3.112 \(\int \frac {(a+b x^3)^{5/3}}{(c+d x^3)^3} \, dx\)

Optimal. Leaf size=217 \[ \frac {5 a^2 \log \left (c+d x^3\right )}{54 c^{8/3} \sqrt [3]{b c-a d}}-\frac {5 a^2 \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} \sqrt [3]{b c-a d}}+\frac {5 a^2 \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} \sqrt [3]{b c-a d}}+\frac {5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}+\frac {x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2} \]

[Out]

1/6*x*(b*x^3+a)^(5/3)/c/(d*x^3+c)^2+5/18*a*x*(b*x^3+a)^(2/3)/c^2/(d*x^3+c)+5/54*a^2*ln(d*x^3+c)/c^(8/3)/(-a*d+
b*c)^(1/3)-5/18*a^2*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(8/3)/(-a*d+b*c)^(1/3)+5/27*a^2*arctan(1/
3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))/c^(8/3)/(-a*d+b*c)^(1/3)*3^(1/2)

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Rubi [A]  time = 0.24, antiderivative size = 276, normalized size of antiderivative = 1.27, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {378, 377, 200, 31, 634, 617, 204, 628} \[ -\frac {5 a^2 \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} \sqrt [3]{b c-a d}}+\frac {5 a^2 \log \left (\frac {x^2 (b c-a d)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+c^{2/3}\right )}{54 c^{8/3} \sqrt [3]{b c-a d}}+\frac {5 a^2 \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+\sqrt [3]{c}}{\sqrt {3} \sqrt [3]{c}}\right )}{9 \sqrt {3} c^{8/3} \sqrt [3]{b c-a d}}+\frac {5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}+\frac {x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(5/3)/(c + d*x^3)^3,x]

[Out]

(x*(a + b*x^3)^(5/3))/(6*c*(c + d*x^3)^2) + (5*a*x*(a + b*x^3)^(2/3))/(18*c^2*(c + d*x^3)) + (5*a^2*ArcTan[(c^
(1/3) + (2*(b*c - a*d)^(1/3)*x)/(a + b*x^3)^(1/3))/(Sqrt[3]*c^(1/3))])/(9*Sqrt[3]*c^(8/3)*(b*c - a*d)^(1/3)) -
(5*a^2*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(a + b*x^3)^(1/3)])/(27*c^(8/3)*(b*c - a*d)^(1/3)) + (5*a^2*Log[c^
(2/3) + ((b*c - a*d)^(2/3)*x^2)/(a + b*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(a + b*x^3)^(1/3)])/(54*c^(8
/3)*(b*c - a*d)^(1/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 377

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

Rule 378

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

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^{5/3}}{\left (c+d x^3\right )^3} \, dx &=\frac {x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac {(5 a) \int \frac {\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^2} \, dx}{6 c}\\ &=\frac {x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac {5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}+\frac {\left (5 a^2\right ) \int \frac {1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{9 c^2}\\ &=\frac {x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac {5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}+\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{9 c^2}\\ &=\frac {x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac {5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}+\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{c}-\sqrt [3]{b c-a d} x} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3}}+\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {2 \sqrt [3]{c}+\sqrt [3]{b c-a d} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3}}\\ &=\frac {x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac {5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}-\frac {5 a^2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} \sqrt [3]{b c-a d}}+\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{18 c^{7/3}}+\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{c} \sqrt [3]{b c-a d}+2 (b c-a d)^{2/3} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{54 c^{8/3} \sqrt [3]{b c-a d}}\\ &=\frac {x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac {5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}-\frac {5 a^2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} \sqrt [3]{b c-a d}}+\frac {5 a^2 \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{54 c^{8/3} \sqrt [3]{b c-a d}}-\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{9 c^{8/3} \sqrt [3]{b c-a d}}\\ &=\frac {x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac {5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}+\frac {5 a^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} \sqrt [3]{b c-a d}}-\frac {5 a^2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} \sqrt [3]{b c-a d}}+\frac {5 a^2 \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{54 c^{8/3} \sqrt [3]{b c-a d}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 79, normalized size = 0.36 \[ \frac {a x \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac {5}{3},\frac {1}{3};\frac {4}{3};\frac {(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{c^3 \left (\frac {b x^3}{a}+1\right )^{2/3} \sqrt [3]{\frac {d x^3}{c}+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(5/3)/(c + d*x^3)^3,x]

[Out]

(a*x*(a + b*x^3)^(2/3)*Hypergeometric2F1[-5/3, 1/3, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(c^3*(1 + (b*x
^3)/a)^(2/3)*(1 + (d*x^3)/c)^(1/3))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(5/3)/(d*x^3+c)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}}}{{\left (d x^{3} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(5/3)/(d*x^3+c)^3,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(5/3)/(d*x^3 + c)^3, x)

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maple [F]  time = 0.57, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {5}{3}}}{\left (d \,x^{3}+c \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(5/3)/(d*x^3+c)^3,x)

[Out]

int((b*x^3+a)^(5/3)/(d*x^3+c)^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}}}{{\left (d x^{3} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(5/3)/(d*x^3+c)^3,x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(5/3)/(d*x^3 + c)^3, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,x^3+a\right )}^{5/3}}{{\left (d\,x^3+c\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(5/3)/(c + d*x^3)^3,x)

[Out]

int((a + b*x^3)^(5/3)/(c + d*x^3)^3, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(5/3)/(d*x**3+c)**3,x)

[Out]

Timed out

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