Optimal. Leaf size=93 \[ \frac {(b c-a d) x^{1+m}}{2 a b \left (a+b x^2\right )}+\frac {(a d (1+m)+b (c-c m)) x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{2 a^2 b (1+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {468, 371}
\begin {gather*} \frac {x^{m+1} (a d (m+1)+b (c-c m)) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{2 a^2 b (m+1)}+\frac {x^{m+1} (b c-a d)}{2 a b \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 468
Rubi steps
\begin {align*} \int \frac {x^m \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac {(b c-a d) x^{1+m}}{2 a b \left (a+b x^2\right )}+\frac {(a d (1+m)+b (c-c m)) \int \frac {x^m}{a+b x^2} \, dx}{2 a b}\\ &=\frac {(b c-a d) x^{1+m}}{2 a b \left (a+b x^2\right )}+\frac {(a d (1+m)+b (c-c m)) x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{2 a^2 b (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 80, normalized size = 0.86 \begin {gather*} \frac {x^{1+m} \left (a d \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )+(b c-a d) \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )\right )}{a^2 b (1+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 14.98, size = 906, normalized size = 9.74 \begin {gather*} c \left (- \frac {a m^{2} x x^{m} \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 )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {2 a m x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {a x x^{m} \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 )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {2 a x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} - \frac {b m^{2} x^{3} x^{m} \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 )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {b x^{3} x^{m} \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 )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}\right ) + d \left (- \frac {a m^{2} x^{3} x^{m} \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 )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {4 a m x^{3} x^{m} \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 )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {2 a m x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {3 a x^{3} x^{m} \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 )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {6 a x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {b m^{2} x^{5} x^{m} \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 )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {4 b m x^{5} x^{m} \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 )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {3 b x^{5} x^{m} \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 )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^m\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________