3.3.54 \(\int \frac {(c \sqrt {a+b x^2})^{3/2}}{x^3} \, dx\) [254]

Optimal. Leaf size=133 \[ -\frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{2 x^2}+\frac {3 b \left (c \sqrt {a+b x^2}\right )^{3/2} \tan ^{-1}\left (\sqrt [4]{1+\frac {b x^2}{a}}\right )}{4 a \left (1+\frac {b x^2}{a}\right )^{3/4}}-\frac {3 b \left (c \sqrt {a+b x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt [4]{1+\frac {b x^2}{a}}\right )}{4 a \left (1+\frac {b x^2}{a}\right )^{3/4}} \]

[Out]

-1/2*(c*(b*x^2+a)^(1/2))^(3/2)/x^2+3/4*b*arctan((1+b*x^2/a)^(1/4))*(c*(b*x^2+a)^(1/2))^(3/2)/a/(1+b*x^2/a)^(3/
4)-3/4*b*arctanh((1+b*x^2/a)^(1/4))*(c*(b*x^2+a)^(1/2))^(3/2)/a/(1+b*x^2/a)^(3/4)

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Rubi [A]
time = 0.05, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1973, 272, 43, 65, 304, 209, 212} \begin {gather*} \frac {3 b \text {ArcTan}\left (\sqrt [4]{\frac {b x^2}{a}+1}\right ) \left (c \sqrt {a+b x^2}\right )^{3/2}}{4 a \left (\frac {b x^2}{a}+1\right )^{3/4}}-\frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{2 x^2}-\frac {3 b \left (c \sqrt {a+b x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt [4]{\frac {b x^2}{a}+1}\right )}{4 a \left (\frac {b x^2}{a}+1\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Sqrt[a + b*x^2])^(3/2)/x^3,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

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

Rule 212

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

Rule 272

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

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 1973

Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Dist[Simp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/
a))^(p*q)], Int[u*(1 + b*(x^n/a))^(p*q), x], x] /; FreeQ[{a, b, c, n, p, q}, x] &&  !GeQ[a, 0]

Rubi steps

\begin {align*} \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^3} \, dx &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \int \frac {\left (a+b x^2\right )^{3/4}}{x^3} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \text {Subst}\left (\int \frac {(a+b x)^{3/4}}{x^2} \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{2 x^2}+\frac {\left (3 b c \sqrt {c \sqrt {a+b x^2}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt [4]{a+b x}} \, dx,x,x^2\right )}{8 \sqrt [4]{a+b x^2}}\\ &=-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{2 x^2}+\frac {\left (3 c \sqrt {c \sqrt {a+b x^2}}\right ) \text {Subst}\left (\int \frac {x^2}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{2 \sqrt [4]{a+b x^2}}\\ &=-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{2 x^2}-\frac {\left (3 b c \sqrt {c \sqrt {a+b x^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 \sqrt [4]{a+b x^2}}+\frac {\left (3 b c \sqrt {c \sqrt {a+b x^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 \sqrt [4]{a+b x^2}}\\ &=-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{2 x^2}+\frac {3 b c \sqrt {c \sqrt {a+b x^2}} \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{4 \sqrt [4]{a} \sqrt [4]{a+b x^2}}-\frac {3 b c \sqrt {c \sqrt {a+b x^2}} \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{4 \sqrt [4]{a} \sqrt [4]{a+b x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 107, normalized size = 0.80 \begin {gather*} -\frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (2 \sqrt [4]{a} \left (a+b x^2\right )^{3/4}-3 b x^2 \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )+3 b x^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right )}{4 \sqrt [4]{a} x^2 \left (a+b x^2\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 138, normalized size = 1.04 \begin {gather*} \frac {{\left (3 \, c^{4} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {\sqrt {b x^{2} + a} c}}{\left (a c^{2}\right )^{\frac {1}{4}}}\right )}{\left (a c^{2}\right )^{\frac {1}{4}}} + \frac {\log \left (\frac {\sqrt {\sqrt {b x^{2} + a} c} - \left (a c^{2}\right )^{\frac {1}{4}}}{\sqrt {\sqrt {b x^{2} + a} c} + \left (a c^{2}\right )^{\frac {1}{4}}}\right )}{\left (a c^{2}\right )^{\frac {1}{4}}}\right )} - \frac {4 \, \left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}} c^{4}}{{\left (b x^{2} + a\right )} c^{2} - a c^{2}}\right )} b}{8 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*c^4*(2*arctan(sqrt(sqrt(b*x^2 + a)*c)/(a*c^2)^(1/4))/(a*c^2)^(1/4) + log((sqrt(sqrt(b*x^2 + a)*c) - (a*
c^2)^(1/4))/(sqrt(sqrt(b*x^2 + a)*c) + (a*c^2)^(1/4)))/(a*c^2)^(1/4)) - 4*(sqrt(b*x^2 + a)*c)^(3/2)*c^4/((b*x^
2 + a)*c^2 - a*c^2))*b/c^2

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.13, size = 212, normalized size = 1.59 \begin {gather*} \frac {{\left (\frac {6 \, \sqrt {2} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {1}{4}}} + \frac {6 \, \sqrt {2} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {1}{4}}} + \frac {3 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} b^{2} \log \left (\sqrt {2} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{2} + a} + \sqrt {-a}\right )}{a} + \frac {3 \, \sqrt {2} b^{2} \log \left (-\sqrt {2} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{2} + a} + \sqrt {-a}\right )}{\left (-a\right )^{\frac {1}{4}}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} b}{x^{2}}\right )} c^{\frac {3}{2}}}{16 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(6*sqrt(2)*b^2*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(b*x^2 + a)^(1/4))/(-a)^(1/4))/(-a)^(1/4) + 6*s
qrt(2)*b^2*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(b*x^2 + a)^(1/4))/(-a)^(1/4))/(-a)^(1/4) + 3*sqrt(2)*(
-a)^(3/4)*b^2*log(sqrt(2)*(b*x^2 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^2 + a) + sqrt(-a))/a + 3*sqrt(2)*b^2*log(-sq
rt(2)*(b*x^2 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^2 + a) + sqrt(-a))/(-a)^(1/4) - 8*(b*x^2 + a)^(3/4)*b/x^2)*c^(3/
2)/b

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,\sqrt {b\,x^2+a}\right )}^{3/2}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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