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3.3.7 \(\int \frac {x^{9/2}}{\sqrt {a x^2+b x^5}} \, dx\)

Optimal. Leaf size=65 \[ \frac {\sqrt {x} \sqrt {a x^2+b x^5}}{3 b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 b^{3/2}} \]

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Rubi [A]  time = 0.09, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2024, 2029, 206} \begin {gather*} \frac {\sqrt {x} \sqrt {a x^2+b x^5}}{3 b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(9/2)/Sqrt[a*x^2 + b*x^5],x]

[Out]

(Sqrt[x]*Sqrt[a*x^2 + b*x^5])/(3*b) - (a*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a*x^2 + b*x^5]])/(3*b^(3/2))

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 2024

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n +
 1)*(a*x^j + b*x^n)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^(n - j)*(m + j*p - n + j + 1))/(b*(m + n*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]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2029

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

Rubi steps

\begin {align*} \int \frac {x^{9/2}}{\sqrt {a x^2+b x^5}} \, dx &=\frac {\sqrt {x} \sqrt {a x^2+b x^5}}{3 b}-\frac {a \int \frac {x^{3/2}}{\sqrt {a x^2+b x^5}} \, dx}{2 b}\\ &=\frac {\sqrt {x} \sqrt {a x^2+b x^5}}{3 b}-\frac {a \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 b}\\ &=\frac {\sqrt {x} \sqrt {a x^2+b x^5}}{3 b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 b^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 81, normalized size = 1.25 \begin {gather*} \frac {\sqrt {b} x^{5/2} \left (a+b x^3\right )-a x \sqrt {a+b x^3} \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a+b x^3}}\right )}{3 b^{3/2} \sqrt {x^2 \left (a+b x^3\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(9/2)/Sqrt[a*x^2 + b*x^5],x]

[Out]

(Sqrt[b]*x^(5/2)*(a + b*x^3) - a*x*Sqrt[a + b*x^3]*ArcTanh[(Sqrt[b]*x^(3/2))/Sqrt[a + b*x^3]])/(3*b^(3/2)*Sqrt
[x^2*(a + b*x^3)])

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IntegrateAlgebraic [A]  time = 0.29, size = 82, normalized size = 1.26 \begin {gather*} -\frac {a \log \left (\sqrt {a x^2+b x^5}+\sqrt {b} x^{5/2}\right )}{3 b^{3/2}}+\frac {2 a \log \left (\sqrt {x}\right )}{3 b^{3/2}}+\frac {\sqrt {x} \sqrt {a x^2+b x^5}}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(9/2)/Sqrt[a*x^2 + b*x^5],x]

[Out]

(Sqrt[x]*Sqrt[a*x^2 + b*x^5])/(3*b) + (2*a*Log[Sqrt[x]])/(3*b^(3/2)) - (a*Log[Sqrt[b]*x^(5/2) + Sqrt[a*x^2 + b
*x^5]])/(3*b^(3/2))

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fricas [A]  time = 0.56, size = 148, normalized size = 2.28 \begin {gather*} \left [\frac {a \sqrt {b} \log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} + 4 \, \sqrt {b x^{5} + a x^{2}} {\left (2 \, b x^{3} + a\right )} \sqrt {b} \sqrt {x} - a^{2}\right ) + 4 \, \sqrt {b x^{5} + a x^{2}} b \sqrt {x}}{12 \, b^{2}}, \frac {a \sqrt {-b} \arctan \left (\frac {2 \, \sqrt {b x^{5} + a x^{2}} \sqrt {-b} \sqrt {x}}{2 \, b x^{3} + a}\right ) + 2 \, \sqrt {b x^{5} + a x^{2}} b \sqrt {x}}{6 \, b^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="fricas")

[Out]

[1/12*(a*sqrt(b)*log(-8*b^2*x^6 - 8*a*b*x^3 + 4*sqrt(b*x^5 + a*x^2)*(2*b*x^3 + a)*sqrt(b)*sqrt(x) - a^2) + 4*s
qrt(b*x^5 + a*x^2)*b*sqrt(x))/b^2, 1/6*(a*sqrt(-b)*arctan(2*sqrt(b*x^5 + a*x^2)*sqrt(-b)*sqrt(x)/(2*b*x^3 + a)
) + 2*sqrt(b*x^5 + a*x^2)*b*sqrt(x))/b^2]

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giac [A]  time = 0.25, size = 44, normalized size = 0.68 \begin {gather*} \frac {\sqrt {b x^{3} + a} x^{\frac {3}{2}}}{3 \, b} + \frac {a \log \left ({\left | -\sqrt {b} x^{\frac {3}{2}} + \sqrt {b x^{3} + a} \right |}\right )}{3 \, b^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="giac")

[Out]

1/3*sqrt(b*x^3 + a)*x^(3/2)/b + 1/3*a*log(abs(-sqrt(b)*x^(3/2) + sqrt(b*x^3 + a)))/b^(3/2)

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maple [C]  time = 1.14, size = 3347, normalized size = 51.49 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(9/2)/(b*x^5+a*x^2)^(1/2),x)

[Out]

1/3/(b*x^5+a*x^2)^(1/2)*x^(3/2)*(b*x^3+a)/b^3*(6*I*3^(1/2)*(-(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^2*a*b^2-6*I*3^(1/2)*(-(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)*EllipticPi((-(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)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2)
)*x^2*a*b^2-12*I*3^(1/2)*(-a*b^2)^(1/3)*(-(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*a*b+12*I*3^(
1/2)*(-a*b^2)^(1/3)*(-(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)*EllipticPi((-(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)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*x*a*b
+6*I*3^(1/2)*(-a*b^2)^(2/3)*(-(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-6*I*3^(1/2)*(-a*b^2)^(2/
3)*(-(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)*EllipticPi((-(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)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*a-6*(-(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^2*a*b^2+6*(-(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)*EllipticPi((-(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)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2
))/(I*3^(1/2)-3))^(1/2))*x^2*a*b^2+12*(-a*b^2)^(1/3)*(-(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
*a*b-12*(-a*b^2)^(1/3)*(-(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)*EllipticPi((-(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)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*x*
a*b+I*3^(1/2)*((b*x^3+a)*x)^(1/2)*((-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)*x*b^2-6*(-a*b^2)^(2/3)*(-(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)*El
lipticF((-(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+6*(-a*b^2)^(2/3)*(-(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)*EllipticPi((-(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)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I
*3^(1/2)-3))^(1/2))*a-3*x*((b*x^3+a)*x)^(1/2)*b^2*((-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 \begin {gather*} \int \frac {x^{\frac {9}{2}}}{\sqrt {b x^{5} + a x^{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^(9/2)/sqrt(b*x^5 + a*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{9/2}}{\sqrt {b\,x^5+a\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(9/2)/(a*x^2 + b*x^5)^(1/2),x)

[Out]

int(x^(9/2)/(a*x^2 + b*x^5)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {9}{2}}}{\sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(9/2)/(b*x**5+a*x**2)**(1/2),x)

[Out]

Integral(x**(9/2)/sqrt(x**2*(a + b*x**3)), x)

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