3.1053 \(\int \frac{(a+b x^4)^{5/4}}{x^5} \, dx\)

Optimal. Leaf size=91 \[ -\frac{\left (a+b x^4\right )^{5/4}}{4 x^4}+\frac{5}{4} b \sqrt [4]{a+b x^4}-\frac{5}{8} \sqrt [4]{a} b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{5}{8} \sqrt [4]{a} b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right ) \]

[Out]

(5*b*(a + b*x^4)^(1/4))/4 - (a + b*x^4)^(5/4)/(4*x^4) - (5*a^(1/4)*b*ArcTan[(a + b*x^4)^(1/4)/a^(1/4)])/8 - (5
*a^(1/4)*b*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4)])/8

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Rubi [A]  time = 0.0529128, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {266, 47, 50, 63, 212, 206, 203} \[ -\frac{\left (a+b x^4\right )^{5/4}}{4 x^4}+\frac{5}{4} b \sqrt [4]{a+b x^4}-\frac{5}{8} \sqrt [4]{a} b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{5}{8} \sqrt [4]{a} b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(5/4)/x^5,x]

[Out]

(5*b*(a + b*x^4)^(1/4))/4 - (a + b*x^4)^(5/4)/(4*x^4) - (5*a^(1/4)*b*ArcTan[(a + b*x^4)^(1/4)/a^(1/4)])/8 - (5
*a^(1/4)*b*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4)])/8

Rule 266

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 47

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 212

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

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 203

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x^4\right )^{5/4}}{x^5} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{(a+b x)^{5/4}}{x^2} \, dx,x,x^4\right )\\ &=-\frac{\left (a+b x^4\right )^{5/4}}{4 x^4}+\frac{1}{16} (5 b) \operatorname{Subst}\left (\int \frac{\sqrt [4]{a+b x}}{x} \, dx,x,x^4\right )\\ &=\frac{5}{4} b \sqrt [4]{a+b x^4}-\frac{\left (a+b x^4\right )^{5/4}}{4 x^4}+\frac{1}{16} (5 a b) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=\frac{5}{4} b \sqrt [4]{a+b x^4}-\frac{\left (a+b x^4\right )^{5/4}}{4 x^4}+\frac{1}{4} (5 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )\\ &=\frac{5}{4} b \sqrt [4]{a+b x^4}-\frac{\left (a+b x^4\right )^{5/4}}{4 x^4}-\frac{1}{8} \left (5 \sqrt{a} b\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )-\frac{1}{8} \left (5 \sqrt{a} b\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )\\ &=\frac{5}{4} b \sqrt [4]{a+b x^4}-\frac{\left (a+b x^4\right )^{5/4}}{4 x^4}-\frac{5}{8} \sqrt [4]{a} b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{5}{8} \sqrt [4]{a} b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0087062, size = 37, normalized size = 0.41 \[ \frac{b \left (a+b x^4\right )^{9/4} \, _2F_1\left (2,\frac{9}{4};\frac{13}{4};\frac{b x^4}{a}+1\right )}{9 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(5/4)/x^5,x]

[Out]

(b*(a + b*x^4)^(9/4)*Hypergeometric2F1[2, 9/4, 13/4, 1 + (b*x^4)/a])/(9*a^2)

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(5/4)/x^5,x)

[Out]

int((b*x^4+a)^(5/4)/x^5,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.60835, size = 421, normalized size = 4.63 \begin{align*} \frac{20 \, \left (a b^{4}\right )^{\frac{1}{4}} x^{4} \arctan \left (-\frac{\left (a b^{4}\right )^{\frac{3}{4}}{\left (b x^{4} + a\right )}^{\frac{1}{4}} b - \left (a b^{4}\right )^{\frac{3}{4}} \sqrt{\sqrt{b x^{4} + a} b^{2} + \sqrt{a b^{4}}}}{a b^{4}}\right ) - 5 \, \left (a b^{4}\right )^{\frac{1}{4}} x^{4} \log \left (5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b + 5 \, \left (a b^{4}\right )^{\frac{1}{4}}\right ) + 5 \, \left (a b^{4}\right )^{\frac{1}{4}} x^{4} \log \left (5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b - 5 \, \left (a b^{4}\right )^{\frac{1}{4}}\right ) + 4 \,{\left (4 \, b x^{4} - a\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{16 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^5,x, algorithm="fricas")

[Out]

1/16*(20*(a*b^4)^(1/4)*x^4*arctan(-((a*b^4)^(3/4)*(b*x^4 + a)^(1/4)*b - (a*b^4)^(3/4)*sqrt(sqrt(b*x^4 + a)*b^2
+ sqrt(a*b^4)))/(a*b^4)) - 5*(a*b^4)^(1/4)*x^4*log(5*(b*x^4 + a)^(1/4)*b + 5*(a*b^4)^(1/4)) + 5*(a*b^4)^(1/4)
*x^4*log(5*(b*x^4 + a)^(1/4)*b - 5*(a*b^4)^(1/4)) + 4*(4*b*x^4 - a)*(b*x^4 + a)^(1/4))/x^4

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Sympy [C]  time = 1.92512, size = 42, normalized size = 0.46 \begin{align*} - \frac{b^{\frac{5}{4}} x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 \Gamma \left (\frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(5/4)/x**5,x)

[Out]

-b**(5/4)*x*gamma(-1/4)*hyper((-5/4, -1/4), (3/4,), a*exp_polar(I*pi)/(b*x**4))/(4*gamma(3/4))

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Giac [B]  time = 1.11314, size = 278, normalized size = 3.05 \begin{align*} -\frac{1}{32} \,{\left (10 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) + 10 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) + 5 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \log \left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right ) - 5 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \log \left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right ) - 32 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} + \frac{8 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}{b x^{4}}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(5/4)/x^5,x, algorithm="giac")

[Out]

-1/32*(10*sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(b*x^4 + a)^(1/4))/(-a)^(1/4)) + 10*sq
rt(2)*(-a)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(b*x^4 + a)^(1/4))/(-a)^(1/4)) + 5*sqrt(2)*(-a)^(
1/4)*log(sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a)) - 5*sqrt(2)*(-a)^(1/4)*log(-sqrt(2
)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a)) - 32*(b*x^4 + a)^(1/4) + 8*(b*x^4 + a)^(1/4)*a/(b
*x^4))*b