3.135 \(\int \frac{(3 a+b x^2)^4}{(a-b x^2)^{7/3}} \, dx\)

Optimal. Leaf size=653 \[ \frac{12312 \sqrt{2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),4 \sqrt{3}-7\right )}{91 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{36936 a^2 x}{91 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{18468 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{91 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac{9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}-\frac{81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{3240}{91} a x \left (a-b x^2\right )^{2/3} \]

[Out]

(-3240*a*x*(a - b*x^2)^(2/3))/91 - (81*x*(a - b*x^2)^(2/3)*(3*a + b*x^2))/13 - (9*x*(3*a + b*x^2)^2)/(2*(a - b
*x^2)^(1/3)) + (3*x*(3*a + b*x^2)^3)/(2*(a - b*x^2)^(4/3)) - (36936*a^2*x)/(91*((1 - Sqrt[3])*a^(1/3) - (a - b
*x^2)^(1/3))) - (18468*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(7/3)*(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[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]])/(91*b*x*S
qrt[-((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)]) + (12312*Sqrt[2
]*3^(3/4)*a^(7/3)*(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]])/(91*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)])

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Rubi [A]  time = 0.512337, antiderivative size = 653, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {413, 526, 528, 388, 235, 304, 219, 1879} \[ -\frac{36936 a^2 x}{91 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{12312 \sqrt{2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{91 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{18468 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{91 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac{9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}-\frac{81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{3240}{91} a x \left (a-b x^2\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[(3*a + b*x^2)^4/(a - b*x^2)^(7/3),x]

[Out]

(-3240*a*x*(a - b*x^2)^(2/3))/91 - (81*x*(a - b*x^2)^(2/3)*(3*a + b*x^2))/13 - (9*x*(3*a + b*x^2)^2)/(2*(a - b
*x^2)^(1/3)) + (3*x*(3*a + b*x^2)^3)/(2*(a - b*x^2)^(4/3)) - (36936*a^2*x)/(91*((1 - Sqrt[3])*a^(1/3) - (a - b
*x^2)^(1/3))) - (18468*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(7/3)*(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[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]])/(91*b*x*S
qrt[-((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)]) + (12312*Sqrt[2
]*3^(3/4)*a^(7/3)*(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]])/(91*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 413

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 526

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

Rule 528

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

Rule 388

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 235

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 304

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, -Dist[(S
qrt[2]*s)/(Sqrt[2 - Sqrt[3]]*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 219

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

Rule 1879

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]*Elli
pticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[-((s
*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), 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 )^4}{\left (a-b x^2\right )^{7/3}} \, dx &=\frac{3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac{3 \int \frac{\left (3 a+b x^2\right )^2 \left (-12 a^2 b+20 a b^2 x^2\right )}{\left (a-b x^2\right )^{4/3}} \, dx}{8 a b}\\ &=-\frac{9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac{3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac{9 \int \frac{\left (3 a+b x^2\right ) \left (-48 a^3 b^2-48 a^2 b^3 x^2\right )}{\sqrt [3]{a-b x^2}} \, dx}{16 a^2 b^2}\\ &=-\frac{81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac{3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}+\frac{27 \int \frac{768 a^4 b^3+640 a^3 b^4 x^2}{\sqrt [3]{a-b x^2}} \, dx}{208 a^2 b^3}\\ &=-\frac{3240}{91} a x \left (a-b x^2\right )^{2/3}-\frac{81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac{3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}+\frac{1}{91} \left (12312 a^2\right ) \int \frac{1}{\sqrt [3]{a-b x^2}} \, dx\\ &=-\frac{3240}{91} a x \left (a-b x^2\right )^{2/3}-\frac{81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac{3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac{\left (18468 a^2 \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{91 b x}\\ &=-\frac{3240}{91} a x \left (a-b x^2\right )^{2/3}-\frac{81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac{3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}+\frac{\left (18468 a^2 \sqrt{-b x^2}\right ) \operatorname{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 )}{91 b x}-\frac{\left (18468 \sqrt{2 \left (2+\sqrt{3}\right )} a^{7/3} \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{91 b x}\\ &=-\frac{3240}{91} a x \left (a-b x^2\right )^{2/3}-\frac{81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac{3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac{36936 a^2 x}{91 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{18468 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \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 )}{91 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{12312 \sqrt{2} 3^{3/4} a^{7/3} \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 )}{91 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*}

Mathematica [C]  time = 5.07125, size = 96, normalized size = 0.15 \[ -\frac{3 \left (-4104 a^2 x \left (a-b x^2\right ) \sqrt [3]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )-4743 a^2 b x^3+1647 a^3 x+177 a b^2 x^5+7 b^3 x^7\right )}{91 \left (a-b x^2\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(3*a + b*x^2)^4/(a - b*x^2)^(7/3),x]

[Out]

(-3*(1647*a^3*x - 4743*a^2*b*x^3 + 177*a*b^2*x^5 + 7*b^3*x^7 - 4104*a^2*x*(a - b*x^2)*(1 - (b*x^2)/a)^(1/3)*Hy
pergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(91*(a - b*x^2)^(4/3))

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{2}+3\,a \right ) ^{4} \left ( -b{x}^{2}+a \right ) ^{-{\frac{7}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+3*a)^4/(-b*x^2+a)^(7/3),x)

[Out]

int((b*x^2+3*a)^4/(-b*x^2+a)^(7/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac{7}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+3*a)^4/(-b*x^2+a)^(7/3),x, algorithm="maxima")

[Out]

integrate((b*x^2 + 3*a)^4/(-b*x^2 + a)^(7/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (b^{4} x^{8} + 12 \, a b^{3} x^{6} + 54 \, a^{2} b^{2} x^{4} + 108 \, a^{3} b x^{2} + 81 \, a^{4}\right )}{\left (-b x^{2} + a\right )}^{\frac{2}{3}}}{b^{3} x^{6} - 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+3*a)^4/(-b*x^2+a)^(7/3),x, algorithm="fricas")

[Out]

integral(-(b^4*x^8 + 12*a*b^3*x^6 + 54*a^2*b^2*x^4 + 108*a^3*b*x^2 + 81*a^4)*(-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., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 a + b x^{2}\right )^{4}}{\left (a - b x^{2}\right )^{\frac{7}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+3*a)**4/(-b*x**2+a)**(7/3),x)

[Out]

Integral((3*a + b*x**2)**4/(a - b*x**2)**(7/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac{7}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+3*a)^4/(-b*x^2+a)^(7/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + 3*a)^4/(-b*x^2 + a)^(7/3), x)