\(\int \frac {(a-b x^3)^2}{(a+b x^3)^{17/3}} \, dx\) [55]

Optimal result

Integrand size = 22, antiderivative size = 93 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{17/3}} \, dx=\frac {x \left (a-b x^3\right )}{7 \left (a+b x^3\right )^{14/3}}+\frac {9 x}{77 \left (a+b x^3\right )^{11/3}}+\frac {57 x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {11}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{77 a^3 \left (a+b x^3\right )^{2/3}} \]

[Out]

1/7*x*(-b*x^3+a)/(b*x^3+a)^(14/3)+9/77*x/(b*x^3+a)^(11/3)+57/77*x*(1+b*x^3/a)^(2/3)*hypergeom([1/3, 11/3],[4/3
],-b*x^3/a)/a^3/(b*x^3+a)^(2/3)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {424, 393, 252, 251} \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{17/3}} \, dx=\frac {57 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {11}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{77 a^3 \left (a+b x^3\right )^{2/3}}+\frac {9 x}{77 \left (a+b x^3\right )^{11/3}}+\frac {x \left (a-b x^3\right )}{7 \left (a+b x^3\right )^{14/3}} \]

[In]

Int[(a - b*x^3)^2/(a + b*x^3)^(17/3),x]

[Out]

(x*(a - b*x^3))/(7*(a + b*x^3)^(14/3)) + (9*x)/(77*(a + b*x^3)^(11/3)) + (57*x*(1 + (b*x^3)/a)^(2/3)*Hypergeom
etric2F1[1/3, 11/3, 4/3, -((b*x^3)/a)])/(77*a^3*(a + b*x^3)^(2/3))

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 393

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

Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a-b x^3\right )}{7 \left (a+b x^3\right )^{14/3}}+\frac {\int \frac {12 a^2 b-6 a b^2 x^3}{\left (a+b x^3\right )^{14/3}} \, dx}{14 a b} \\ & = \frac {x \left (a-b x^3\right )}{7 \left (a+b x^3\right )^{14/3}}+\frac {9 x}{77 \left (a+b x^3\right )^{11/3}}+\frac {57}{77} \int \frac {1}{\left (a+b x^3\right )^{11/3}} \, dx \\ & = \frac {x \left (a-b x^3\right )}{7 \left (a+b x^3\right )^{14/3}}+\frac {9 x}{77 \left (a+b x^3\right )^{11/3}}+\frac {\left (57 \left (1+\frac {b x^3}{a}\right )^{2/3}\right ) \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{11/3}} \, dx}{77 a^3 \left (a+b x^3\right )^{2/3}} \\ & = \frac {x \left (a-b x^3\right )}{7 \left (a+b x^3\right )^{14/3}}+\frac {9 x}{77 \left (a+b x^3\right )^{11/3}}+\frac {57 x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {11}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{77 a^3 \left (a+b x^3\right )^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{17/3}} \, dx=\frac {x \left (2282 a^4+4879 a^3 b x^3+6270 a^2 b^2 x^6+3591 a b^3 x^9+798 b^4 x^{12}+798 \left (a+b x^3\right )^4 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )\right )}{3080 a^3 \left (a+b x^3\right )^{14/3}} \]

[In]

Integrate[(a - b*x^3)^2/(a + b*x^3)^(17/3),x]

[Out]

(x*(2282*a^4 + 4879*a^3*b*x^3 + 6270*a^2*b^2*x^6 + 3591*a*b^3*x^9 + 798*b^4*x^12 + 798*(a + b*x^3)^4*(1 + (b*x
^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)]))/(3080*a^3*(a + b*x^3)^(14/3))

Maple [F]

\[\int \frac {\left (-b \,x^{3}+a \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {17}{3}}}d x\]

[In]

int((-b*x^3+a)^2/(b*x^3+a)^(17/3),x)

[Out]

int((-b*x^3+a)^2/(b*x^3+a)^(17/3),x)

Fricas [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{17/3}} \, dx=\int { \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {17}{3}}} \,d x } \]

[In]

integrate((-b*x^3+a)^2/(b*x^3+a)^(17/3),x, algorithm="fricas")

[Out]

integral((b^2*x^6 - 2*a*b*x^3 + a^2)*(b*x^3 + a)^(1/3)/(b^6*x^18 + 6*a*b^5*x^15 + 15*a^2*b^4*x^12 + 20*a^3*b^3
*x^9 + 15*a^4*b^2*x^6 + 6*a^5*b*x^3 + a^6), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{17/3}} \, dx=\text {Timed out} \]

[In]

integrate((-b*x**3+a)**2/(b*x**3+a)**(17/3),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{17/3}} \, dx=\int { \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {17}{3}}} \,d x } \]

[In]

integrate((-b*x^3+a)^2/(b*x^3+a)^(17/3),x, algorithm="maxima")

[Out]

integrate((b*x^3 - a)^2/(b*x^3 + a)^(17/3), x)

Giac [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{17/3}} \, dx=\int { \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {17}{3}}} \,d x } \]

[In]

integrate((-b*x^3+a)^2/(b*x^3+a)^(17/3),x, algorithm="giac")

[Out]

integrate((b*x^3 - a)^2/(b*x^3 + a)^(17/3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{17/3}} \, dx=\int \frac {{\left (a-b\,x^3\right )}^2}{{\left (b\,x^3+a\right )}^{17/3}} \,d x \]

[In]

int((a - b*x^3)^2/(a + b*x^3)^(17/3),x)

[Out]

int((a - b*x^3)^2/(a + b*x^3)^(17/3), x)