3.18.88 \(\int \frac {1}{(-b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx\) [1788]

Optimal. Leaf size=121 \[ \frac {\text {ArcTan}\left (\frac {2^{3/4} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{b^2 x+a^2 x^3}}{b+a x}\right )}{2 \sqrt [4]{2} a^{3/4} b^{3/4}}-\frac {\tanh ^{-1}\left (\frac {2^{3/4} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{b^2 x+a^2 x^3}}{b+a x}\right )}{2 \sqrt [4]{2} a^{3/4} b^{3/4}} \]

[Out]

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

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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 0.19, antiderivative size = 150, normalized size of antiderivative = 1.24, number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2081, 973, 477, 525, 524} \begin {gather*} -\frac {4 a x^2 \sqrt [4]{\frac {a^2 x^2}{b^2}+1} F_1\left (\frac {7}{8};1,\frac {1}{4};\frac {15}{8};\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{7 b^2 \sqrt [4]{a^2 x^3+b^2 x}}-\frac {4 x \sqrt [4]{\frac {a^2 x^2}{b^2}+1} F_1\left (\frac {3}{8};1,\frac {1}{4};\frac {11}{8};\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{3 b \sqrt [4]{a^2 x^3+b^2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*x*(1 + (a^2*x^2)/b^2)^(1/4)*AppellF1[3/8, 1, 1/4, 11/8, (a^2*x^2)/b^2, -((a^2*x^2)/b^2)])/(3*b*(b^2*x + a^
2*x^3)^(1/4)) - (4*a*x^2*(1 + (a^2*x^2)/b^2)^(1/4)*AppellF1[7/8, 1, 1/4, 15/8, (a^2*x^2)/b^2, -((a^2*x^2)/b^2)
])/(7*b^2*(b^2*x + a^2*x^3)^(1/4))

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 973

Int[(((g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d*((g*x)^n/x^n), In
t[(x^n*(a + c*x^2)^p)/(d^2 - e^2*x^2), x], x] - Dist[e*((g*x)^n/x^n), Int[(x^(n + 1)*(a + c*x^2)^p)/(d^2 - e^2
*x^2), x], x] /; FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] &&  !IntegersQ[n, 2
*p]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps

\begin {align*} \int \frac {1}{(-b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x) \sqrt [4]{b^2+a^2 x^2}} \, dx}{\sqrt [4]{b^2 x+a^2 x^3}}\\ &=-\frac {\left (a \sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \int \frac {x^{3/4}}{\left (b^2-a^2 x^2\right ) \sqrt [4]{b^2+a^2 x^2}} \, dx}{\sqrt [4]{b^2 x+a^2 x^3}}-\frac {\left (b \sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \int \frac {1}{\sqrt [4]{x} \left (b^2-a^2 x^2\right ) \sqrt [4]{b^2+a^2 x^2}} \, dx}{\sqrt [4]{b^2 x+a^2 x^3}}\\ &=-\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^6}{\left (b^2-a^2 x^8\right ) \sqrt [4]{b^2+a^2 x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b^2 x+a^2 x^3}}-\frac {\left (4 b \sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b^2-a^2 x^8\right ) \sqrt [4]{b^2+a^2 x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b^2 x+a^2 x^3}}\\ &=-\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{1+\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {x^6}{\left (b^2-a^2 x^8\right ) \sqrt [4]{1+\frac {a^2 x^8}{b^2}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b^2 x+a^2 x^3}}-\frac {\left (4 b \sqrt [4]{x} \sqrt [4]{1+\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b^2-a^2 x^8\right ) \sqrt [4]{1+\frac {a^2 x^8}{b^2}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b^2 x+a^2 x^3}}\\ &=-\frac {4 x \sqrt [4]{1+\frac {a^2 x^2}{b^2}} F_1\left (\frac {3}{8};1,\frac {1}{4};\frac {11}{8};\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{3 b \sqrt [4]{b^2 x+a^2 x^3}}-\frac {4 a x^2 \sqrt [4]{1+\frac {a^2 x^2}{b^2}} F_1\left (\frac {7}{8};1,\frac {1}{4};\frac {15}{8};\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{7 b^2 \sqrt [4]{b^2 x+a^2 x^3}}\\ \end {align*}

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Mathematica [F]
time = 20.45, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(-b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a x -b \right ) \left (a^{2} x^{3}+b^{2} x \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x-b)/(a^2*x^3+b^2*x)^(1/4),x)

[Out]

int(1/(a*x-b)/(a^2*x^3+b^2*x)^(1/4),x)

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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/(a*x-b)/(a^2*x^3+b^2*x)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((a^2*x^3 + b^2*x)^(1/4)*(a*x - b)), x)

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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(1/(a*x-b)/(a^2*x^3+b^2*x)^(1/4),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 {1}{\sqrt [4]{x \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-b)/(a**2*x**3+b**2*x)**(1/4),x)

[Out]

Integral(1/((x*(a**2*x**2 + b**2))**(1/4)*(a*x - b)), x)

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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/(a*x-b)/(a^2*x^3+b^2*x)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((a^2*x^3 + b^2*x)^(1/4)*(a*x - b)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((b^2*x + a^2*x^3)^(1/4)*(b - a*x)),x)

[Out]

-int(1/((b^2*x + a^2*x^3)^(1/4)*(b - a*x)), x)

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