Integrand size = 24, antiderivative size = 192 \[ \int \frac {(e x)^m (c+d x)^2}{\sqrt {a+b x^2}} \, dx=\frac {d^2 (e x)^{1+m} \sqrt {a+b x^2}}{b e (2+m)}+\frac {\left (\frac {c^2}{1+m}-\frac {a d^2}{b (2+m)}\right ) (e x)^{1+m} \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{e \sqrt {a+b x^2}}+\frac {2 c d (e x)^{2+m} \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )}{e^2 (2+m) \sqrt {a+b x^2}} \] Output:
d^2*(e*x)^(1+m)*(b*x^2+a)^(1/2)/b/e/(2+m)+(c^2/(1+m)-a*d^2/b/(2+m))*(e*x)^ (1+m)*(1+b*x^2/a)^(1/2)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/e /(b*x^2+a)^(1/2)+2*c*d*(e*x)^(2+m)*(1+b*x^2/a)^(1/2)*hypergeom([1/2, 1+1/2 *m],[2+1/2*m],-b*x^2/a)/e^2/(2+m)/(b*x^2+a)^(1/2)
Time = 0.07 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.83 \[ \int \frac {(e x)^m (c+d x)^2}{\sqrt {a+b x^2}} \, dx=\frac {x (e x)^m \sqrt {1+\frac {b x^2}{a}} \left (c^2 \left (6+5 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )+d (1+m) x \left (2 c (3+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-\frac {b x^2}{a}\right )+d (2+m) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},-\frac {b x^2}{a}\right )\right )\right )}{(1+m) (2+m) (3+m) \sqrt {a+b x^2}} \] Input:
Integrate[((e*x)^m*(c + d*x)^2)/Sqrt[a + b*x^2],x]
Output:
(x*(e*x)^m*Sqrt[1 + (b*x^2)/a]*(c^2*(6 + 5*m + m^2)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + d*(1 + m)*x*(2*c*(3 + m)*Hypergeome tric2F1[1/2, (2 + m)/2, (4 + m)/2, -((b*x^2)/a)] + d*(2 + m)*x*Hypergeomet ric2F1[1/2, (3 + m)/2, (5 + m)/2, -((b*x^2)/a)])))/((1 + m)*(2 + m)*(3 + m )*Sqrt[a + b*x^2])
Time = 0.35 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {559, 25, 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.
\(\displaystyle \int \frac {(c+d x)^2 (e x)^m}{\sqrt {a+b x^2}} \, dx\) |
\(\Big \downarrow \) 559 |
\(\displaystyle \frac {\int -\frac {(e x)^m \left (-b (m+2) c^2-2 b d (m+2) x c+a d^2 (m+1)\right )}{\sqrt {b x^2+a}}dx}{b (m+2)}+\frac {d^2 \sqrt {a+b x^2} (e x)^{m+1}}{b e (m+2)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {d^2 \sqrt {a+b x^2} (e x)^{m+1}}{b e (m+2)}-\frac {\int \frac {(e x)^m \left (-b (m+2) c^2-2 b d (m+2) x c+a d^2 (m+1)\right )}{\sqrt {b x^2+a}}dx}{b (m+2)}\) |
\(\Big \downarrow \) 557 |
\(\displaystyle \frac {d^2 \sqrt {a+b x^2} (e x)^{m+1}}{b e (m+2)}-\frac {\left (a d^2 (m+1)-b c^2 (m+2)\right ) \int \frac {(e x)^m}{\sqrt {b x^2+a}}dx-\frac {2 b c d (m+2) \int \frac {(e x)^{m+1}}{\sqrt {b x^2+a}}dx}{e}}{b (m+2)}\) |
\(\Big \downarrow \) 279 |
\(\displaystyle \frac {d^2 \sqrt {a+b x^2} (e x)^{m+1}}{b e (m+2)}-\frac {\frac {\sqrt {\frac {b x^2}{a}+1} \left (a d^2 (m+1)-b c^2 (m+2)\right ) \int \frac {(e x)^m}{\sqrt {\frac {b x^2}{a}+1}}dx}{\sqrt {a+b x^2}}-\frac {2 b c d (m+2) \sqrt {\frac {b x^2}{a}+1} \int \frac {(e x)^{m+1}}{\sqrt {\frac {b x^2}{a}+1}}dx}{e \sqrt {a+b x^2}}}{b (m+2)}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {d^2 \sqrt {a+b x^2} (e x)^{m+1}}{b e (m+2)}-\frac {\frac {\sqrt {\frac {b x^2}{a}+1} (e x)^{m+1} \left (a d^2 (m+1)-b c^2 (m+2)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{e (m+1) \sqrt {a+b x^2}}-\frac {2 b c d \sqrt {\frac {b x^2}{a}+1} (e x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-\frac {b x^2}{a}\right )}{e^2 \sqrt {a+b x^2}}}{b (m+2)}\) |
Input:
Int[((e*x)^m*(c + d*x)^2)/Sqrt[a + b*x^2],x]
Output:
(d^2*(e*x)^(1 + m)*Sqrt[a + b*x^2])/(b*e*(2 + m)) - (((a*d^2*(1 + m) - b*c ^2*(2 + m))*(e*x)^(1 + m)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(e*(1 + m)*Sqrt[a + b*x^2]) - (2*b*c*d*(e* x)^(2 + m)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2 , -((b*x^2)/a)])/(e^2*Sqrt[a + b*x^2]))/(b*(2 + m))
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]
\[\int \frac {\left (e x \right )^{m} \left (d x +c \right )^{2}}{\sqrt {b \,x^{2}+a}}d x\]
Input:
int((e*x)^m*(d*x+c)^2/(b*x^2+a)^(1/2),x)
Output:
int((e*x)^m*(d*x+c)^2/(b*x^2+a)^(1/2),x)
\[ \int \frac {(e x)^m (c+d x)^2}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{2} \left (e x\right )^{m}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
integral((d^2*x^2 + 2*c*d*x + c^2)*(e*x)^m/sqrt(b*x^2 + a), x)
Result contains complex when optimal does not.
Time = 1.92 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.90 \[ \int \frac {(e x)^m (c+d x)^2}{\sqrt {a+b x^2}} \, dx=\frac {c^{2} e^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {c d e^{m} x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (\frac {m}{2} + 2\right )} + \frac {d^{2} e^{m} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} \] Input:
integrate((e*x)**m*(d*x+c)**2/(b*x**2+a)**(1/2),x)
Output:
c**2*e**m*x**(m + 1)*gamma(m/2 + 1/2)*hyper((1/2, m/2 + 1/2), (m/2 + 3/2,) , b*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*gamma(m/2 + 3/2)) + c*d*e**m*x**(m + 2)*gamma(m/2 + 1)*hyper((1/2, m/2 + 1), (m/2 + 2,), b*x**2*exp_polar(I*p i)/a)/(sqrt(a)*gamma(m/2 + 2)) + d**2*e**m*x**(m + 3)*gamma(m/2 + 3/2)*hyp er((1/2, m/2 + 3/2), (m/2 + 5/2,), b*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*ga mma(m/2 + 5/2))
\[ \int \frac {(e x)^m (c+d x)^2}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{2} \left (e x\right )^{m}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^2*(e*x)^m/sqrt(b*x^2 + a), x)
\[ \int \frac {(e x)^m (c+d x)^2}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{2} \left (e x\right )^{m}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((e*x)^m*(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x + c)^2*(e*x)^m/sqrt(b*x^2 + a), x)
Timed out. \[ \int \frac {(e x)^m (c+d x)^2}{\sqrt {a+b x^2}} \, dx=\int \frac {{\left (e\,x\right )}^m\,{\left (c+d\,x\right )}^2}{\sqrt {b\,x^2+a}} \,d x \] Input:
int(((e*x)^m*(c + d*x)^2)/(a + b*x^2)^(1/2),x)
Output:
int(((e*x)^m*(c + d*x)^2)/(a + b*x^2)^(1/2), x)
\[ \int \frac {(e x)^m (c+d x)^2}{\sqrt {a+b x^2}} \, dx=e^{m} \left (\left (\int \frac {x^{m}}{\sqrt {b \,x^{2}+a}}d x \right ) c^{2}+\left (\int \frac {x^{m} x^{2}}{\sqrt {b \,x^{2}+a}}d x \right ) d^{2}+2 \left (\int \frac {x^{m} x}{\sqrt {b \,x^{2}+a}}d x \right ) c d \right ) \] Input:
int((e*x)^m*(d*x+c)^2/(b*x^2+a)^(1/2),x)
Output:
e**m*(int(x**m/sqrt(a + b*x**2),x)*c**2 + int((x**m*x**2)/sqrt(a + b*x**2) ,x)*d**2 + 2*int((x**m*x)/sqrt(a + b*x**2),x)*c*d)