3.117 \(\int \frac{x^{5/2}}{\sqrt{b \sqrt{x}+a x}} \, dx\)

Optimal. Leaf size=204 \[ \frac{33 b^2 x^{3/2} \sqrt{a x+b \sqrt{x}}}{80 a^3}-\frac{231 b^5 \sqrt{a x+b \sqrt{x}}}{256 a^6}+\frac{77 b^4 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{128 a^5}-\frac{77 b^3 x \sqrt{a x+b \sqrt{x}}}{160 a^4}+\frac{231 b^6 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{256 a^{13/2}}-\frac{11 b x^2 \sqrt{a x+b \sqrt{x}}}{30 a^2}+\frac{x^{5/2} \sqrt{a x+b \sqrt{x}}}{3 a} \]

[Out]

(-231*b^5*Sqrt[b*Sqrt[x] + a*x])/(256*a^6) + (77*b^4*Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])/(128*a^5) - (77*b^3*x*Sqrt
[b*Sqrt[x] + a*x])/(160*a^4) + (33*b^2*x^(3/2)*Sqrt[b*Sqrt[x] + a*x])/(80*a^3) - (11*b*x^2*Sqrt[b*Sqrt[x] + a*
x])/(30*a^2) + (x^(5/2)*Sqrt[b*Sqrt[x] + a*x])/(3*a) + (231*b^6*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a*x
]])/(256*a^(13/2))

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Rubi [A]  time = 0.17155, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2018, 670, 640, 620, 206} \[ \frac{33 b^2 x^{3/2} \sqrt{a x+b \sqrt{x}}}{80 a^3}-\frac{231 b^5 \sqrt{a x+b \sqrt{x}}}{256 a^6}+\frac{77 b^4 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{128 a^5}-\frac{77 b^3 x \sqrt{a x+b \sqrt{x}}}{160 a^4}+\frac{231 b^6 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{256 a^{13/2}}-\frac{11 b x^2 \sqrt{a x+b \sqrt{x}}}{30 a^2}+\frac{x^{5/2} \sqrt{a x+b \sqrt{x}}}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)/Sqrt[b*Sqrt[x] + a*x],x]

[Out]

(-231*b^5*Sqrt[b*Sqrt[x] + a*x])/(256*a^6) + (77*b^4*Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])/(128*a^5) - (77*b^3*x*Sqrt
[b*Sqrt[x] + a*x])/(160*a^4) + (33*b^2*x^(3/2)*Sqrt[b*Sqrt[x] + a*x])/(80*a^3) - (11*b*x^2*Sqrt[b*Sqrt[x] + a*
x])/(30*a^2) + (x^(5/2)*Sqrt[b*Sqrt[x] + a*x])/(3*a) + (231*b^6*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a*x
]])/(256*a^(13/2))

Rule 2018

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)
/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^{5/2}}{\sqrt{b \sqrt{x}+a x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^6}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}-\frac{(11 b) \operatorname{Subst}\left (\int \frac{x^5}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{6 a}\\ &=-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}+\frac{\left (33 b^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{20 a^2}\\ &=\frac{33 b^2 x^{3/2} \sqrt{b \sqrt{x}+a x}}{80 a^3}-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}-\frac{\left (231 b^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{160 a^3}\\ &=-\frac{77 b^3 x \sqrt{b \sqrt{x}+a x}}{160 a^4}+\frac{33 b^2 x^{3/2} \sqrt{b \sqrt{x}+a x}}{80 a^3}-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}+\frac{\left (77 b^4\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{64 a^4}\\ &=\frac{77 b^4 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{128 a^5}-\frac{77 b^3 x \sqrt{b \sqrt{x}+a x}}{160 a^4}+\frac{33 b^2 x^{3/2} \sqrt{b \sqrt{x}+a x}}{80 a^3}-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}-\frac{\left (231 b^5\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{256 a^5}\\ &=-\frac{231 b^5 \sqrt{b \sqrt{x}+a x}}{256 a^6}+\frac{77 b^4 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{128 a^5}-\frac{77 b^3 x \sqrt{b \sqrt{x}+a x}}{160 a^4}+\frac{33 b^2 x^{3/2} \sqrt{b \sqrt{x}+a x}}{80 a^3}-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}+\frac{\left (231 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^2}} \, dx,x,\sqrt{x}\right )}{512 a^6}\\ &=-\frac{231 b^5 \sqrt{b \sqrt{x}+a x}}{256 a^6}+\frac{77 b^4 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{128 a^5}-\frac{77 b^3 x \sqrt{b \sqrt{x}+a x}}{160 a^4}+\frac{33 b^2 x^{3/2} \sqrt{b \sqrt{x}+a x}}{80 a^3}-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}+\frac{\left (231 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{b \sqrt{x}+a x}}\right )}{256 a^6}\\ &=-\frac{231 b^5 \sqrt{b \sqrt{x}+a x}}{256 a^6}+\frac{77 b^4 \sqrt{x} \sqrt{b \sqrt{x}+a x}}{128 a^5}-\frac{77 b^3 x \sqrt{b \sqrt{x}+a x}}{160 a^4}+\frac{33 b^2 x^{3/2} \sqrt{b \sqrt{x}+a x}}{80 a^3}-\frac{11 b x^2 \sqrt{b \sqrt{x}+a x}}{30 a^2}+\frac{x^{5/2} \sqrt{b \sqrt{x}+a x}}{3 a}+\frac{231 b^6 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b \sqrt{x}+a x}}\right )}{256 a^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.179312, size = 164, normalized size = 0.8 \[ \frac{\left (a \sqrt{x}+b\right ) \left (\sqrt{a} \sqrt{x} \sqrt{\frac{a \sqrt{x}}{b}+1} \left (1584 a^3 b^2 x^{3/2}-1848 a^2 b^3 x-1408 a^4 b x^2+1280 a^5 x^{5/2}+2310 a b^4 \sqrt{x}-3465 b^5\right )+3465 b^{11/2} \sqrt [4]{x} \sinh ^{-1}\left (\frac{\sqrt{a} \sqrt [4]{x}}{\sqrt{b}}\right )\right )}{3840 a^{13/2} \sqrt{\frac{a \sqrt{x}}{b}+1} \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)/Sqrt[b*Sqrt[x] + a*x],x]

[Out]

((b + a*Sqrt[x])*(Sqrt[a]*Sqrt[1 + (a*Sqrt[x])/b]*Sqrt[x]*(-3465*b^5 + 2310*a*b^4*Sqrt[x] - 1848*a^2*b^3*x + 1
584*a^3*b^2*x^(3/2) - 1408*a^4*b*x^2 + 1280*a^5*x^(5/2)) + 3465*b^(11/2)*x^(1/4)*ArcSinh[(Sqrt[a]*x^(1/4))/Sqr
t[b]]))/(3840*a^(13/2)*Sqrt[1 + (a*Sqrt[x])/b]*Sqrt[b*Sqrt[x] + a*x])

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Maple [A]  time = 0.01, size = 245, normalized size = 1.2 \begin{align*}{\frac{1}{7680}\sqrt{b\sqrt{x}+ax} \left ( 2560\,{x}^{3/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{11/2}+8544\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{7/2}\sqrt{x}{b}^{2}-5376\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{9/2}xb-12240\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{5/2}{b}^{3}+16860\,\sqrt{b\sqrt{x}+ax}{a}^{5/2}\sqrt{x}{b}^{4}+8430\,\sqrt{b\sqrt{x}+ax}{a}^{3/2}{b}^{5}-15360\,{a}^{3/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }{b}^{5}+7680\,a\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 ){b}^{6}-4215\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ) a{b}^{6} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{a}^{-{\frac{15}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x)

[Out]

1/7680*(b*x^(1/2)+a*x)^(1/2)*(2560*x^(3/2)*(b*x^(1/2)+a*x)^(3/2)*a^(11/2)+8544*(b*x^(1/2)+a*x)^(3/2)*a^(7/2)*x
^(1/2)*b^2-5376*(b*x^(1/2)+a*x)^(3/2)*a^(9/2)*x*b-12240*(b*x^(1/2)+a*x)^(3/2)*a^(5/2)*b^3+16860*(b*x^(1/2)+a*x
)^(1/2)*a^(5/2)*x^(1/2)*b^4+8430*(b*x^(1/2)+a*x)^(1/2)*a^(3/2)*b^5-15360*a^(3/2)*(x^(1/2)*(b+a*x^(1/2)))^(1/2)
*b^5+7680*a*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))*b^6-4215*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*b^6)/(x^(1/2)*(b+a*x^(1/2)))^(1/2)/a^(15/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{5}{2}}}{\sqrt{a x + b \sqrt{x}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^(5/2)/sqrt(a*x + b*sqrt(x)), x)

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Fricas [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(x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{5}{2}}}{\sqrt{a x + b \sqrt{x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)/(b*x**(1/2)+a*x)**(1/2),x)

[Out]

Integral(x**(5/2)/sqrt(a*x + b*sqrt(x)), x)

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Giac [A]  time = 1.32523, size = 169, normalized size = 0.83 \begin{align*} \frac{1}{3840} \, \sqrt{a x + b \sqrt{x}}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, \sqrt{x}{\left (\frac{10 \, \sqrt{x}}{a} - \frac{11 \, b}{a^{2}}\right )} + \frac{99 \, b^{2}}{a^{3}}\right )} \sqrt{x} - \frac{231 \, b^{3}}{a^{4}}\right )} \sqrt{x} + \frac{1155 \, b^{4}}{a^{5}}\right )} \sqrt{x} - \frac{3465 \, b^{5}}{a^{6}}\right )} - \frac{231 \, b^{6} \log \left ({\left | -2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} - b \right |}\right )}{512 \, a^{\frac{13}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="giac")

[Out]

1/3840*sqrt(a*x + b*sqrt(x))*(2*(4*(2*(8*sqrt(x)*(10*sqrt(x)/a - 11*b/a^2) + 99*b^2/a^3)*sqrt(x) - 231*b^3/a^4
)*sqrt(x) + 1155*b^4/a^5)*sqrt(x) - 3465*b^5/a^6) - 231/512*b^6*log(abs(-2*sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(a*x
+ b*sqrt(x))) - b))/a^(13/2)