Integrand size = 20, antiderivative size = 363 \[ \int (a-b x)^{4/3} (a+b x)^{7/3} \, dx=\frac {24}{55} a^3 x \sqrt [3]{a-b x} \sqrt [3]{a+b x}+\frac {3}{11} a x (a-b x)^{4/3} (a+b x)^{4/3}-\frac {3 (a-b x)^{7/3} (a+b x)^{7/3}}{14 b}+\frac {16\ 3^{3/4} \sqrt {2-\sqrt {3}} a^7 \left (1-\frac {b^2 x^2}{a^2}\right )^{2/3} \left (1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right ) \sqrt {\frac {1+\sqrt [3]{1-\frac {b^2 x^2}{a^2}}+\left (1-\frac {b^2 x^2}{a^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}\right ),-7+4 \sqrt {3}\right )}{55 b^2 x (a-b x)^{2/3} (a+b x)^{2/3} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}}} \] Output:
24/55*a^3*x*(-b*x+a)^(1/3)*(b*x+a)^(1/3)+3/11*a*x*(-b*x+a)^(4/3)*(b*x+a)^( 4/3)-3/14*(-b*x+a)^(7/3)*(b*x+a)^(7/3)/b+16/55*3^(3/4)*(1/2*6^(1/2)-1/2*2^ (1/2))*a^7*(1-b^2*x^2/a^2)^(2/3)*(1-(1-b^2*x^2/a^2)^(1/3))*((1+(1-b^2*x^2/ a^2)^(1/3)+(1-b^2*x^2/a^2)^(2/3))/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/3))^2)^(1/ 2)*EllipticF((1+3^(1/2)-(1-b^2*x^2/a^2)^(1/3))/(1-3^(1/2)-(1-b^2*x^2/a^2)^ (1/3)),2*I-I*3^(1/2))/b^2/x/(-b*x+a)^(2/3)/(b*x+a)^(2/3)/(-(1-(1-b^2*x^2/a ^2)^(1/3))/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/3))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.19 \[ \int (a-b x)^{4/3} (a+b x)^{7/3} \, dx=-\frac {12 \sqrt [3]{2} a^2 (a-b x)^{7/3} \sqrt [3]{a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {7}{3},\frac {7}{3},\frac {10}{3},\frac {a-b x}{2 a}\right )}{7 b \sqrt [3]{\frac {a+b x}{a}}} \] Input:
Integrate[(a - b*x)^(4/3)*(a + b*x)^(7/3),x]
Output:
(-12*2^(1/3)*a^2*(a - b*x)^(7/3)*(a + b*x)^(1/3)*Hypergeometric2F1[-7/3, 7 /3, 10/3, (a - b*x)/(2*a)])/(7*b*((a + b*x)/a)^(1/3))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.19, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {80, 27, 79}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (a-b x)^{4/3} (a+b x)^{7/3} \, dx\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {4 \sqrt [3]{2} a^2 \sqrt [3]{a+b x} \int \frac {(a-b x)^{4/3} \left (\frac {b x}{a}+1\right )^{7/3}}{4 \sqrt [3]{2}}dx}{\sqrt [3]{\frac {a+b x}{a}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^2 \sqrt [3]{a+b x} \int (a-b x)^{4/3} \left (\frac {b x}{a}+1\right )^{7/3}dx}{\sqrt [3]{\frac {a+b x}{a}}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {12 \sqrt [3]{2} a^2 (a-b x)^{7/3} \sqrt [3]{a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {7}{3},\frac {7}{3},\frac {10}{3},\frac {a-b x}{2 a}\right )}{7 b \sqrt [3]{\frac {a+b x}{a}}}\) |
Input:
Int[(a - b*x)^(4/3)*(a + b*x)^(7/3),x]
Output:
(-12*2^(1/3)*a^2*(a - b*x)^(7/3)*(a + b*x)^(1/3)*Hypergeometric2F1[-7/3, 7 /3, 10/3, (a - b*x)/(2*a)])/(7*b*((a + b*x)/a)^(1/3))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
\[\int \left (-b x +a \right )^{\frac {4}{3}} \left (b x +a \right )^{\frac {7}{3}}d x\]
Input:
int((-b*x+a)^(4/3)*(b*x+a)^(7/3),x)
Output:
int((-b*x+a)^(4/3)*(b*x+a)^(7/3),x)
\[ \int (a-b x)^{4/3} (a+b x)^{7/3} \, dx=\int { {\left (b x + a\right )}^{\frac {7}{3}} {\left (-b x + a\right )}^{\frac {4}{3}} \,d x } \] Input:
integrate((-b*x+a)^(4/3)*(b*x+a)^(7/3),x, algorithm="fricas")
Output:
integral(-(b^3*x^3 + a*b^2*x^2 - a^2*b*x - a^3)*(b*x + a)^(1/3)*(-b*x + a) ^(1/3), x)
\[ \int (a-b x)^{4/3} (a+b x)^{7/3} \, dx=\int \left (a - b x\right )^{\frac {4}{3}} \left (a + b x\right )^{\frac {7}{3}}\, dx \] Input:
integrate((-b*x+a)**(4/3)*(b*x+a)**(7/3),x)
Output:
Integral((a - b*x)**(4/3)*(a + b*x)**(7/3), x)
\[ \int (a-b x)^{4/3} (a+b x)^{7/3} \, dx=\int { {\left (b x + a\right )}^{\frac {7}{3}} {\left (-b x + a\right )}^{\frac {4}{3}} \,d x } \] Input:
integrate((-b*x+a)^(4/3)*(b*x+a)^(7/3),x, algorithm="maxima")
Output:
integrate((b*x + a)^(7/3)*(-b*x + a)^(4/3), x)
\[ \int (a-b x)^{4/3} (a+b x)^{7/3} \, dx=\int { {\left (b x + a\right )}^{\frac {7}{3}} {\left (-b x + a\right )}^{\frac {4}{3}} \,d x } \] Input:
integrate((-b*x+a)^(4/3)*(b*x+a)^(7/3),x, algorithm="giac")
Output:
integrate((b*x + a)^(7/3)*(-b*x + a)^(4/3), x)
Timed out. \[ \int (a-b x)^{4/3} (a+b x)^{7/3} \, dx=\int {\left (a+b\,x\right )}^{7/3}\,{\left (a-b\,x\right )}^{4/3} \,d x \] Input:
int((a + b*x)^(7/3)*(a - b*x)^(4/3),x)
Output:
int((a + b*x)^(7/3)*(a - b*x)^(4/3), x)
\[ \int (a-b x)^{4/3} (a+b x)^{7/3} \, dx=\frac {-165 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +a \right )^{\frac {1}{3}} a^{4}+546 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +a \right )^{\frac {1}{3}} a^{3} b x +330 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +a \right )^{\frac {1}{3}} a^{2} b^{2} x^{2}-210 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +a \right )^{\frac {1}{3}} a \,b^{3} x^{3}-165 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +a \right )^{\frac {1}{3}} b^{4} x^{4}+224 \left (\int \frac {\left (b x +a \right )^{\frac {1}{3}} \left (-b x +a \right )^{\frac {1}{3}}}{-b^{2} x^{2}+a^{2}}d x \right ) a^{5} b}{770 b} \] Input:
int((-b*x+a)^(4/3)*(b*x+a)^(7/3),x)
Output:
( - 165*(a + b*x)**(1/3)*(a - b*x)**(1/3)*a**4 + 546*(a + b*x)**(1/3)*(a - b*x)**(1/3)*a**3*b*x + 330*(a + b*x)**(1/3)*(a - b*x)**(1/3)*a**2*b**2*x* *2 - 210*(a + b*x)**(1/3)*(a - b*x)**(1/3)*a*b**3*x**3 - 165*(a + b*x)**(1 /3)*(a - b*x)**(1/3)*b**4*x**4 + 224*int(((a + b*x)**(1/3)*(a - b*x)**(1/3 ))/(a**2 - b**2*x**2),x)*a**5*b)/(770*b)