Optimal. Leaf size=67 \[ -\frac {2 (d (b+2 c x))^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right ) d (1+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {708, 371}
\begin {gather*} -\frac {2 (d (b+2 c x))^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 708
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^m}{a+b x+c x^2} \, dx &=\frac {\text {Subst}\left (\int \frac {x^m}{a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=-\frac {2 (d (b+2 c x))^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right ) d (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(186\) vs. \(2(67)=134\).
time = 0.20, size = 186, normalized size = 2.78 \begin {gather*} \frac {(d (b+2 c x))^m \left (\left (\frac {b+2 c x}{b-\sqrt {b^2-4 a c}+2 c x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac {\sqrt {b^2-4 a c}}{-b+\sqrt {b^2-4 a c}-2 c x}\right )-\left (\frac {b+2 c x}{b+\sqrt {b^2-4 a c}+2 c x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac {\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}+2 c x}\right )\right )}{\sqrt {b^2-4 a c} m} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \frac {\left (2 c d x +b d \right )^{m}}{c \,x^{2}+b x +a}\, dx\]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d \left (b + 2 c x\right )\right )^{m}}{a + b x + c x^{2}}\, dx \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 {{\left (b\,d+2\,c\,d\,x\right )}^m}{c\,x^2+b\,x+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________