3.2.36 \(\int \frac {(3 a+b x^2)^3}{(a-b x^2)^{7/3}} \, dx\) [136]

Optimal. Leaf size=596 \[ -\frac {27}{14} x \left (a-b x^2\right )^{2/3}+\frac {3 x \left (3 a+b x^2\right )^2}{2 \left (a-b x^2\right )^{4/3}}-\frac {324 a x}{7 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {162 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {108 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]

[Out]

-27/14*x*(-b*x^2+a)^(2/3)+3/2*x*(b*x^2+3*a)^2/(-b*x^2+a)^(4/3)-324/7*a*x/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)
))+108/7*3^(3/4)*a^(4/3)*(a^(1/3)-(-b*x^2+a)^(1/3))*EllipticF((-(-b*x^2+a)^(1/3)+a^(1/3)*(1+3^(1/2)))/(-(-b*x^
2+a)^(1/3)+a^(1/3)*(1-3^(1/2))),2*I-I*3^(1/2))*2^(1/2)*((a^(2/3)+a^(1/3)*(-b*x^2+a)^(1/3)+(-b*x^2+a)^(2/3))/(-
(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)/b/x/(-a^(1/3)*(a^(1/3)-(-b*x^2+a)^(1/3))/(-(-b*x^2+a)^(1/3)+a^(
1/3)*(1-3^(1/2)))^2)^(1/2)-162/7*3^(1/4)*a^(4/3)*(a^(1/3)-(-b*x^2+a)^(1/3))*EllipticE((-(-b*x^2+a)^(1/3)+a^(1/
3)*(1+3^(1/2)))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2))),2*I-I*3^(1/2))*((a^(2/3)+a^(1/3)*(-b*x^2+a)^(1/3)+(-b*
x^2+a)^(2/3))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))/b/x/(-a^(1/3)*(a^(1/3
)-(-b*x^2+a)^(1/3))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)

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Rubi [A]
time = 0.37, antiderivative size = 596, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {424, 21, 396, 241, 310, 225, 1893} \begin {gather*} \frac {108 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\text {ArcSin}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {162 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\text {ArcSin}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {3 x \left (3 a+b x^2\right )^2}{2 \left (a-b x^2\right )^{4/3}}-\frac {27}{14} x \left (a-b x^2\right )^{2/3}-\frac {324 a x}{7 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3*a + b*x^2)^3/(a - b*x^2)^(7/3),x]

[Out]

(-27*x*(a - b*x^2)^(2/3))/14 + (3*x*(3*a + b*x^2)^2)/(2*(a - b*x^2)^(4/3)) - (324*a*x)/(7*((1 - Sqrt[3])*a^(1/
3) - (a - b*x^2)^(1/3))) - (162*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3)
+ a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcS
in[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/
(7*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + (108*
Sqrt[2]*3^(3/4)*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^
(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1
/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(7*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x
^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 225

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r
*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 241

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

Rule 310

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 + Sqrt[3]))*(s/r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x]
, x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 396

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

Rule 424

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a*d - c*b)*x*(a + b*x^n)^(
p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 1))), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rule 1893

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[(1 + Sqrt[3])*(d/c)]]
, s = Denom[Simplify[(1 + Sqrt[3])*(d/c)]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x
] + Simp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(r^2
*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/(
(1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {\left (3 a+b x^2\right )^3}{\left (a-b x^2\right )^{7/3}} \, dx &=\frac {3 x \left (3 a+b x^2\right )^2}{2 \left (a-b x^2\right )^{4/3}}-\frac {3 \int \frac {\left (3 a+b x^2\right ) \left (-12 a^2 b+12 a b^2 x^2\right )}{\left (a-b x^2\right )^{4/3}} \, dx}{8 a b}\\ &=\frac {3 x \left (3 a+b x^2\right )^2}{2 \left (a-b x^2\right )^{4/3}}+\frac {9}{2} \int \frac {3 a+b x^2}{\sqrt [3]{a-b x^2}} \, dx\\ &=-\frac {27}{14} x \left (a-b x^2\right )^{2/3}+\frac {3 x \left (3 a+b x^2\right )^2}{2 \left (a-b x^2\right )^{4/3}}+\frac {1}{7} (108 a) \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx\\ &=-\frac {27}{14} x \left (a-b x^2\right )^{2/3}+\frac {3 x \left (3 a+b x^2\right )^2}{2 \left (a-b x^2\right )^{4/3}}-\frac {\left (162 a \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{7 b x}\\ &=-\frac {27}{14} x \left (a-b x^2\right )^{2/3}+\frac {3 x \left (3 a+b x^2\right )^2}{2 \left (a-b x^2\right )^{4/3}}+\frac {\left (162 a \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{7 b x}-\frac {\left (162 \sqrt {2 \left (2+\sqrt {3}\right )} a^{4/3} \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{7 b x}\\ &=-\frac {27}{14} x \left (a-b x^2\right )^{2/3}+\frac {3 x \left (3 a+b x^2\right )^2}{2 \left (a-b x^2\right )^{4/3}}-\frac {324 a x}{7 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {162 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {108 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 15.06, size = 83, normalized size = 0.14 \begin {gather*} \frac {81 a^2 x+90 a b x^3-3 b^2 x^5+108 a x \left (a-b x^2\right ) \sqrt [3]{1-\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\frac {b x^2}{a}\right )}{7 \left (a-b x^2\right )^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*a + b*x^2)^3/(a - b*x^2)^(7/3),x]

[Out]

(81*a^2*x + 90*a*b*x^3 - 3*b^2*x^5 + 108*a*x*(a - b*x^2)*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2
, (b*x^2)/a])/(7*(a - b*x^2)^(4/3))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{2}+3 a \right )^{3}}{\left (-b \,x^{2}+a \right )^{\frac {7}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+3*a)^3/(-b*x^2+a)^(7/3),x)

[Out]

int((b*x^2+3*a)^3/(-b*x^2+a)^(7/3),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((b*x^2+3*a)^3/(-b*x^2+a)^(7/3),x, algorithm="maxima")

[Out]

integrate((b*x^2 + 3*a)^3/(-b*x^2 + a)^(7/3), x)

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Fricas [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((b*x^2+3*a)^3/(-b*x^2+a)^(7/3),x, algorithm="fricas")

[Out]

integral(-(b^3*x^6 + 9*a*b^2*x^4 + 27*a^2*b*x^2 + 27*a^3)*(-b*x^2 + a)^(2/3)/(b^3*x^6 - 3*a*b^2*x^4 + 3*a^2*b*
x^2 - a^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 a + b x^{2}\right )^{3}}{\left (a - b x^{2}\right )^{\frac {7}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+3*a)**3/(-b*x**2+a)**(7/3),x)

[Out]

Integral((3*a + b*x**2)**3/(a - b*x**2)**(7/3), 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((b*x^2+3*a)^3/(-b*x^2+a)^(7/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + 3*a)^3/(-b*x^2 + a)^(7/3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (b\,x^2+3\,a\right )}^3}{{\left (a-b\,x^2\right )}^{7/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*a + b*x^2)^3/(a - b*x^2)^(7/3),x)

[Out]

int((3*a + b*x^2)^3/(a - b*x^2)^(7/3), x)

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