3.9.36 \(\int \frac {\sqrt [4]{b x^3+a x^4}}{x (b+a x+x^2)} \, dx\) [836]

Optimal. Leaf size=63 \[ -\text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-a+2 \text {$\#$1}^4}\& \right ] \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(182\) vs. \(2(63)=126\).
time = 0.33, antiderivative size = 182, normalized size of antiderivative = 2.89, number of steps used = 9, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2081, 925, 129, 525, 524} \begin {gather*} -\frac {8 \sqrt [4]{a x^4+b x^3} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 x}{a-\sqrt {a^2-4 b}},-\frac {a x}{b}\right )}{3 \left (-a \sqrt {a^2-4 b}+a^2-4 b\right ) \sqrt [4]{\frac {a x}{b}+1}}-\frac {8 \sqrt [4]{a x^4+b x^3} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {2 x}{a+\sqrt {a^2-4 b}},-\frac {a x}{b}\right )}{3 \left (a \sqrt {a^2-4 b}+a^2-4 b\right ) \sqrt [4]{\frac {a x}{b}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

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 925

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x
] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[m] &&  !IntegerQ[n]

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

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Mathematica [A]
time = 0.00, size = 92, normalized size = 1.46 \begin {gather*} -\frac {\sqrt [4]{x^3 (b+a x)} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt [4]{x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}}{-a+2 \text {$\#$1}^4}\&\right ]}{x^{3/4} \sqrt [4]{b+a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((x^3*(b + a*x))^(1/4)*RootSum[b - a*#1^4 + #1^8 & , (-(Log[x^(1/4)]*#1) + Log[(b + a*x)^(1/4) - x^(1/4)*#1]
*#1)/(-a + 2*#1^4) & ])/(x^(3/4)*(b + a*x)^(1/4)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.45, size = 1918, normalized size = 30.44 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(2)*sqrt(sqrt(2)*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8
*a^2*b + 16*b^2)))*arctan(-1/8*sqrt(2)*(sqrt(2)*(a*x^4 + b*x^3)^(1/4)*(a^4 - 8*a^2*b + 16*b^2 - (a^7 - 12*a^5*
b + 48*a^3*b^2 - 64*a*b^3)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/sqrt
(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)) - ((a^4 - 8*a^2*b + 16*b^2)*x - (a^7 - 12*a^
5*b + 48*a^3*b^2 - 64*a*b^3)*x/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/
sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2))*sqrt((sqrt(2)*(a^2 - 4*b)*x^2*sqrt((a +
(a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)) + 2*sqrt(a*x^4
+ b*x^3))/x^2))*sqrt(sqrt(2)*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a
^4 - 8*a^2*b + 16*b^2)))/(b*x)) - 2*sqrt(2)*sqrt(sqrt(2)*sqrt((a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*
b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))*arctan(1/8*sqrt(2)*(sqrt(2)*(a*x^4 + b*x^3)^(1/4)*(a^4 -
8*a^2*b + 16*b^2 + (a^7 - 12*a^5*b + 48*a^3*b^2 - 64*a*b^3)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))*sqrt((
a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)) - ((a^4 - 8
*a^2*b + 16*b^2)*x + (a^7 - 12*a^5*b + 48*a^3*b^2 - 64*a*b^3)*x/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))*sq
rt((a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2))*sqrt((s
qrt(2)*(a^2 - 4*b)*x^2*sqrt((a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8
*a^2*b + 16*b^2)) + 2*sqrt(a*x^4 + b*x^3))/x^2))*sqrt(sqrt(2)*sqrt((a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12
*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))/(b*x)) + 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((a + (a^4 - 8
*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))*log((sqrt(2)*(a^4 - 8*
a^2*b + 16*b^2)*x*sqrt(sqrt(2)*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/
(a^4 - 8*a^2*b + 16*b^2)))/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3) + 2*(a*x^4 + b*x^3)^(1/4))/x) - 1/2*sqrt
(2)*sqrt(sqrt(2)*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b
+ 16*b^2)))*log(-(sqrt(2)*(a^4 - 8*a^2*b + 16*b^2)*x*sqrt(sqrt(2)*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6
- 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3) - 2*
(a*x^4 + b*x^3)^(1/4))/x) - 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b +
48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))*log((sqrt(2)*(a^4 - 8*a^2*b + 16*b^2)*x*sqrt(sqrt(2)*sqrt((a
- (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))/sqrt(a^6 - 1
2*a^4*b + 48*a^2*b^2 - 64*b^3) + 2*(a*x^4 + b*x^3)^(1/4))/x) + 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((a - (a^4 - 8*a^2
*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))*log(-(sqrt(2)*(a^4 - 8*a^2
*b + 16*b^2)*x*sqrt(sqrt(2)*sqrt((a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^
4 - 8*a^2*b + 16*b^2)))/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3) - 2*(a*x^4 + b*x^3)^(1/4))/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (a\,x^4+b\,x^3\right )}^{1/4}}{x\,\left (x^2+a\,x+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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