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

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

[Out]

231/256*b^6*arctanh(a^(1/2)*x^(1/2)/(b*x^(1/2)+a*x)^(1/2))/a^(13/2)-231/256*b^5*(b*x^(1/2)+a*x)^(1/2)/a^6-77/1
60*b^3*x*(b*x^(1/2)+a*x)^(1/2)/a^4+33/80*b^2*x^(3/2)*(b*x^(1/2)+a*x)^(1/2)/a^3-11/30*b*x^2*(b*x^(1/2)+a*x)^(1/
2)/a^2+1/3*x^(5/2)*(b*x^(1/2)+a*x)^(1/2)/a+77/128*b^4*x^(1/2)*(b*x^(1/2)+a*x)^(1/2)/a^5

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Rubi [A]  time = 0.17, antiderivative size = 204, normalized size of antiderivative = 1.00, 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 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])

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 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 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 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]

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*}

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Mathematica [A]  time = 0.26, size = 164, normalized size = 0.80 \[ \frac {\left (a \sqrt {x}+b\right ) \left (\sqrt {a} \sqrt {x} \sqrt {\frac {a \sqrt {x}}{b}+1} \left (1280 a^5 x^{5/2}-1408 a^4 b x^2+1584 a^3 b^2 x^{3/2}-1848 a^2 b^3 x+2310 a b^4 \sqrt {x}-3465 b^5\right )+3465 b^{11/2} \sqrt [4]{x} \sinh ^{-1}\left (\frac {\sqrt {a} \sqrt [4]{x}}{\sqrt {b}}\right )\right )}{3840 a^{13/2} \sqrt {\frac {a \sqrt {x}}{b}+1} \sqrt {a x+b \sqrt {x}}} \]

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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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [A]  time = 0.28, size = 125, normalized size = 0.61 \[ \frac {1}{3840} \, \sqrt {a x + b \sqrt {x}} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, \sqrt {x} {\left (\frac {10 \, \sqrt {x}}{a} - \frac {11 \, b}{a^{2}}\right )} + \frac {99 \, b^{2}}{a^{3}}\right )} \sqrt {x} - \frac {231 \, b^{3}}{a^{4}}\right )} \sqrt {x} + \frac {1155 \, b^{4}}{a^{5}}\right )} \sqrt {x} - \frac {3465 \, b^{5}}{a^{6}}\right )} - \frac {231 \, b^{6} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{512 \, a^{\frac {13}{2}}} \]

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)

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maple [A]  time = 0.06, size = 245, normalized size = 1.20 \[ \frac {\sqrt {a x +b \sqrt {x}}\, \left (7680 a \,b^{6} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-4215 a \,b^{6} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2560 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{\frac {3}{2}}+16860 \sqrt {a x +b \sqrt {x}}\, a^{\frac {5}{2}} b^{4} \sqrt {x}-5376 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} b x -15360 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {3}{2}} b^{5}+8430 \sqrt {a x +b \sqrt {x}}\, a^{\frac {3}{2}} b^{5}+8544 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} \sqrt {x}-12240 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3}\right )}{7680 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {5}{2}}}{\sqrt {a x + b \sqrt {x}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{5/2}}{\sqrt {a\,x+b\,\sqrt {x}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {5}{2}}}{\sqrt {a x + b \sqrt {x}}}\, dx \]

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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