3.7.26 \(\int \frac {1}{(c x)^{3/2} (a+b x^2)^{3/2}} \, dx\) [626]

Optimal. Leaf size=296 \[ \frac {1}{a c \sqrt {c x} \sqrt {a+b x^2}}-\frac {3 \sqrt {a+b x^2}}{a^2 c \sqrt {c x}}+\frac {3 \sqrt {b} \sqrt {c x} \sqrt {a+b x^2}}{a^2 c^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {3 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{a^{7/4} c^{3/2} \sqrt {a+b x^2}}+\frac {3 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{2 a^{7/4} c^{3/2} \sqrt {a+b x^2}} \]

[Out]

1/a/c/(c*x)^(1/2)/(b*x^2+a)^(1/2)-3*(b*x^2+a)^(1/2)/a^2/c/(c*x)^(1/2)+3*b^(1/2)*(c*x)^(1/2)*(b*x^2+a)^(1/2)/a^
2/c^2/(a^(1/2)+x*b^(1/2))-3*b^(1/4)*(cos(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)))^2)^(1/2)/cos(2*arctan(
b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)))*EllipticE(sin(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))),1/2*2^(1/2)
)*(a^(1/2)+x*b^(1/2))*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/a^(7/4)/c^(3/2)/(b*x^2+a)^(1/2)+3/2*b^(1/4)*(cos
(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)))*El
lipticF(sin(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))),1/2*2^(1/2))*(a^(1/2)+x*b^(1/2))*((b*x^2+a)/(a^(1/2
)+x*b^(1/2))^2)^(1/2)/a^(7/4)/c^(3/2)/(b*x^2+a)^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {296, 331, 335, 311, 226, 1210} \begin {gather*} \frac {3 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{2 a^{7/4} c^{3/2} \sqrt {a+b x^2}}-\frac {3 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{a^{7/4} c^{3/2} \sqrt {a+b x^2}}+\frac {3 \sqrt {b} \sqrt {c x} \sqrt {a+b x^2}}{a^2 c^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {3 \sqrt {a+b x^2}}{a^2 c \sqrt {c x}}+\frac {1}{a c \sqrt {c x} \sqrt {a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

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 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.01, size = 57, normalized size = 0.19 \begin {gather*} -\frac {2 x \sqrt {1+\frac {b x^2}{a}} \, _2F_1\left (-\frac {1}{4},\frac {3}{2};\frac {3}{4};-\frac {b x^2}{a}\right )}{a (c x)^{3/2} \sqrt {a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-1/4, 3/2, 3/4, -((b*x^2)/a)])/(a*(c*x)^(3/2)*Sqrt[a + b*x^2])

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Maple [A]
time = 0.08, size = 197, normalized size = 0.67

method	result	size
default	\(\frac {6 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a -3 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a -6 b \,x^{2}-4 a}{2 \sqrt {b \,x^{2}+a}\, c \sqrt {c x}\, a^{2}}\)	\(197\)
elliptic	\(\frac {\sqrt {c x \left (b \,x^{2}+a \right )}\, \left (-\frac {b \,x^{2}}{c \,a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}-\frac {2 \left (c \,x^{2} b +a c \right )}{a^{2} c^{2} \sqrt {x \left (c \,x^{2} b +a c \right )}}+\frac {3 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a^{2} c \sqrt {b c \,x^{3}+a c x}}\right )}{\sqrt {c x}\, \sqrt {b \,x^{2}+a}}\)	\(251\)
risch	\(-\frac {2 \sqrt {b \,x^{2}+a}}{a^{2} c \sqrt {c x}}+\frac {b^{2} \left (\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b^{2} \sqrt {b c \,x^{3}+a c x}}-\frac {a \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a b \sqrt {b c \,x^{3}+a c x}}\right )}{b}\right ) \sqrt {c x \left (b \,x^{2}+a \right )}}{a^{2} c \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\)	\(412\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(3/2)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*(6*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(
1/2))^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a-3*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2
))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/
2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a-6*b*x^2-4*a)/(b*x^2+a)^(1/2)/c/(c*x)^(1/2)/a^2

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.32, size = 83, normalized size = 0.28 \begin {gather*} -\frac {3 \, {\left (b x^{3} + a x\right )} \sqrt {b c} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (3 \, b x^{2} + 2 \, a\right )} \sqrt {b x^{2} + a} \sqrt {c x}}{a^{2} b c^{2} x^{3} + a^{3} c^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*(b*x^3 + a*x)*sqrt(b*c)*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b, 0, x)) + (3*b*x^2 + 2*a)*sq
rt(b*x^2 + a)*sqrt(c*x))/(a^2*b*c^2*x^3 + a^3*c^2*x)

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Sympy [C] Result contains complex when optimal does not.
time = 1.61, size = 48, normalized size = 0.16 \begin {gather*} \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} c^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(3/2)/(b*x**2+a)**(3/2),x)

[Out]

gamma(-1/4)*hyper((-1/4, 3/2), (3/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(3/2)*c**(3/2)*sqrt(x)*gamma(3/4))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (c\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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