3.1079 \(\int (a+b x^4)^{7/4} \, dx\)

Optimal. Leaf size=96 \[ \frac{21 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 \sqrt [4]{b}}+\frac{21 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 \sqrt [4]{b}}+\frac{7}{32} a x \left (a+b x^4\right )^{3/4}+\frac{1}{8} x \left (a+b x^4\right )^{7/4} \]

[Out]

(7*a*x*(a + b*x^4)^(3/4))/32 + (x*(a + b*x^4)^(7/4))/8 + (21*a^2*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^
(1/4)) + (21*a^2*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(1/4))

________________________________________________________________________________________

Rubi [A]  time = 0.0256012, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {195, 240, 212, 206, 203} \[ \frac{21 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 \sqrt [4]{b}}+\frac{21 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 \sqrt [4]{b}}+\frac{7}{32} a x \left (a+b x^4\right )^{3/4}+\frac{1}{8} x \left (a+b x^4\right )^{7/4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(7/4),x]

[Out]

(7*a*x*(a + b*x^4)^(3/4))/32 + (x*(a + b*x^4)^(7/4))/8 + (21*a^2*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^
(1/4)) + (21*a^2*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(1/4))

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

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 \left (a+b x^4\right )^{7/4} \, dx &=\frac{1}{8} x \left (a+b x^4\right )^{7/4}+\frac{1}{8} (7 a) \int \left (a+b x^4\right )^{3/4} \, dx\\ &=\frac{7}{32} a x \left (a+b x^4\right )^{3/4}+\frac{1}{8} x \left (a+b x^4\right )^{7/4}+\frac{1}{32} \left (21 a^2\right ) \int \frac{1}{\sqrt [4]{a+b x^4}} \, dx\\ &=\frac{7}{32} a x \left (a+b x^4\right )^{3/4}+\frac{1}{8} x \left (a+b x^4\right )^{7/4}+\frac{1}{32} \left (21 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )\\ &=\frac{7}{32} a x \left (a+b x^4\right )^{3/4}+\frac{1}{8} x \left (a+b x^4\right )^{7/4}+\frac{1}{64} \left (21 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )+\frac{1}{64} \left (21 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )\\ &=\frac{7}{32} a x \left (a+b x^4\right )^{3/4}+\frac{1}{8} x \left (a+b x^4\right )^{7/4}+\frac{21 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 \sqrt [4]{b}}+\frac{21 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 \sqrt [4]{b}}\\ \end{align*}

Mathematica [C]  time = 0.0106926, size = 47, normalized size = 0.49 \[ \frac{a x \left (a+b x^4\right )^{3/4} \, _2F_1\left (-\frac{7}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )}{\left (\frac{b x^4}{a}+1\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(7/4),x]

[Out]

(a*x*(a + b*x^4)^(3/4)*Hypergeometric2F1[-7/4, 1/4, 5/4, -((b*x^4)/a)])/(1 + (b*x^4)/a)^(3/4)

________________________________________________________________________________________

Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(7/4),x)

[Out]

int((b*x^4+a)^(7/4),x)

________________________________________________________________________________________

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)^(7/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.6182, size = 477, normalized size = 4.97 \begin{align*} \frac{1}{32} \,{\left (4 \, b x^{5} + 11 \, a x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}} + \frac{21}{32} \, \left (\frac{a^{8}}{b}\right )^{\frac{1}{4}} \arctan \left (-\frac{\left (\frac{a^{8}}{b}\right )^{\frac{1}{4}}{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6} - \left (\frac{a^{8}}{b}\right )^{\frac{1}{4}} x \sqrt{\frac{\sqrt{b x^{4} + a} a^{12} + \sqrt{\frac{a^{8}}{b}} a^{8} b x^{2}}{x^{2}}}}{a^{8} x}\right ) + \frac{21}{128} \, \left (\frac{a^{8}}{b}\right )^{\frac{1}{4}} \log \left (\frac{9261 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6} + \left (\frac{a^{8}}{b}\right )^{\frac{3}{4}} b x\right )}}{x}\right ) - \frac{21}{128} \, \left (\frac{a^{8}}{b}\right )^{\frac{1}{4}} \log \left (\frac{9261 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6} - \left (\frac{a^{8}}{b}\right )^{\frac{3}{4}} b x\right )}}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(4*b*x^5 + 11*a*x)*(b*x^4 + a)^(3/4) + 21/32*(a^8/b)^(1/4)*arctan(-((a^8/b)^(1/4)*(b*x^4 + a)^(1/4)*a^6 -
(a^8/b)^(1/4)*x*sqrt((sqrt(b*x^4 + a)*a^12 + sqrt(a^8/b)*a^8*b*x^2)/x^2))/(a^8*x)) + 21/128*(a^8/b)^(1/4)*log
(9261*((b*x^4 + a)^(1/4)*a^6 + (a^8/b)^(3/4)*b*x)/x) - 21/128*(a^8/b)^(1/4)*log(9261*((b*x^4 + a)^(1/4)*a^6 -
(a^8/b)^(3/4)*b*x)/x)

________________________________________________________________________________________

Sympy [C]  time = 4.06477, size = 37, normalized size = 0.39 \begin{align*} \frac{a^{\frac{7}{4}} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(7/4),x)

[Out]

a**(7/4)*x*gamma(1/4)*hyper((-7/4, 1/4), (5/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(5/4))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{4} + a\right )}^{\frac{7}{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^4 + a)^(7/4), x)