\frac{4096 a^5 \sqrt{a x+b \sqrt{x}}}{231 b^7 \sqrt{x}}-\frac{2048 a^4 \sqrt{a x+b \sqrt{x}}}{231 b^6 x}+\frac{160 a \sqrt{a x+b \sqrt{x}}}{33 b^3 x^{5/2}}-\frac{48 \sqrt{a x+b \sqrt{x}}}{11 b^2 x^3}+\frac{4}{b x^{5/2} \sqrt{a x+b \sqrt{x}}} \]

[Out]

4/(b*x^(5/2)*Sqrt[b*Sqrt[x] + a*x]) - (48*Sqrt[b*Sqrt[x] + a*x])/(11*b^2*x^3) + (160*a*Sqrt[b*Sqrt[x] + a*x])/
(33*b^3*x^(5/2)) - (1280*a^2*Sqrt[b*Sqrt[x] + a*x])/(231*b^4*x^2) + (512*a^3*Sqrt[b*Sqrt[x] + a*x])/(77*b^5*x^
(3/2)) - (2048*a^4*Sqrt[b*Sqrt[x] + a*x])/(231*b^6*x) + (4096*a^5*Sqrt[b*Sqrt[x] + a*x])/(231*b^7*Sqrt[x])

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Rubi [A]  time = 0.299999, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2015, 2016, 2014} \[ \frac{512 a^3 \sqrt{a x+b \sqrt{x}}}{77 b^5 x^{3/2}}-\frac{1280 a^2 \sqrt{a x+b \sqrt{x}}}{231 b^4 x^2}+\frac{4096 a^5 \sqrt{a x+b \sqrt{x}}}{231 b^7 \sqrt{x}}-\frac{2048 a^4 \sqrt{a x+b \sqrt{x}}}{231 b^6 x}+\frac{160 a \sqrt{a x+b \sqrt{x}}}{33 b^3 x^{5/2}}-\frac{48 \sqrt{a x+b \sqrt{x}}}{11 b^2 x^3}+\frac{4}{b x^{5/2} \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(b*x^(5/2)*Sqrt[b*Sqrt[x] + a*x]) - (48*Sqrt[b*Sqrt[x] + a*x])/(11*b^2*x^3) + (160*a*Sqrt[b*Sqrt[x] + a*x])/
(33*b^3*x^(5/2)) - (1280*a^2*Sqrt[b*Sqrt[x] + a*x])/(231*b^4*x^2) + (512*a^3*Sqrt[b*Sqrt[x] + a*x])/(77*b^5*x^
(3/2)) - (2048*a^4*Sqrt[b*Sqrt[x] + a*x])/(231*b^6*x) + (4096*a^5*Sqrt[b*Sqrt[x] + a*x])/(231*b^7*Sqrt[x])

Rule 2015

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 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^3 \left (b \sqrt{x}+a x\right )^{3/2}} \, dx &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}+\frac{12 \int \frac{1}{x^{7/2} \sqrt{b \sqrt{x}+a x}} \, dx}{b}\\ &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}-\frac{48 \sqrt{b \sqrt{x}+a x}}{11 b^2 x^3}-\frac{(120 a) \int \frac{1}{x^3 \sqrt{b \sqrt{x}+a x}} \, dx}{11 b^2}\\ &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}-\frac{48 \sqrt{b \sqrt{x}+a x}}{11 b^2 x^3}+\frac{160 a \sqrt{b \sqrt{x}+a x}}{33 b^3 x^{5/2}}+\frac{\left (320 a^2\right ) \int \frac{1}{x^{5/2} \sqrt{b \sqrt{x}+a x}} \, dx}{33 b^3}\\ &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}-\frac{48 \sqrt{b \sqrt{x}+a x}}{11 b^2 x^3}+\frac{160 a \sqrt{b \sqrt{x}+a x}}{33 b^3 x^{5/2}}-\frac{1280 a^2 \sqrt{b \sqrt{x}+a x}}{231 b^4 x^2}-\frac{\left (640 a^3\right ) \int \frac{1}{x^2 \sqrt{b \sqrt{x}+a x}} \, dx}{77 b^4}\\ &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}-\frac{48 \sqrt{b \sqrt{x}+a x}}{11 b^2 x^3}+\frac{160 a \sqrt{b \sqrt{x}+a x}}{33 b^3 x^{5/2}}-\frac{1280 a^2 \sqrt{b \sqrt{x}+a x}}{231 b^4 x^2}+\frac{512 a^3 \sqrt{b \sqrt{x}+a x}}{77 b^5 x^{3/2}}+\frac{\left (512 a^4\right ) \int \frac{1}{x^{3/2} \sqrt{b \sqrt{x}+a x}} \, dx}{77 b^5}\\ &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}-\frac{48 \sqrt{b \sqrt{x}+a x}}{11 b^2 x^3}+\frac{160 a \sqrt{b \sqrt{x}+a x}}{33 b^3 x^{5/2}}-\frac{1280 a^2 \sqrt{b \sqrt{x}+a x}}{231 b^4 x^2}+\frac{512 a^3 \sqrt{b \sqrt{x}+a x}}{77 b^5 x^{3/2}}-\frac{2048 a^4 \sqrt{b \sqrt{x}+a x}}{231 b^6 x}-\frac{\left (1024 a^5\right ) \int \frac{1}{x \sqrt{b \sqrt{x}+a x}} \, dx}{231 b^6}\\ &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}-\frac{48 \sqrt{b \sqrt{x}+a x}}{11 b^2 x^3}+\frac{160 a \sqrt{b \sqrt{x}+a x}}{33 b^3 x^{5/2}}-\frac{1280 a^2 \sqrt{b \sqrt{x}+a x}}{231 b^4 x^2}+\frac{512 a^3 \sqrt{b \sqrt{x}+a x}}{77 b^5 x^{3/2}}-\frac{2048 a^4 \sqrt{b \sqrt{x}+a x}}{231 b^6 x}+\frac{4096 a^5 \sqrt{b \sqrt{x}+a x}}{231 b^7 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0627457, size = 96, normalized size = 0.49 \[ \frac{4 \left (-128 a^4 b^2 x^2+64 a^3 b^3 x^{3/2}-40 a^2 b^4 x+512 a^5 b x^{5/2}+1024 a^6 x^3+28 a b^5 \sqrt{x}-21 b^6\right )}{231 b^7 x^{5/2} \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-21*b^6 + 28*a*b^5*Sqrt[x] - 40*a^2*b^4*x + 64*a^3*b^3*x^(3/2) - 128*a^4*b^2*x^2 + 512*a^5*b*x^(5/2) + 102
4*a^6*x^3))/(231*b^7*x^(5/2)*Sqrt[b*Sqrt[x] + a*x])

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Maple [C]  time = 0.012, size = 614, normalized size = 3.2 \begin{align*}{\frac{1}{231\,{b}^{8}}\sqrt{b\sqrt{x}+ax} \left ( 1155\,\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}^{15/2}{a}^{8}b-2310\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ){x}^{7}{a}^{7}{b}^{2}-924\,{a}^{15/2}{x}^{13/2} \left ( \sqrt{x} \left ( b+a\sqrt{x} \right ) \right ) ^{3/2}+2310\,\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}^{7}{a}^{7}{b}^{2}+256\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{7/2}{x}^{9/2}{b}^{4}-1155\,\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}^{3}+1155\,\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}^{3}-160\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{5/2}{x}^{4}{b}^{5}+112\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{3/2}{x}^{7/2}{b}^{6}-84\, \left ( b\sqrt{x}+ax \right ) ^{3/2}\sqrt{a}{x}^{3}{b}^{7}-512\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{9/2}{x}^{5}{b}^{3}+2048\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{11/2}{x}^{11/2}{b}^{2}-4620\,{a}^{15/2}{x}^{7}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }b+8716\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{13/2}{x}^{6}b-4620\,\sqrt{b\sqrt{x}+ax}{a}^{15/2}{x}^{7}b-2310\,\sqrt{b\sqrt{x}+ax}{a}^{13/2}{x}^{13/2}{b}^{2}-2310\,{a}^{13/2}{x}^{13/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }{b}^{2}+5544\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{15/2}{x}^{13/2}-2310\,\sqrt{b\sqrt{x}+ax}{a}^{17/2}{x}^{15/2}-2310\,{a}^{17/2}{x}^{15/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }-1155\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ){x}^{15/2}{a}^{8}b \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{\frac{1}{\sqrt{a}}}{x}^{-{\frac{13}{2}}} \left ( b+a\sqrt{x} \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/231*(b*x^(1/2)+a*x)^(1/2)*(1155*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^(1
5/2)*a^8*b-2310*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^7*a^7*b^2-924*a^(15/2)*x^(13
/2)*(x^(1/2)*(b+a*x^(1/2)))^(3/2)+2310*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^7*a^7*b^2+256*(b*x^(1/2)+a*x)^(3/2)*a^(7/2)*x^(9/2)*b^4-1155*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^3+1155*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^3-160*(b*x^(1/2)+a*x)^(3/2)*a^(5/2)*x^4*b^5+112*(b*x^(1/2)+a*x)^(3/2)*a^(3/2)*x^(7/2)*b^6-84*
(b*x^(1/2)+a*x)^(3/2)*a^(1/2)*x^3*b^7-512*(b*x^(1/2)+a*x)^(3/2)*a^(9/2)*x^5*b^3+2048*(b*x^(1/2)+a*x)^(3/2)*a^(
11/2)*x^(11/2)*b^2-4620*a^(15/2)*x^7*(x^(1/2)*(b+a*x^(1/2)))^(1/2)*b+8716*(b*x^(1/2)+a*x)^(3/2)*a^(13/2)*x^6*b
-4620*(b*x^(1/2)+a*x)^(1/2)*a^(15/2)*x^7*b-2310*(b*x^(1/2)+a*x)^(1/2)*a^(13/2)*x^(13/2)*b^2-2310*a^(13/2)*x^(1
3/2)*(x^(1/2)*(b+a*x^(1/2)))^(1/2)*b^2+5544*(b*x^(1/2)+a*x)^(3/2)*a^(15/2)*x^(13/2)-2310*(b*x^(1/2)+a*x)^(1/2)
*a^(17/2)*x^(15/2)-2310*a^(17/2)*x^(15/2)*(x^(1/2)*(b+a*x^(1/2)))^(1/2)-1155*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^(15/2)*a^8*b)/(x^(1/2)*(b+a*x^(1/2)))^(1/2)/b^8/a^(1/2)/x^(13/2)/(b+a*x^(1/2)
)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.89979, size = 246, normalized size = 1.26 \begin{align*} -\frac{4 \,{\left (512 \, a^{6} b x^{3} - 192 \, a^{4} b^{3} x^{2} - 68 \, a^{2} b^{5} x - 21 \, b^{7} -{\left (1024 \, a^{7} x^{3} - 640 \, a^{5} b^{2} x^{2} - 104 \, a^{3} b^{4} x - 49 \, a b^{6}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{231 \,{\left (a^{2} b^{7} x^{4} - b^{9} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/231*(512*a^6*b*x^3 - 192*a^4*b^3*x^2 - 68*a^2*b^5*x - 21*b^7 - (1024*a^7*x^3 - 640*a^5*b^2*x^2 - 104*a^3*b^
4*x - 49*a*b^6)*sqrt(x))*sqrt(a*x + b*sqrt(x))/(a^2*b^7*x^4 - b^9*x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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