Integrand size = 22, antiderivative size = 168 \[ \int \frac {(e x)^m (c+d x)^3}{a+b x^2} \, dx=\frac {3 c d^2 (e x)^{1+m}}{b e (1+m)}+\frac {d^3 (e x)^{2+m}}{b e^2 (2+m)}+\frac {c \left (b c^2-3 a d^2\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a b e (1+m)}+\frac {d \left (3 b c^2-a d^2\right ) (e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )}{a b e^2 (2+m)} \] Output:
3*c*d^2*(e*x)^(1+m)/b/e/(1+m)+d^3*(e*x)^(2+m)/b/e^2/(2+m)+c*(-3*a*d^2+b*c^ 2)*(e*x)^(1+m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/a/b/e/(1+m)+ d*(-a*d^2+3*b*c^2)*(e*x)^(2+m)*hypergeom([1, 1+1/2*m],[2+1/2*m],-b*x^2/a)/ a/b/e^2/(2+m)
Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.76 \[ \int \frac {(e x)^m (c+d x)^3}{a+b x^2} \, dx=\frac {x (e x)^m \left (c \left (b c^2-3 a d^2\right ) (2+m) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )+d \left (a d (3 c (2+m)+d (1+m) x)+\left (3 b c^2-a d^2\right ) (1+m) x \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )\right )\right )}{a b (1+m) (2+m)} \] Input:
Integrate[((e*x)^m*(c + d*x)^3)/(a + b*x^2),x]
Output:
(x*(e*x)^m*(c*(b*c^2 - 3*a*d^2)*(2 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + d*(a*d*(3*c*(2 + m) + d*(1 + m)*x) + (3*b*c^2 - a *d^2)*(1 + m)*x*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -((b*x^2)/a)])) )/(a*b*(1 + m)*(2 + m))
Time = 0.46 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {559, 2333, 2009}
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.
| \(\displaystyle \int \frac {(c+d x)^3 (e x)^m}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 559 |
| \(\displaystyle \frac {\int \frac {(e x)^m \left (b (m+2) c^3+3 b d^2 (m+2) x^2 c+d \left (3 b c^2-a d^2\right ) (m+2) x\right )}{b x^2+a}dx}{b (m+2)}+\frac {d^3 (e x)^{m+2}}{b e^2 (m+2)}\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \frac {\int \left (3 c d^2 (m+2) (e x)^m+\frac {\left (c \left (b c^2-3 a d^2\right ) (m+2)+d \left (3 b c^2-a d^2\right ) x (m+2)\right ) (e x)^m}{b x^2+a}\right )dx}{b (m+2)}+\frac {d^3 (e x)^{m+2}}{b e^2 (m+2)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {d (e x)^{m+2} \left (3 b c^2-a d^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},-\frac {b x^2}{a}\right )}{a e^2}+\frac {c (m+2) (e x)^{m+1} \left (b c^2-3 a d^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{a e (m+1)}+\frac {3 c d^2 (m+2) (e x)^{m+1}}{e (m+1)}}{b (m+2)}+\frac {d^3 (e x)^{m+2}}{b e^2 (m+2)}\) |
Input:
Int[((e*x)^m*(c + d*x)^3)/(a + b*x^2),x]
Output:
(d^3*(e*x)^(2 + m))/(b*e^2*(2 + m)) + ((3*c*d^2*(2 + m)*(e*x)^(1 + m))/(e* (1 + m)) + (c*(b*c^2 - 3*a*d^2)*(2 + m)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*e*(1 + m)) + (d*(3*b*c^2 - a*d^2) *(e*x)^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -((b*x^2)/a)])/( a*e^2))/(b*(2 + m))
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] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
\[\int \frac {\left (e x \right )^{m} \left (d x +c \right )^{3}}{b \,x^{2}+a}d x\]
Input:
int((e*x)^m*(d*x+c)^3/(b*x^2+a),x)
Output:
int((e*x)^m*(d*x+c)^3/(b*x^2+a),x)
\[ \int \frac {(e x)^m (c+d x)^3}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3} \left (e x\right )^{m}}{b x^{2} + a} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)^3/(b*x^2+a),x, algorithm="fricas")
Output:
integral((d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*(e*x)^m/(b*x^2 + a), x)
Result contains complex when optimal does not.
Time = 3.01 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.41 \[ \int \frac {(e x)^m (c+d x)^3}{a+b x^2} \, dx=\frac {c^{3} e^{m} m x^{m + 1} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {c^{3} e^{m} x^{m + 1} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {3 c^{2} d e^{m} m x^{m + 2} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{4 a \Gamma \left (\frac {m}{2} + 2\right )} + \frac {3 c^{2} d e^{m} x^{m + 2} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{2 a \Gamma \left (\frac {m}{2} + 2\right )} + \frac {3 c d^{2} e^{m} m x^{m + 3} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {9 c d^{2} e^{m} x^{m + 3} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {d^{3} e^{m} m x^{m + 4} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + 2\right ) \Gamma \left (\frac {m}{2} + 2\right )}{4 a \Gamma \left (\frac {m}{2} + 3\right )} + \frac {d^{3} e^{m} x^{m + 4} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + 2\right ) \Gamma \left (\frac {m}{2} + 2\right )}{a \Gamma \left (\frac {m}{2} + 3\right )} \] Input:
integrate((e*x)**m*(d*x+c)**3/(b*x**2+a),x)
Output:
c**3*e**m*m*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*ga mma(m/2 + 1/2)/(4*a*gamma(m/2 + 3/2)) + c**3*e**m*x**(m + 1)*lerchphi(b*x* *2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(4*a*gamma(m/2 + 3/2) ) + 3*c**2*d*e**m*m*x**(m + 2)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(4*a*gamma(m/2 + 2)) + 3*c**2*d*e**m*x**(m + 2)*lerchph i(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1)*gamma(m/2 + 1)/(2*a*gamma(m/2 + 2) ) + 3*c*d**2*e**m*m*x**(m + 3)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*a*gamma(m/2 + 5/2)) + 9*c*d**2*e**m*x**(m + 3)*l erchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*a*gamm a(m/2 + 5/2)) + d**3*e**m*m*x**(m + 4)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 2)*gamma(m/2 + 2)/(4*a*gamma(m/2 + 3)) + d**3*e**m*x**(m + 4)*ler chphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 2)*gamma(m/2 + 2)/(a*gamma(m/2 + 3))
\[ \int \frac {(e x)^m (c+d x)^3}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3} \left (e x\right )^{m}}{b x^{2} + a} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)^3/(b*x^2+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^3*(e*x)^m/(b*x^2 + a), x)
\[ \int \frac {(e x)^m (c+d x)^3}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3} \left (e x\right )^{m}}{b x^{2} + a} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)^3/(b*x^2+a),x, algorithm="giac")
Output:
integrate((d*x + c)^3*(e*x)^m/(b*x^2 + a), x)
Timed out. \[ \int \frac {(e x)^m (c+d x)^3}{a+b x^2} \, dx=\int \frac {{\left (e\,x\right )}^m\,{\left (c+d\,x\right )}^3}{b\,x^2+a} \,d x \] Input:
int(((e*x)^m*(c + d*x)^3)/(a + b*x^2),x)
Output:
int(((e*x)^m*(c + d*x)^3)/(a + b*x^2), x)
\[ \int \frac {(e x)^m (c+d x)^3}{a+b x^2} \, dx=\frac {e^{m} \left (-x^{m} a \,d^{3} m^{2}-3 x^{m} a \,d^{3} m -2 x^{m} a \,d^{3}+3 x^{m} b \,c^{2} d \,m^{2}+9 x^{m} b \,c^{2} d m +6 x^{m} b \,c^{2} d +3 x^{m} b c \,d^{2} m^{2} x +6 x^{m} b c \,d^{2} m x +x^{m} b \,d^{3} m^{2} x^{2}+x^{m} b \,d^{3} m \,x^{2}+\left (\int \frac {x^{m}}{b \,x^{3}+a x}d x \right ) a^{2} d^{3} m^{3}+3 \left (\int \frac {x^{m}}{b \,x^{3}+a x}d x \right ) a^{2} d^{3} m^{2}+2 \left (\int \frac {x^{m}}{b \,x^{3}+a x}d x \right ) a^{2} d^{3} m -3 \left (\int \frac {x^{m}}{b \,x^{3}+a x}d x \right ) a b \,c^{2} d \,m^{3}-9 \left (\int \frac {x^{m}}{b \,x^{3}+a x}d x \right ) a b \,c^{2} d \,m^{2}-6 \left (\int \frac {x^{m}}{b \,x^{3}+a x}d x \right ) a b \,c^{2} d m -3 \left (\int \frac {x^{m}}{b \,x^{2}+a}d x \right ) a b c \,d^{2} m^{3}-9 \left (\int \frac {x^{m}}{b \,x^{2}+a}d x \right ) a b c \,d^{2} m^{2}-6 \left (\int \frac {x^{m}}{b \,x^{2}+a}d x \right ) a b c \,d^{2} m +\left (\int \frac {x^{m}}{b \,x^{2}+a}d x \right ) b^{2} c^{3} m^{3}+3 \left (\int \frac {x^{m}}{b \,x^{2}+a}d x \right ) b^{2} c^{3} m^{2}+2 \left (\int \frac {x^{m}}{b \,x^{2}+a}d x \right ) b^{2} c^{3} m \right )}{b^{2} m \left (m^{2}+3 m +2\right )} \] Input:
int((e*x)^m*(d*x+c)^3/(b*x^2+a),x)
Output:
(e**m*( - x**m*a*d**3*m**2 - 3*x**m*a*d**3*m - 2*x**m*a*d**3 + 3*x**m*b*c* *2*d*m**2 + 9*x**m*b*c**2*d*m + 6*x**m*b*c**2*d + 3*x**m*b*c*d**2*m**2*x + 6*x**m*b*c*d**2*m*x + x**m*b*d**3*m**2*x**2 + x**m*b*d**3*m*x**2 + int(x* *m/(a*x + b*x**3),x)*a**2*d**3*m**3 + 3*int(x**m/(a*x + b*x**3),x)*a**2*d* *3*m**2 + 2*int(x**m/(a*x + b*x**3),x)*a**2*d**3*m - 3*int(x**m/(a*x + b*x **3),x)*a*b*c**2*d*m**3 - 9*int(x**m/(a*x + b*x**3),x)*a*b*c**2*d*m**2 - 6 *int(x**m/(a*x + b*x**3),x)*a*b*c**2*d*m - 3*int(x**m/(a + b*x**2),x)*a*b* c*d**2*m**3 - 9*int(x**m/(a + b*x**2),x)*a*b*c*d**2*m**2 - 6*int(x**m/(a + b*x**2),x)*a*b*c*d**2*m + int(x**m/(a + b*x**2),x)*b**2*c**3*m**3 + 3*int (x**m/(a + b*x**2),x)*b**2*c**3*m**2 + 2*int(x**m/(a + b*x**2),x)*b**2*c** 3*m))/(b**2*m*(m**2 + 3*m + 2))