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3.304 \(\int \frac {1}{x^{5/2} \sqrt {a x^2+b x^5}} \, dx\)

Optimal. Leaf size=235 \[ -\frac {2 b x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt [4]{3} a^{4/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}}-\frac {2 \sqrt {a x^2+b x^5}}{5 a x^{7/2}} \]

[Out]

-2/5*(b*x^5+a*x^2)^(1/2)/a/x^(7/2)-2/15*b*x^(3/2)*(a^(1/3)+b^(1/3)*x)*((a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^(1
/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)/(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))*(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))*EllipticF(
(1-(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*((a^(2/
3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)*3^(3/4)/a^(4/3)/(b*x^5+a*x^2)^(1/2)
/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)

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Rubi [A]  time = 0.22, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2025, 2032, 329, 225} \[ -\frac {2 b x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt [4]{3} a^{4/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}}-\frac {2 \sqrt {a x^2+b x^5}}{5 a x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*Sqrt[a*x^2 + b*x^5]),x]

[Out]

(-2*Sqrt[a*x^2 + b*x^5])/(5*a*x^(7/2)) - (2*b*x^(3/2)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^
(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(5*3^(1/4)*a^(4/3)*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))
/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a*x^2 + b*x^5])

Rule 225

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(x*(s
+ r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2
)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[(r*x^2*(s + r*x^2))/(s + (1
+ Sqrt[3])*r*x^2)^2]), x]] /; FreeQ[{a, b}, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
 && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2025

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, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

Rule 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[m]*(a*x^j + b*x^n)^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rubi steps

\begin {align*} \int \frac {1}{x^{5/2} \sqrt {a x^2+b x^5}} \, dx &=-\frac {2 \sqrt {a x^2+b x^5}}{5 a x^{7/2}}-\frac {(2 b) \int \frac {\sqrt {x}}{\sqrt {a x^2+b x^5}} \, dx}{5 a}\\ &=-\frac {2 \sqrt {a x^2+b x^5}}{5 a x^{7/2}}-\frac {\left (2 b x \sqrt {a+b x^3}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^3}} \, dx}{5 a \sqrt {a x^2+b x^5}}\\ &=-\frac {2 \sqrt {a x^2+b x^5}}{5 a x^{7/2}}-\frac {\left (4 b x \sqrt {a+b x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^6}} \, dx,x,\sqrt {x}\right )}{5 a \sqrt {a x^2+b x^5}}\\ &=-\frac {2 \sqrt {a x^2+b x^5}}{5 a x^{7/2}}-\frac {2 b x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt [4]{3} a^{4/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 57, normalized size = 0.24 \[ -\frac {2 \sqrt {\frac {b x^3}{a}+1} \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {1}{6};-\frac {b x^3}{a}\right )}{5 x^{3/2} \sqrt {x^2 \left (a+b x^3\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*Sqrt[a*x^2 + b*x^5]),x]

[Out]

(-2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-5/6, 1/2, 1/6, -((b*x^3)/a)])/(5*x^(3/2)*Sqrt[x^2*(a + b*x^3)])

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fricas [F]  time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{5} + a x^{2}} \sqrt {x}}{b x^{8} + a x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*x^5 + a*x^2)*sqrt(x)/(b*x^8 + a*x^5), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{5} + a x^{2}} x^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.08, size = 1795, normalized size = 7.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5/(b*x^5+a*x^2)^(1/2)/x^(3/2)*(b*x^3+a)/(-a*b^2)^(1/3)/a*(4*I*(-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(
1/3))*b*x)^(1/2)*((2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3))/(1+I*3^(1/2))/(-b*x+(-a*b^2)^(1/3)))^(1/2)*(
(-2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(
1/2)-3)/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3))*b*x)^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3
))^(1/2))*3^(1/2)*x^5*b^2-8*I*(-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3))*b*x)^(1/2)*((2*b*x+I*3^(1/2)
*(-a*b^2)^(1/3)+(-a*b^2)^(1/3))/(1+I*3^(1/2))/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((-2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)-(
-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-b*x+(-a*b^
2)^(1/3))*b*x)^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-a*b^2)^(1/3)*3^(1/2)*x
^4*b+4*I*(-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3))*b*x)^(1/2)*((2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b
^2)^(1/3))/(1+I*3^(1/2))/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((-2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)-(-a*b^2)^(1/3))/(I*3^(
1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3))*b*x)^(1/2),
((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-a*b^2)^(2/3)*3^(1/2)*x^3-4*(-(I*3^(1/2)-3)/
(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3))*b*x)^(1/2)*((2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3))/(1+I*3^(1/2))/
(-b*x+(-a*b^2)^(1/3)))^(1/2)*((-2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1
/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3))*b*x)^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)
-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*x^5*b^2+8*(-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3))*b*x)^(1/
2)*((2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3))/(1+I*3^(1/2))/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((-2*b*x+I*3^(1
/2)*(-a*b^2)^(1/3)-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)/(I*3^(
1/2)-1)/(-b*x+(-a*b^2)^(1/3))*b*x)^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-a*
b^2)^(1/3)*x^4*b-4*(-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3))*b*x)^(1/2)*((2*b*x+I*3^(1/2)*(-a*b^2)^(
1/3)+(-a*b^2)^(1/3))/(1+I*3^(1/2))/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((-2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)-(-a*b^2)^(1/
3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3))*b
*x)^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-a*b^2)^(2/3)*x^3-I*((-b*x+(-a*b^2
)^(1/3))*(2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3))*(-2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)-(-a*b^2)^(1/3))/b^2*
x)^(1/2)*(-a*b^2)^(1/3)*3^(1/2)*((b*x^3+a)*x)^(1/2)+3*((b*x^3+a)*x)^(1/2)*(-a*b^2)^(1/3)*((-b*x+(-a*b^2)^(1/3)
)*(2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3))*(-2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)-(-a*b^2)^(1/3))/b^2*x)^(1/2
))/((b*x^3+a)*x)^(1/2)/(I*3^(1/2)-3)/((-b*x+(-a*b^2)^(1/3))*(2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3))*(-
2*b*x+I*3^(1/2)*(-a*b^2)^(1/3)-(-a*b^2)^(1/3))/b^2*x)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{5} + a x^{2}} x^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{\frac {5}{2}} \sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**(5/2)*sqrt(x**2*(a + b*x**3))), x)

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