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

Optimal. Leaf size=187 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8 (-d)+2 \text {$\#$1}^4 a d-a^2 d+b^2 c\& ,\frac {-\text {$\#$1}^4 a d \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 a d \log (x)+a^2 d \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-b^2 c \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-a^2 d \log (x)+b^2 c \log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]}{2 d^2}+\frac {4 \sqrt [4]{a x^4+b x^3}}{d x} \]

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Rubi [B]  time = 1.13, antiderivative size = 406, normalized size of antiderivative = 2.17, number of steps used = 13, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2056, 908, 37, 6725, 93, 298, 205, 208} \begin {gather*} \frac {\sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{a x+b}}+\frac {\sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {c}} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{a x+b}}-\frac {\sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{a x+b}}-\frac {\sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{d^{9/8} x^{3/4} \sqrt [4]{a x+b}}+\frac {4 \sqrt [4]{a x^4+b x^3}}{d x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 205

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

Rule 208

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

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 908

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

Rule 2056

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]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 137, normalized size = 0.73 \begin {gather*} \frac {2 x^2 \left (x \left (b \sqrt {c}-a \sqrt {d}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (a-\frac {b \sqrt {c}}{\sqrt {d}}\right ) x}{b+a x}\right )-x \left (a \sqrt {d}+b \sqrt {c}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (a+\frac {b \sqrt {c}}{\sqrt {d}}\right ) x}{b+a x}\right )+6 \sqrt {d} (a x+b)\right )}{3 d^{3/2} \left (x^3 (a x+b)\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^2*(6*Sqrt[d]*(b + a*x) + (b*Sqrt[c] - a*Sqrt[d])*x*Hypergeometric2F1[3/4, 1, 7/4, ((a - (b*Sqrt[c])/Sqrt[
d])*x)/(b + a*x)] - (b*Sqrt[c] + a*Sqrt[d])*x*Hypergeometric2F1[3/4, 1, 7/4, ((a + (b*Sqrt[c])/Sqrt[d])*x)/(b
+ a*x)]))/(3*d^(3/2)*(x^3*(b + a*x))^(3/4))

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IntegrateAlgebraic [A]  time = 0.00, size = 186, normalized size = 0.99 \begin {gather*} \frac {4 \sqrt [4]{b x^3+a x^4}}{d x}+\frac {\text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {-b^2 c \log (x)+a^2 d \log (x)+b^2 c \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-a^2 d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-a d \log (x) \text {$\#$1}^4+a d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(b*x^3 + a*x^4)^(1/4))/(d*x) + RootSum[b^2*c - a^2*d + 2*a*d*#1^4 - d*#1^8 & , (-(b^2*c*Log[x]) + a^2*d*Log
[x] + b^2*c*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a^2*d*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a*d*Log[x]*#1^4 + a*
d*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/(2*d^2)

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fricas [B]  time = 0.53, size = 730, normalized size = 3.90 \begin {gather*} -\frac {4 \, d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (d^{8} x \sqrt {\frac {b^{2} c}{d^{9}}} - a d^{4} x\right )} \sqrt {\frac {d^{2} x^{2} \sqrt {\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}} + \sqrt {a x^{4} + b x^{3}}}{x^{2}}} \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {3}{4}} - {\left (d^{8} \sqrt {\frac {b^{2} c}{d^{9}}} - a d^{4}\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {3}{4}}}{{\left (b^{2} c - a^{2} d\right )} x}\right ) - 4 \, d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (d^{8} x \sqrt {\frac {b^{2} c}{d^{9}}} + a d^{4} x\right )} \sqrt {\frac {d^{2} x^{2} \sqrt {-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}} + \sqrt {a x^{4} + b x^{3}}}{x^{2}}} \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {3}{4}} - {\left (d^{8} \sqrt {\frac {b^{2} c}{d^{9}}} + a d^{4}\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {3}{4}}}{{\left (b^{2} c - a^{2} d\right )} x}\right ) + d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {d x \left (\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} + a}{d^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {d x \left (-\frac {d^{4} \sqrt {\frac {b^{2} c}{d^{9}}} - a}{d^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 8 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{2 \, d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*d*x*((d^4*sqrt(b^2*c/d^9) + a)/d^4)^(1/4)*arctan(-((d^8*x*sqrt(b^2*c/d^9) - a*d^4*x)*sqrt((d^2*x^2*sqr
t((d^4*sqrt(b^2*c/d^9) + a)/d^4) + sqrt(a*x^4 + b*x^3))/x^2)*((d^4*sqrt(b^2*c/d^9) + a)/d^4)^(3/4) - (d^8*sqrt
(b^2*c/d^9) - a*d^4)*(a*x^4 + b*x^3)^(1/4)*((d^4*sqrt(b^2*c/d^9) + a)/d^4)^(3/4))/((b^2*c - a^2*d)*x)) - 4*d*x
*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(1/4)*arctan(-((d^8*x*sqrt(b^2*c/d^9) + a*d^4*x)*sqrt((d^2*x^2*sqrt(-(d^4*sq
rt(b^2*c/d^9) - a)/d^4) + sqrt(a*x^4 + b*x^3))/x^2)*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(3/4) - (d^8*sqrt(b^2*c/d
^9) + a*d^4)*(a*x^4 + b*x^3)^(1/4)*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(3/4))/((b^2*c - a^2*d)*x)) + d*x*((d^4*sq
rt(b^2*c/d^9) + a)/d^4)^(1/4)*log((d*x*((d^4*sqrt(b^2*c/d^9) + a)/d^4)^(1/4) + (a*x^4 + b*x^3)^(1/4))/x) - d*x
*((d^4*sqrt(b^2*c/d^9) + a)/d^4)^(1/4)*log(-(d*x*((d^4*sqrt(b^2*c/d^9) + a)/d^4)^(1/4) - (a*x^4 + b*x^3)^(1/4)
)/x) + d*x*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(1/4)*log((d*x*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(1/4) + (a*x^4 + b
*x^3)^(1/4))/x) - d*x*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(1/4)*log(-(d*x*(-(d^4*sqrt(b^2*c/d^9) - a)/d^4)^(1/4)
- (a*x^4 + b*x^3)^(1/4))/x) - 8*(a*x^4 + b*x^3)^(1/4))/(d*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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