3.130 \(\int \frac{1}{x^{7/2} (b \sqrt{x}+a x)^{3/2}} \, dx\)

Optimal. Leaf size=223 \[ -\frac{1024 a^4 \sqrt{a x+b \sqrt{x}}}{143 b^6 x^{3/2}}+\frac{2560 a^3 \sqrt{a x+b \sqrt{x}}}{429 b^5 x^2}-\frac{2240 a^2 \sqrt{a x+b \sqrt{x}}}{429 b^4 x^{5/2}}-\frac{8192 a^6 \sqrt{a x+b \sqrt{x}}}{429 b^8 \sqrt{x}}+\frac{4096 a^5 \sqrt{a x+b \sqrt{x}}}{429 b^7 x}+\frac{672 a \sqrt{a x+b \sqrt{x}}}{143 b^3 x^3}-\frac{56 \sqrt{a x+b \sqrt{x}}}{13 b^2 x^{7/2}}+\frac{4}{b x^3 \sqrt{a x+b \sqrt{x}}} \]

[Out]

4/(b*x^3*Sqrt[b*Sqrt[x] + a*x]) - (56*Sqrt[b*Sqrt[x] + a*x])/(13*b^2*x^(7/2)) + (672*a*Sqrt[b*Sqrt[x] + a*x])/
(143*b^3*x^3) - (2240*a^2*Sqrt[b*Sqrt[x] + a*x])/(429*b^4*x^(5/2)) + (2560*a^3*Sqrt[b*Sqrt[x] + a*x])/(429*b^5
*x^2) - (1024*a^4*Sqrt[b*Sqrt[x] + a*x])/(143*b^6*x^(3/2)) + (4096*a^5*Sqrt[b*Sqrt[x] + a*x])/(429*b^7*x) - (8
192*a^6*Sqrt[b*Sqrt[x] + a*x])/(429*b^8*Sqrt[x])

________________________________________________________________________________________

Rubi [A]  time = 0.353146, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2015, 2016, 2014} \[ -\frac{1024 a^4 \sqrt{a x+b \sqrt{x}}}{143 b^6 x^{3/2}}+\frac{2560 a^3 \sqrt{a x+b \sqrt{x}}}{429 b^5 x^2}-\frac{2240 a^2 \sqrt{a x+b \sqrt{x}}}{429 b^4 x^{5/2}}-\frac{8192 a^6 \sqrt{a x+b \sqrt{x}}}{429 b^8 \sqrt{x}}+\frac{4096 a^5 \sqrt{a x+b \sqrt{x}}}{429 b^7 x}+\frac{672 a \sqrt{a x+b \sqrt{x}}}{143 b^3 x^3}-\frac{56 \sqrt{a x+b \sqrt{x}}}{13 b^2 x^{7/2}}+\frac{4}{b x^3 \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(7/2)*(b*Sqrt[x] + a*x)^(3/2)),x]

[Out]

4/(b*x^3*Sqrt[b*Sqrt[x] + a*x]) - (56*Sqrt[b*Sqrt[x] + a*x])/(13*b^2*x^(7/2)) + (672*a*Sqrt[b*Sqrt[x] + a*x])/
(143*b^3*x^3) - (2240*a^2*Sqrt[b*Sqrt[x] + a*x])/(429*b^4*x^(5/2)) + (2560*a^3*Sqrt[b*Sqrt[x] + a*x])/(429*b^5
*x^2) - (1024*a^4*Sqrt[b*Sqrt[x] + a*x])/(143*b^6*x^(3/2)) + (4096*a^5*Sqrt[b*Sqrt[x] + a*x])/(429*b^7*x) - (8
192*a^6*Sqrt[b*Sqrt[x] + a*x])/(429*b^8*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^{7/2} \left (b \sqrt{x}+a x\right )^{3/2}} \, dx &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}+\frac{14 \int \frac{1}{x^4 \sqrt{b \sqrt{x}+a x}} \, dx}{b}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}-\frac{(168 a) \int \frac{1}{x^{7/2} \sqrt{b \sqrt{x}+a x}} \, dx}{13 b^2}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}+\frac{672 a \sqrt{b \sqrt{x}+a x}}{143 b^3 x^3}+\frac{\left (1680 a^2\right ) \int \frac{1}{x^3 \sqrt{b \sqrt{x}+a x}} \, dx}{143 b^3}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}+\frac{672 a \sqrt{b \sqrt{x}+a x}}{143 b^3 x^3}-\frac{2240 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^4 x^{5/2}}-\frac{\left (4480 a^3\right ) \int \frac{1}{x^{5/2} \sqrt{b \sqrt{x}+a x}} \, dx}{429 b^4}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}+\frac{672 a \sqrt{b \sqrt{x}+a x}}{143 b^3 x^3}-\frac{2240 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^4 x^{5/2}}+\frac{2560 a^3 \sqrt{b \sqrt{x}+a x}}{429 b^5 x^2}+\frac{\left (1280 a^4\right ) \int \frac{1}{x^2 \sqrt{b \sqrt{x}+a x}} \, dx}{143 b^5}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}+\frac{672 a \sqrt{b \sqrt{x}+a x}}{143 b^3 x^3}-\frac{2240 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^4 x^{5/2}}+\frac{2560 a^3 \sqrt{b \sqrt{x}+a x}}{429 b^5 x^2}-\frac{1024 a^4 \sqrt{b \sqrt{x}+a x}}{143 b^6 x^{3/2}}-\frac{\left (1024 a^5\right ) \int \frac{1}{x^{3/2} \sqrt{b \sqrt{x}+a x}} \, dx}{143 b^6}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}+\frac{672 a \sqrt{b \sqrt{x}+a x}}{143 b^3 x^3}-\frac{2240 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^4 x^{5/2}}+\frac{2560 a^3 \sqrt{b \sqrt{x}+a x}}{429 b^5 x^2}-\frac{1024 a^4 \sqrt{b \sqrt{x}+a x}}{143 b^6 x^{3/2}}+\frac{4096 a^5 \sqrt{b \sqrt{x}+a x}}{429 b^7 x}+\frac{\left (2048 a^6\right ) \int \frac{1}{x \sqrt{b \sqrt{x}+a x}} \, dx}{429 b^7}\\ &=\frac{4}{b x^3 \sqrt{b \sqrt{x}+a x}}-\frac{56 \sqrt{b \sqrt{x}+a x}}{13 b^2 x^{7/2}}+\frac{672 a \sqrt{b \sqrt{x}+a x}}{143 b^3 x^3}-\frac{2240 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^4 x^{5/2}}+\frac{2560 a^3 \sqrt{b \sqrt{x}+a x}}{429 b^5 x^2}-\frac{1024 a^4 \sqrt{b \sqrt{x}+a x}}{143 b^6 x^{3/2}}+\frac{4096 a^5 \sqrt{b \sqrt{x}+a x}}{429 b^7 x}-\frac{8192 a^6 \sqrt{b \sqrt{x}+a x}}{429 b^8 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0629794, size = 107, normalized size = 0.48 \[ -\frac{4 \left (-256 a^5 b^2 x^{5/2}+128 a^4 b^3 x^2-80 a^3 b^4 x^{3/2}+56 a^2 b^5 x+1024 a^6 b x^3+2048 a^7 x^{7/2}-42 a b^6 \sqrt{x}+33 b^7\right )}{429 b^8 x^3 \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(7/2)*(b*Sqrt[x] + a*x)^(3/2)),x]

[Out]

(-4*(33*b^7 - 42*a*b^6*Sqrt[x] + 56*a^2*b^5*x - 80*a^3*b^4*x^(3/2) + 128*a^4*b^3*x^2 - 256*a^5*b^2*x^(5/2) + 1
024*a^6*b*x^3 + 2048*a^7*x^(7/2)))/(429*b^8*x^3*Sqrt[b*Sqrt[x] + a*x])

________________________________________________________________________________________

Maple [C]  time = 0.011, size = 636, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(7/2)/(b*x^(1/2)+a*x)^(3/2),x)

[Out]

-2/429*(b*x^(1/2)+a*x)^(1/2)*(2048*(b*x^(1/2)+a*x)^(3/2)*a^(13/2)*x^(13/2)*b^2+9244*(b*x^(1/2)+a*x)^(3/2)*a^(1
5/2)*x^7*b+66*(b*x^(1/2)+a*x)^(3/2)*a^(1/2)*x^(7/2)*b^8-160*(b*x^(1/2)+a*x)^(3/2)*a^(7/2)*x^5*b^5-2574*(b*x^(1
/2)+a*x)^(1/2)*a^(15/2)*x^(15/2)*b^2-2574*a^(15/2)*x^(15/2)*(x^(1/2)*(b+a*x^(1/2)))^(1/2)*b^2-1287*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^(17/2)*a^9*b-2574*a^(19/2)*x^(17/2)*(x^(1/2)*(b+a*x^(1/
2)))^(1/2)+1287*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^(17/2)*a^9*b-2574*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^8*a^8*b^2-858*a^(17/2)*x^(15/2)*(x^(1/2)*(b+a*
x^(1/2)))^(3/2)+2574*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^8*a^8*b^2-1287*
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^7*b^3+1287*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^(15/2)*a^7*b^3+112*(b*x^(1/2)+a*x)^(3/2)*a^(5/2)*x^(9/2)*b^6
-84*(b*x^(1/2)+a*x)^(3/2)*a^(3/2)*x^4*b^7+6006*(b*x^(1/2)+a*x)^(3/2)*a^(17/2)*x^(15/2)-2574*(b*x^(1/2)+a*x)^(1
/2)*a^(19/2)*x^(17/2)-5148*(b*x^(1/2)+a*x)^(1/2)*a^(17/2)*x^8*b-5148*a^(17/2)*x^8*(x^(1/2)*(b+a*x^(1/2)))^(1/2
)*b+256*(b*x^(1/2)+a*x)^(3/2)*a^(9/2)*x^(11/2)*b^4-512*(b*x^(1/2)+a*x)^(3/2)*a^(11/2)*x^6*b^3)/(x^(1/2)*(b+a*x
^(1/2)))^(1/2)/b^9/a^(1/2)/x^(15/2)/(b+a*x^(1/2))^2

________________________________________________________________________________________

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^{\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)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(7/2)), x)

________________________________________________________________________________________

Fricas [A]  time = 2.36309, size = 278, normalized size = 1.25 \begin{align*} \frac{4 \,{\left (1024 \, a^{7} b x^{4} - 384 \, a^{5} b^{3} x^{3} - 136 \, a^{3} b^{5} x^{2} - 75 \, a b^{7} x -{\left (2048 \, a^{8} x^{4} - 1280 \, a^{6} b^{2} x^{3} - 208 \, a^{4} b^{4} x^{2} - 98 \, a^{2} b^{6} x - 33 \, b^{8}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{429 \,{\left (a^{2} b^{8} x^{5} - b^{10} x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="fricas")

[Out]

4/429*(1024*a^7*b*x^4 - 384*a^5*b^3*x^3 - 136*a^3*b^5*x^2 - 75*a*b^7*x - (2048*a^8*x^4 - 1280*a^6*b^2*x^3 - 20
8*a^4*b^4*x^2 - 98*a^2*b^6*x - 33*b^8)*sqrt(x))*sqrt(a*x + b*sqrt(x))/(a^2*b^8*x^5 - b^10*x^4)

________________________________________________________________________________________

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)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

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^{\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)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(7/2)), x)