3.19.25 \(\int \frac {(-d+2 c x) \sqrt [4]{b x^3+a x^4}}{x} \, dx\)

Optimal. Leaf size=124 \[ \frac {\left (4 a b d+3 b^2 c\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{8 a^{7/4}}+\frac {\left (-4 a b d-3 b^2 c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{8 a^{7/4}}+\frac {\sqrt [4]{a x^4+b x^3} (4 a c x-4 a d+b c)}{4 a} \]

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Rubi [A]  time = 0.29, antiderivative size = 193, normalized size of antiderivative = 1.56, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2039, 2021, 2032, 63, 331, 298, 203, 206} \begin {gather*} \frac {b x^{9/4} (a x+b)^{3/4} (4 a d+3 b c) \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{8 a^{7/4} \left (a x^4+b x^3\right )^{3/4}}-\frac {b x^{9/4} (a x+b)^{3/4} (4 a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{8 a^{7/4} \left (a x^4+b x^3\right )^{3/4}}-\frac {\sqrt [4]{a x^4+b x^3} (4 a d+3 b c)}{4 a}+\frac {c \left (a x^4+b x^3\right )^{5/4}}{a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((3*b*c + 4*a*d)*(b*x^3 + a*x^4)^(1/4))/a + (c*(b*x^3 + a*x^4)^(5/4))/(a*x^3) + (b*(3*b*c + 4*a*d)*x^(9/4
)*(b + a*x)^(3/4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(8*a^(7/4)*(b*x^3 + a*x^4)^(3/4)) - (b*(3*b*c + 4
*a*d)*x^(9/4)*(b + a*x)^(3/4)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(8*a^(7/4)*(b*x^3 + a*x^4)^(3/4))

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 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])

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 298

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

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]

Rule 2021

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

Rule 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[m]*(a*x^j + b*x^n)^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rule 2039

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

Rubi steps

\begin {align*} \int \frac {(-d+2 c x) \sqrt [4]{b x^3+a x^4}}{x} \, dx &=\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}-\frac {\left (\frac {3 b c}{2}+2 a d\right ) \int \frac {\sqrt [4]{b x^3+a x^4}}{x} \, dx}{2 a}\\ &=-\frac {(3 b c+4 a d) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}-\frac {(b (3 b c+4 a d)) \int \frac {x^2}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{16 a}\\ &=-\frac {(3 b c+4 a d) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}-\frac {\left (b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{16 a \left (b x^3+a x^4\right )^{3/4}}\\ &=-\frac {(3 b c+4 a d) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}-\frac {\left (b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{4 a \left (b x^3+a x^4\right )^{3/4}}\\ &=-\frac {(3 b c+4 a d) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}-\frac {\left (b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4 a \left (b x^3+a x^4\right )^{3/4}}\\ &=-\frac {(3 b c+4 a d) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}-\frac {\left (b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{3/2} \left (b x^3+a x^4\right )^{3/4}}+\frac {\left (b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{3/2} \left (b x^3+a x^4\right )^{3/4}}\\ &=-\frac {(3 b c+4 a d) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {c \left (b x^3+a x^4\right )^{5/4}}{a x^3}+\frac {b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{7/4} \left (b x^3+a x^4\right )^{3/4}}-\frac {b (3 b c+4 a d) x^{9/4} (b+a x)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a^{7/4} \left (b x^3+a x^4\right )^{3/4}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 81, normalized size = 0.65 \begin {gather*} \frac {\sqrt [4]{x^3 (a x+b)} \left (3 c (a x+b) \sqrt [4]{\frac {a x}{b}+1}-(4 a d+3 b c) \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-\frac {a x}{b}\right )\right )}{3 a \sqrt [4]{\frac {a x}{b}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.67, size = 124, normalized size = 1.00 \begin {gather*} \frac {(b c-4 a d+4 a c x) \sqrt [4]{b x^3+a x^4}}{4 a}+\frac {\left (3 b^2 c+4 a b d\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{8 a^{7/4}}+\frac {\left (-3 b^2 c-4 a b d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{8 a^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - 4*a*d + 4*a*c*x)*(b*x^3 + a*x^4)^(1/4))/(4*a) + ((3*b^2*c + 4*a*b*d)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4
)^(1/4)])/(8*a^(7/4)) + ((-3*b^2*c - 4*a*b*d)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(8*a^(7/4))

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fricas [B]  time = 0.48, size = 771, normalized size = 6.22 \begin {gather*} \frac {4 \, a \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {1}{4}} \arctan \left (\frac {a^{5} x \sqrt {\frac {a^{4} x^{2} \sqrt {\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}} + {\left (9 \, b^{4} c^{2} + 24 \, a b^{3} c d + 16 \, a^{2} b^{2} d^{2}\right )} \sqrt {a x^{4} + b x^{3}}}{x^{2}}} \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {3}{4}} - {\left (3 \, a^{5} b^{2} c + 4 \, a^{6} b d\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {3}{4}}}{{\left (81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}\right )} x}\right ) - a \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {1}{4}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (3 \, b^{2} c + 4 \, a b d\right )}}{x}\right ) + a \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{2} x \left (\frac {81 \, b^{8} c^{4} + 432 \, a b^{7} c^{3} d + 864 \, a^{2} b^{6} c^{2} d^{2} + 768 \, a^{3} b^{5} c d^{3} + 256 \, a^{4} b^{4} d^{4}}{a^{7}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (3 \, b^{2} c + 4 \, a b d\right )}}{x}\right ) + 4 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (4 \, a c x + b c - 4 \, a d\right )}}{16 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*a*((81*b^8*c^4 + 432*a*b^7*c^3*d + 864*a^2*b^6*c^2*d^2 + 768*a^3*b^5*c*d^3 + 256*a^4*b^4*d^4)/a^7)^(1/
4)*arctan((a^5*x*sqrt((a^4*x^2*sqrt((81*b^8*c^4 + 432*a*b^7*c^3*d + 864*a^2*b^6*c^2*d^2 + 768*a^3*b^5*c*d^3 +
256*a^4*b^4*d^4)/a^7) + (9*b^4*c^2 + 24*a*b^3*c*d + 16*a^2*b^2*d^2)*sqrt(a*x^4 + b*x^3))/x^2)*((81*b^8*c^4 + 4
32*a*b^7*c^3*d + 864*a^2*b^6*c^2*d^2 + 768*a^3*b^5*c*d^3 + 256*a^4*b^4*d^4)/a^7)^(3/4) - (3*a^5*b^2*c + 4*a^6*
b*d)*(a*x^4 + b*x^3)^(1/4)*((81*b^8*c^4 + 432*a*b^7*c^3*d + 864*a^2*b^6*c^2*d^2 + 768*a^3*b^5*c*d^3 + 256*a^4*
b^4*d^4)/a^7)^(3/4))/((81*b^8*c^4 + 432*a*b^7*c^3*d + 864*a^2*b^6*c^2*d^2 + 768*a^3*b^5*c*d^3 + 256*a^4*b^4*d^
4)*x)) - a*((81*b^8*c^4 + 432*a*b^7*c^3*d + 864*a^2*b^6*c^2*d^2 + 768*a^3*b^5*c*d^3 + 256*a^4*b^4*d^4)/a^7)^(1
/4)*log((a^2*x*((81*b^8*c^4 + 432*a*b^7*c^3*d + 864*a^2*b^6*c^2*d^2 + 768*a^3*b^5*c*d^3 + 256*a^4*b^4*d^4)/a^7
)^(1/4) + (a*x^4 + b*x^3)^(1/4)*(3*b^2*c + 4*a*b*d))/x) + a*((81*b^8*c^4 + 432*a*b^7*c^3*d + 864*a^2*b^6*c^2*d
^2 + 768*a^3*b^5*c*d^3 + 256*a^4*b^4*d^4)/a^7)^(1/4)*log(-(a^2*x*((81*b^8*c^4 + 432*a*b^7*c^3*d + 864*a^2*b^6*
c^2*d^2 + 768*a^3*b^5*c*d^3 + 256*a^4*b^4*d^4)/a^7)^(1/4) - (a*x^4 + b*x^3)^(1/4)*(3*b^2*c + 4*a*b*d))/x) + 4*
(a*x^4 + b*x^3)^(1/4)*(4*a*c*x + b*c - 4*a*d))/a

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giac [B]  time = 0.22, size = 322, normalized size = 2.60 \begin {gather*} \frac {\frac {2 \, \sqrt {2} {\left (3 \, b^{3} c + 4 \, a b^{2} d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a} + \frac {2 \, \sqrt {2} {\left (3 \, b^{3} c + 4 \, a b^{2} d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a} + \frac {\sqrt {2} {\left (3 \, b^{3} c + 4 \, a b^{2} d\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a} - \frac {\sqrt {2} {\left (3 \, b^{3} c + 4 \, a b^{2} d\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a} + \frac {8 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} b^{3} c + 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a b^{3} c - 4 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a b^{2} d + 4 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{2} b^{2} d\right )} x^{2}}{a b^{2}}}{32 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(2*sqrt(2)*(3*b^3*c + 4*a*b^2*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/4))/
((-a)^(3/4)*a) + 2*sqrt(2)*(3*b^3*c + 4*a*b^2*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a + b/x)^(1/4))/
(-a)^(1/4))/((-a)^(3/4)*a) + sqrt(2)*(3*b^3*c + 4*a*b^2*d)*log(sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) +
sqrt(a + b/x))/((-a)^(3/4)*a) - sqrt(2)*(3*b^3*c + 4*a*b^2*d)*log(-sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(
-a) + sqrt(a + b/x))/((-a)^(3/4)*a) + 8*((a + b/x)^(5/4)*b^3*c + 3*(a + b/x)^(1/4)*a*b^3*c - 4*(a + b/x)^(5/4)
*a*b^2*d + 4*(a + b/x)^(1/4)*a^2*b^2*d)*x^2/(a*b^2))/b

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (2 c x -d \right ) \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (2 \, c x - d\right )}}{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-int(((a*x^4 + b*x^3)^(1/4)*(d - 2*c*x))/x, 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 )} \left (2 c x - d\right )}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x**3*(a*x + b))**(1/4)*(2*c*x - d)/x, x)

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