Integrand size = 29, antiderivative size = 29 \[ \int \frac {(h x)^m}{\left (a+b x^2\right )^{3/2} \left (d+e x+f x^2\right )^2} \, dx=\text {Int}\left (\frac {(h x)^m}{\left (a+b x^2\right )^{3/2} \left (d+e x+f x^2\right )^2},x\right ) \] Output:
Defer(Int)((h*x)^m/(b*x^2+a)^(3/2)/(f*x^2+e*x+d)^2,x)
Not integrable
Time = 11.59 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {(h x)^m}{\left (a+b x^2\right )^{3/2} \left (d+e x+f x^2\right )^2} \, dx=\int \frac {(h x)^m}{\left (a+b x^2\right )^{3/2} \left (d+e x+f x^2\right )^2} \, dx \] Input:
Integrate[(h*x)^m/((a + b*x^2)^(3/2)*(d + e*x + f*x^2)^2),x]
Output:
Integrate[(h*x)^m/((a + b*x^2)^(3/2)*(d + e*x + f*x^2)^2), x]
Leaf count is larger than twice the leaf count of optimal. \(1413\) vs. \(2(32)=64\).
Time = 3.78 (sec) , antiderivative size = 1413, normalized size of antiderivative = 48.72, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {7293, 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.
\(\displaystyle \int \frac {(h x)^m}{\left (a+b x^2\right )^{3/2} \left (d+e x+f x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 f^2 (h x)^m}{\left (a+b x^2\right )^{3/2} \left (e^2-4 d f\right )^{3/2} \left (\sqrt {e^2-4 d f}-e-2 f x\right )}+\frac {4 f^2 (h x)^m}{\left (a+b x^2\right )^{3/2} \left (e^2-4 d f\right )^{3/2} \left (\sqrt {e^2-4 d f}+e+2 f x\right )}+\frac {4 f^2 (h x)^m}{\left (a+b x^2\right )^{3/2} \left (e^2-4 d f\right ) \left (\sqrt {e^2-4 d f}-e-2 f x\right )^2}+\frac {4 f^2 (h x)^m}{\left (a+b x^2\right )^{3/2} \left (e^2-4 d f\right ) \left (\sqrt {e^2-4 d f}+e+2 f x\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 f^2 x \sqrt {\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {m+1}{2},\frac {3}{2},1,\frac {m+3}{2},-\frac {b x^2}{a},\frac {4 f^2 x^2}{\left (e-\sqrt {e^2-4 d f}\right )^2}\right ) (h x)^m}{a \left (e^2-4 d f\right )^{3/2} \left (e-\sqrt {e^2-4 d f}\right ) (m+1) \sqrt {b x^2+a}}+\frac {4 f^2 x \sqrt {\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {m+1}{2},\frac {3}{2},1,\frac {m+3}{2},-\frac {b x^2}{a},\frac {4 f^2 x^2}{\left (e+\sqrt {e^2-4 d f}\right )^2}\right ) (h x)^m}{a \left (e^2-4 d f\right )^{3/2} \left (e+\sqrt {e^2-4 d f}\right ) (m+1) \sqrt {b x^2+a}}+\frac {f^2 \left (e-\sqrt {e^2-4 d f}\right )^2 x \sqrt {\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {m+1}{2},\frac {3}{2},2,\frac {m+3}{2},-\frac {b x^2}{a},\frac {2 f^2 x^2}{e^2-\sqrt {e^2-4 d f} e-2 d f}\right ) (h x)^m}{a \left (e^2-4 d f\right ) \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )^2 (m+1) \sqrt {b x^2+a}}+\frac {f^2 \left (e+\sqrt {e^2-4 d f}\right )^2 x \sqrt {\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {m+1}{2},\frac {3}{2},2,\frac {m+3}{2},-\frac {b x^2}{a},\frac {2 f^2 x^2}{e^2+\sqrt {e^2-4 d f} e-2 d f}\right ) (h x)^m}{a \left (e^2-4 d f\right ) \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )^2 (m+1) \sqrt {b x^2+a}}+\frac {8 f^3 x^2 \sqrt {\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {m+2}{2},\frac {3}{2},1,\frac {m+4}{2},-\frac {b x^2}{a},\frac {4 f^2 x^2}{\left (e-\sqrt {e^2-4 d f}\right )^2}\right ) (h x)^m}{a \left (e^2-4 d f\right )^{3/2} \left (e-\sqrt {e^2-4 d f}\right )^2 (m+2) \sqrt {b x^2+a}}-\frac {8 f^3 x^2 \sqrt {\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {m+2}{2},\frac {3}{2},1,\frac {m+4}{2},-\frac {b x^2}{a},\frac {4 f^2 x^2}{\left (e+\sqrt {e^2-4 d f}\right )^2}\right ) (h x)^m}{a \left (e^2-4 d f\right )^{3/2} \left (e+\sqrt {e^2-4 d f}\right )^2 (m+2) \sqrt {b x^2+a}}-\frac {4 f^3 \left (e-\sqrt {e^2-4 d f}\right ) x^2 \sqrt {\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {m+2}{2},\frac {3}{2},2,\frac {m+4}{2},-\frac {b x^2}{a},\frac {2 f^2 x^2}{e^2-\sqrt {e^2-4 d f} e-2 d f}\right ) (h x)^m}{a \left (e^2-4 d f\right ) \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )^2 (m+2) \sqrt {b x^2+a}}-\frac {4 f^3 \left (e+\sqrt {e^2-4 d f}\right ) x^2 \sqrt {\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {m+2}{2},\frac {3}{2},2,\frac {m+4}{2},-\frac {b x^2}{a},\frac {2 f^2 x^2}{e^2+\sqrt {e^2-4 d f} e-2 d f}\right ) (h x)^m}{a \left (e^2-4 d f\right ) \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )^2 (m+2) \sqrt {b x^2+a}}+\frac {4 f^4 x^3 \sqrt {\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {m+3}{2},\frac {3}{2},2,\frac {m+5}{2},-\frac {b x^2}{a},\frac {2 f^2 x^2}{e^2-\sqrt {e^2-4 d f} e-2 d f}\right ) (h x)^m}{a \left (e^2-4 d f\right ) \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )^2 (m+3) \sqrt {b x^2+a}}+\frac {4 f^4 x^3 \sqrt {\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {m+3}{2},\frac {3}{2},2,\frac {m+5}{2},-\frac {b x^2}{a},\frac {2 f^2 x^2}{e^2+\sqrt {e^2-4 d f} e-2 d f}\right ) (h x)^m}{a \left (e^2-4 d f\right ) \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )^2 (m+3) \sqrt {b x^2+a}}\) |
Input:
Int[(h*x)^m/((a + b*x^2)^(3/2)*(d + e*x + f*x^2)^2),x]
Output:
(-4*f^2*x*(h*x)^m*Sqrt[1 + (b*x^2)/a]*AppellF1[(1 + m)/2, 3/2, 1, (3 + m)/ 2, -((b*x^2)/a), (4*f^2*x^2)/(e - Sqrt[e^2 - 4*d*f])^2])/(a*(e^2 - 4*d*f)^ (3/2)*(e - Sqrt[e^2 - 4*d*f])*(1 + m)*Sqrt[a + b*x^2]) + (4*f^2*x*(h*x)^m* Sqrt[1 + (b*x^2)/a]*AppellF1[(1 + m)/2, 3/2, 1, (3 + m)/2, -((b*x^2)/a), ( 4*f^2*x^2)/(e + Sqrt[e^2 - 4*d*f])^2])/(a*(e^2 - 4*d*f)^(3/2)*(e + Sqrt[e^ 2 - 4*d*f])*(1 + m)*Sqrt[a + b*x^2]) + (f^2*(e - Sqrt[e^2 - 4*d*f])^2*x*(h *x)^m*Sqrt[1 + (b*x^2)/a]*AppellF1[(1 + m)/2, 3/2, 2, (3 + m)/2, -((b*x^2) /a), (2*f^2*x^2)/(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])])/(a*(e^2 - 4*d*f)*(e ^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])^2*(1 + m)*Sqrt[a + b*x^2]) + (f^2*(e + S qrt[e^2 - 4*d*f])^2*x*(h*x)^m*Sqrt[1 + (b*x^2)/a]*AppellF1[(1 + m)/2, 3/2, 2, (3 + m)/2, -((b*x^2)/a), (2*f^2*x^2)/(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f ])])/(a*(e^2 - 4*d*f)*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])^2*(1 + m)*Sqrt[a + b*x^2]) + (8*f^3*x^2*(h*x)^m*Sqrt[1 + (b*x^2)/a]*AppellF1[(2 + m)/2, 3/ 2, 1, (4 + m)/2, -((b*x^2)/a), (4*f^2*x^2)/(e - Sqrt[e^2 - 4*d*f])^2])/(a* (e^2 - 4*d*f)^(3/2)*(e - Sqrt[e^2 - 4*d*f])^2*(2 + m)*Sqrt[a + b*x^2]) - ( 8*f^3*x^2*(h*x)^m*Sqrt[1 + (b*x^2)/a]*AppellF1[(2 + m)/2, 3/2, 1, (4 + m)/ 2, -((b*x^2)/a), (4*f^2*x^2)/(e + Sqrt[e^2 - 4*d*f])^2])/(a*(e^2 - 4*d*f)^ (3/2)*(e + Sqrt[e^2 - 4*d*f])^2*(2 + m)*Sqrt[a + b*x^2]) - (4*f^3*(e - Sqr t[e^2 - 4*d*f])*x^2*(h*x)^m*Sqrt[1 + (b*x^2)/a]*AppellF1[(2 + m)/2, 3/2, 2 , (4 + m)/2, -((b*x^2)/a), (2*f^2*x^2)/(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*...
Not integrable
Time = 1.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
\[\int \frac {\left (h x \right )^{m}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (f \,x^{2}+e x +d \right )^{2}}d x\]
Input:
int((h*x)^m/(b*x^2+a)^(3/2)/(f*x^2+e*x+d)^2,x)
Output:
int((h*x)^m/(b*x^2+a)^(3/2)/(f*x^2+e*x+d)^2,x)
Not integrable
Time = 3.65 (sec) , antiderivative size = 175, normalized size of antiderivative = 6.03 \[ \int \frac {(h x)^m}{\left (a+b x^2\right )^{3/2} \left (d+e x+f x^2\right )^2} \, dx=\int { \frac {\left (h x\right )^{m}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e x + d\right )}^{2}} \,d x } \] Input:
integrate((h*x)^m/(b*x^2+a)^(3/2)/(f*x^2+e*x+d)^2,x, algorithm="fricas")
Output:
integral(sqrt(b*x^2 + a)*(h*x)^m/(b^2*f^2*x^8 + 2*b^2*e*f*x^7 + (b^2*e^2 + 2*b^2*d*f + 2*a*b*f^2)*x^6 + 2*(b^2*d*e + 2*a*b*e*f)*x^5 + 2*a^2*d*e*x + (b^2*d^2 + 2*a*b*e^2 + 4*a*b*d*f + a^2*f^2)*x^4 + a^2*d^2 + 2*(2*a*b*d*e + a^2*e*f)*x^3 + (2*a*b*d^2 + a^2*e^2 + 2*a^2*d*f)*x^2), x)
Timed out. \[ \int \frac {(h x)^m}{\left (a+b x^2\right )^{3/2} \left (d+e x+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((h*x)**m/(b*x**2+a)**(3/2)/(f*x**2+e*x+d)**2,x)
Output:
Timed out
Not integrable
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(h x)^m}{\left (a+b x^2\right )^{3/2} \left (d+e x+f x^2\right )^2} \, dx=\int { \frac {\left (h x\right )^{m}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e x + d\right )}^{2}} \,d x } \] Input:
integrate((h*x)^m/(b*x^2+a)^(3/2)/(f*x^2+e*x+d)^2,x, algorithm="maxima")
Output:
integrate((h*x)^m/((b*x^2 + a)^(3/2)*(f*x^2 + e*x + d)^2), x)
Not integrable
Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(h x)^m}{\left (a+b x^2\right )^{3/2} \left (d+e x+f x^2\right )^2} \, dx=\int { \frac {\left (h x\right )^{m}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e x + d\right )}^{2}} \,d x } \] Input:
integrate((h*x)^m/(b*x^2+a)^(3/2)/(f*x^2+e*x+d)^2,x, algorithm="giac")
Output:
integrate((h*x)^m/((b*x^2 + a)^(3/2)*(f*x^2 + e*x + d)^2), x)
Not integrable
Time = 44.49 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(h x)^m}{\left (a+b x^2\right )^{3/2} \left (d+e x+f x^2\right )^2} \, dx=\int \frac {{\left (h\,x\right )}^m}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (f\,x^2+e\,x+d\right )}^2} \,d x \] Input:
int((h*x)^m/((a + b*x^2)^(3/2)*(d + e*x + f*x^2)^2),x)
Output:
int((h*x)^m/((a + b*x^2)^(3/2)*(d + e*x + f*x^2)^2), x)
Not integrable
Time = 200.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(h x)^m}{\left (a+b x^2\right )^{3/2} \left (d+e x+f x^2\right )^2} \, dx=\int \frac {\left (h x \right )^{m}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (f \,x^{2}+e x +d \right )^{2}}d x \] Input:
int((h*x)^m/(b*x^2+a)^(3/2)/(f*x^2+e*x+d)^2,x)
Output:
int((h*x)^m/(b*x^2+a)^(3/2)/(f*x^2+e*x+d)^2,x)