Integrand size = 29, antiderivative size = 250 \[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{7/2}}{g^2 (9+m)}+\frac {d^7 (11+4 m) (g x)^{1+m} \sqrt {d^2-e^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{g (1+m) (8+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {d^6 e (29+4 m) (g x)^{2+m} \sqrt {d^2-e^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) (9+m) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
-3*d*(g*x)^(1+m)*(-e^2*x^2+d^2)^(7/2)/g/(8+m)-e*(g*x)^(2+m)*(-e^2*x^2+d^2) ^(7/2)/g^2/(9+m)+d^7*(11+4*m)*(g*x)^(1+m)*hypergeom([-5/2, 1/2+1/2*m],[3/2 +1/2*m],e^2*x^2/d^2)*(-e^2*x^2+d^2)^(1/2)/g/(1+m)/(8+m)/(1-e^2*x^2/d^2)^(1 /2)+d^6*e*(29+4*m)*(g*x)^(2+m)*hypergeom([-5/2, 1+1/2*m],[2+1/2*m],e^2*x^2 /d^2)*(-e^2*x^2+d^2)^(1/2)/g^2/(2+m)/(9+m)/(1-e^2*x^2/d^2)^(1/2)
Time = 0.88 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.80 \[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {d^4 x (g x)^m \sqrt {d^2-e^2 x^2} \left (\frac {d^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{1+m}+e x \left (\frac {3 d^2 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{2+m}+e x \left (\frac {3 d \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3+m}{2},\frac {5+m}{2},\frac {e^2 x^2}{d^2}\right )}{3+m}+\frac {e x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {4+m}{2},\frac {6+m}{2},\frac {e^2 x^2}{d^2}\right )}{4+m}\right )\right )\right )}{\sqrt {1-\frac {e^2 x^2}{d^2}}} \]
(d^4*x*(g*x)^m*Sqrt[d^2 - e^2*x^2]*((d^3*Hypergeometric2F1[-5/2, (1 + m)/2 , (3 + m)/2, (e^2*x^2)/d^2])/(1 + m) + e*x*((3*d^2*Hypergeometric2F1[-5/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2])/(2 + m) + e*x*((3*d*Hypergeometric2 F1[-5/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2])/(3 + m) + (e*x*Hypergeometr ic2F1[-5/2, (4 + m)/2, (6 + m)/2, (e^2*x^2)/d^2])/(4 + m)))))/Sqrt[1 - (e^ 2*x^2)/d^2]
Time = 0.54 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {559, 25, 2340, 25, 27, 557, 279, 278}
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.
| \(\displaystyle \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} (g x)^m \, dx\) |
| \(\Big \downarrow \) 559 |
| \(\displaystyle -\frac {\int -(g x)^m \left (d^2-e^2 x^2\right )^{5/2} \left (3 d (m+9) x^2 e^4+d^2 (4 m+29) x e^3+d^3 (m+9) e^2\right )dx}{e^2 (m+9)}-\frac {e \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+2}}{g^2 (m+9)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \left (3 d (m+9) x^2 e^4+d^2 (4 m+29) x e^3+d^3 (m+9) e^2\right )dx}{e^2 (m+9)}-\frac {e \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+2}}{g^2 (m+9)}\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle \frac {-\frac {\int -d^2 e^4 (g x)^m (d (m+9) (4 m+11)+e (m+8) (4 m+29) x) \left (d^2-e^2 x^2\right )^{5/2}dx}{e^2 (m+8)}-\frac {3 d e^2 (m+9) \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}}{e^2 (m+9)}-\frac {e \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+2}}{g^2 (m+9)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int d^2 e^4 (g x)^m (d (m+9) (4 m+11)+e (m+8) (4 m+29) x) \left (d^2-e^2 x^2\right )^{5/2}dx}{e^2 (m+8)}-\frac {3 d e^2 (m+9) \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}}{e^2 (m+9)}-\frac {e \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+2}}{g^2 (m+9)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d^2 e^2 \int (g x)^m (d (m+9) (4 m+11)+e (m+8) (4 m+29) x) \left (d^2-e^2 x^2\right )^{5/2}dx}{m+8}-\frac {3 d e^2 (m+9) \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}}{e^2 (m+9)}-\frac {e \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+2}}{g^2 (m+9)}\) |
| \(\Big \downarrow \) 557 |
| \(\displaystyle \frac {\frac {d^2 e^2 \left (d (m+9) (4 m+11) \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2}dx+\frac {e (m+8) (4 m+29) \int (g x)^{m+1} \left (d^2-e^2 x^2\right )^{5/2}dx}{g}\right )}{m+8}-\frac {3 d e^2 (m+9) \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}}{e^2 (m+9)}-\frac {e \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+2}}{g^2 (m+9)}\) |
| \(\Big \downarrow \) 279 |
| \(\displaystyle \frac {\frac {d^2 e^2 \left (\frac {d^5 (m+9) (4 m+11) \sqrt {d^2-e^2 x^2} \int (g x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^{5/2}dx}{\sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {d^4 e (m+8) (4 m+29) \sqrt {d^2-e^2 x^2} \int (g x)^{m+1} \left (1-\frac {e^2 x^2}{d^2}\right )^{5/2}dx}{g \sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{m+8}-\frac {3 d e^2 (m+9) \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}}{e^2 (m+9)}-\frac {e \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+2}}{g^2 (m+9)}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {\frac {d^2 e^2 \left (\frac {d^5 (m+9) (4 m+11) \sqrt {d^2-e^2 x^2} (g x)^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{g (m+1) \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {d^4 e (m+8) (4 m+29) \sqrt {d^2-e^2 x^2} (g x)^{m+2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {m+2}{2},\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) \sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{m+8}-\frac {3 d e^2 (m+9) \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}}{e^2 (m+9)}-\frac {e \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+2}}{g^2 (m+9)}\) |
-((e*(g*x)^(2 + m)*(d^2 - e^2*x^2)^(7/2))/(g^2*(9 + m))) + ((-3*d*e^2*(9 + m)*(g*x)^(1 + m)*(d^2 - e^2*x^2)^(7/2))/(g*(8 + m)) + (d^2*e^2*((d^5*(9 + m)*(11 + 4*m)*(g*x)^(1 + m)*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, ( 1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2])/(g*(1 + m)*Sqrt[1 - (e^2*x^2)/d^2]) + (d^4*e*(8 + m)*(29 + 4*m)*(g*x)^(2 + m)*Sqrt[d^2 - e^2*x^2]*Hypergeometri c2F1[-5/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2])/(g^2*(2 + m)*Sqrt[1 - (e^ 2*x^2)/d^2])))/(8 + m))/(e^2*(9 + m))
3.3.26.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(c*x)^m* (1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e Int[(e*x)^( m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d^n*(e*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*e^(n - 1)*( m + n + 2*p + 1))), x] + Simp[1/(b*(m + n + 2*p + 1)) Int[(e*x)^m*(a + b* x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1 )*x^n - a*d^n*(m + n - 1)*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && IGtQ[n, 1] && !IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 )*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m + q + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) *Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ [Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
\[\int \left (g x \right )^{m} \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}d x\]
\[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{3} \left (g x\right )^{m} \,d x } \]
integral((e^7*x^7 + 3*d*e^6*x^6 + d^2*e^5*x^5 - 5*d^3*e^4*x^4 - 5*d^4*e^3* x^3 + d^5*e^2*x^2 + 3*d^6*e*x + d^7)*sqrt(-e^2*x^2 + d^2)*(g*x)^m, x)
Result contains complex when optimal does not.
Time = 15.98 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.01 \[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {d^{8} g^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {3 d^{7} e g^{m} x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {d^{6} e^{2} g^{m} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {5 d^{5} e^{3} g^{m} x^{m + 4} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 3\right )} - \frac {5 d^{4} e^{4} g^{m} x^{m + 5} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {5}{2} \\ \frac {m}{2} + \frac {7}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} + \frac {d^{3} e^{5} g^{m} x^{m + 6} \Gamma \left (\frac {m}{2} + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 3 \\ \frac {m}{2} + 4 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 4\right )} + \frac {3 d^{2} e^{6} g^{m} x^{m + 7} \Gamma \left (\frac {m}{2} + \frac {7}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {7}{2} \\ \frac {m}{2} + \frac {9}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {9}{2}\right )} + \frac {d e^{7} g^{m} x^{m + 8} \Gamma \left (\frac {m}{2} + 4\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 4 \\ \frac {m}{2} + 5 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 5\right )} \]
d**8*g**m*x**(m + 1)*gamma(m/2 + 1/2)*hyper((-1/2, m/2 + 1/2), (m/2 + 3/2, ), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 3/2)) + 3*d**7*e*g**m* x**(m + 2)*gamma(m/2 + 1)*hyper((-1/2, m/2 + 1), (m/2 + 2,), e**2*x**2*exp _polar(2*I*pi)/d**2)/(2*gamma(m/2 + 2)) + d**6*e**2*g**m*x**(m + 3)*gamma( m/2 + 3/2)*hyper((-1/2, m/2 + 3/2), (m/2 + 5/2,), e**2*x**2*exp_polar(2*I* pi)/d**2)/(2*gamma(m/2 + 5/2)) - 5*d**5*e**3*g**m*x**(m + 4)*gamma(m/2 + 2 )*hyper((-1/2, m/2 + 2), (m/2 + 3,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2* gamma(m/2 + 3)) - 5*d**4*e**4*g**m*x**(m + 5)*gamma(m/2 + 5/2)*hyper((-1/2 , m/2 + 5/2), (m/2 + 7/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 7/2)) + d**3*e**5*g**m*x**(m + 6)*gamma(m/2 + 3)*hyper((-1/2, m/2 + 3), (m/2 + 4,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 4)) + 3*d**2 *e**6*g**m*x**(m + 7)*gamma(m/2 + 7/2)*hyper((-1/2, m/2 + 7/2), (m/2 + 9/2 ,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 9/2)) + d*e**7*g**m*x **(m + 8)*gamma(m/2 + 4)*hyper((-1/2, m/2 + 4), (m/2 + 5,), e**2*x**2*exp_ polar(2*I*pi)/d**2)/(2*gamma(m/2 + 5))
\[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{3} \left (g x\right )^{m} \,d x } \]
\[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{3} \left (g x\right )^{m} \,d x } \]
Timed out. \[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int {\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (g\,x\right )}^m\,{\left (d+e\,x\right )}^3 \,d x \]