3.8.92 \(\int \frac {(a+c x^4)^{3/2}}{x^7} \, dx\) [792]

Optimal. Leaf size=68 \[ -\frac {c \sqrt {a+c x^4}}{2 x^2}-\frac {\left (a+c x^4\right )^{3/2}}{6 x^6}+\frac {1}{2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \]

[Out]

-1/6*(c*x^4+a)^(3/2)/x^6+1/2*c^(3/2)*arctanh(x^2*c^(1/2)/(c*x^4+a)^(1/2))-1/2*c*(c*x^4+a)^(1/2)/x^2

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Rubi [A]
time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {281, 283, 223, 212} \begin {gather*} \frac {1}{2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )-\frac {\left (a+c x^4\right )^{3/2}}{6 x^6}-\frac {c \sqrt {a+c x^4}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^(3/2)/x^7,x]

[Out]

-1/2*(c*Sqrt[a + c*x^4])/x^2 - (a + c*x^4)^(3/2)/(6*x^6) + (c^(3/2)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/2

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 281

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

Rule 283

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

Rubi steps

\begin {align*} \int \frac {\left (a+c x^4\right )^{3/2}}{x^7} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\left (a+c x^2\right )^{3/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {\left (a+c x^4\right )^{3/2}}{6 x^6}+\frac {1}{2} c \text {Subst}\left (\int \frac {\sqrt {a+c x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {c \sqrt {a+c x^4}}{2 x^2}-\frac {\left (a+c x^4\right )^{3/2}}{6 x^6}+\frac {1}{2} c^2 \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^2}} \, dx,x,x^2\right )\\ &=-\frac {c \sqrt {a+c x^4}}{2 x^2}-\frac {\left (a+c x^4\right )^{3/2}}{6 x^6}+\frac {1}{2} c^2 \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {a+c x^4}}\right )\\ &=-\frac {c \sqrt {a+c x^4}}{2 x^2}-\frac {\left (a+c x^4\right )^{3/2}}{6 x^6}+\frac {1}{2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 59, normalized size = 0.87 \begin {gather*} \frac {\left (-a-4 c x^4\right ) \sqrt {a+c x^4}}{6 x^6}+\frac {1}{2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {c} x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^(3/2)/x^7,x]

[Out]

((-a - 4*c*x^4)*Sqrt[a + c*x^4])/(6*x^6) + (c^(3/2)*ArcTanh[Sqrt[a + c*x^4]/(Sqrt[c]*x^2)])/2

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Maple [A]
time = 0.14, size = 55, normalized size = 0.81

method	result	size
risch	\(-\frac {\sqrt {x^{4} c +a}\, \left (4 x^{4} c +a \right )}{6 x^{6}}+\frac {c^{\frac {3}{2}} \ln \left (x^{2} \sqrt {c}+\sqrt {x^{4} c +a}\right )}{2}\)	\(47\)
default	\(\frac {c^{\frac {3}{2}} \ln \left (x^{2} \sqrt {c}+\sqrt {x^{4} c +a}\right )}{2}-\frac {a \sqrt {x^{4} c +a}}{6 x^{6}}-\frac {2 c \sqrt {x^{4} c +a}}{3 x^{2}}\)	\(55\)
elliptic	\(\frac {c^{\frac {3}{2}} \ln \left (x^{2} \sqrt {c}+\sqrt {x^{4} c +a}\right )}{2}-\frac {a \sqrt {x^{4} c +a}}{6 x^{6}}-\frac {2 c \sqrt {x^{4} c +a}}{3 x^{2}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(3/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

1/2*c^(3/2)*ln(x^2*c^(1/2)+(c*x^4+a)^(1/2))-1/6*a/x^6*(c*x^4+a)^(1/2)-2/3*c*(c*x^4+a)^(1/2)/x^2

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Maxima [A]
time = 0.51, size = 75, normalized size = 1.10 \begin {gather*} -\frac {1}{4} \, c^{\frac {3}{2}} \log \left (-\frac {\sqrt {c} - \frac {\sqrt {c x^{4} + a}}{x^{2}}}{\sqrt {c} + \frac {\sqrt {c x^{4} + a}}{x^{2}}}\right ) - \frac {\sqrt {c x^{4} + a} c}{2 \, x^{2}} - \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^7,x, algorithm="maxima")

[Out]

-1/4*c^(3/2)*log(-(sqrt(c) - sqrt(c*x^4 + a)/x^2)/(sqrt(c) + sqrt(c*x^4 + a)/x^2)) - 1/2*sqrt(c*x^4 + a)*c/x^2
- 1/6*(c*x^4 + a)^(3/2)/x^6

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Fricas [A]
time = 0.39, size = 116, normalized size = 1.71 \begin {gather*} \left [\frac {3 \, c^{\frac {3}{2}} x^{6} \log \left (-2 \, c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right ) - 2 \, {\left (4 \, c x^{4} + a\right )} \sqrt {c x^{4} + a}}{12 \, x^{6}}, -\frac {3 \, \sqrt {-c} c x^{6} \arctan \left (\frac {\sqrt {-c} x^{2}}{\sqrt {c x^{4} + a}}\right ) + {\left (4 \, c x^{4} + a\right )} \sqrt {c x^{4} + a}}{6 \, x^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^7,x, algorithm="fricas")

[Out]

[1/12*(3*c^(3/2)*x^6*log(-2*c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(c)*x^2 - a) - 2*(4*c*x^4 + a)*sqrt(c*x^4 + a))/x^6,
-1/6*(3*sqrt(-c)*c*x^6*arctan(sqrt(-c)*x^2/sqrt(c*x^4 + a)) + (4*c*x^4 + a)*sqrt(c*x^4 + a))/x^6]

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Sympy [A]
time = 1.21, size = 80, normalized size = 1.18 \begin {gather*} - \frac {a \sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{6 x^{4}} - \frac {2 c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{3} - \frac {c^{\frac {3}{2}} \log {\left (\frac {a}{c x^{4}} \right )}}{4} + \frac {c^{\frac {3}{2}} \log {\left (\sqrt {\frac {a}{c x^{4}} + 1} + 1 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(3/2)/x**7,x)

[Out]

-a*sqrt(c)*sqrt(a/(c*x**4) + 1)/(6*x**4) - 2*c**(3/2)*sqrt(a/(c*x**4) + 1)/3 - c**(3/2)*log(a/(c*x**4))/4 + c*
*(3/2)*log(sqrt(a/(c*x**4) + 1) + 1)/2

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (52) = 104\).
time = 0.70, size = 122, normalized size = 1.79 \begin {gather*} -\frac {1}{4} \, c^{\frac {3}{2}} \log \left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2}\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} a c^{\frac {3}{2}} - 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} a^{2} c^{\frac {3}{2}} + 2 \, a^{3} c^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^7,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^4+a\right )}^{3/2}}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^(3/2)/x^7,x)

[Out]

int((a + c*x^4)^(3/2)/x^7, x)

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