Optimal. Leaf size=178 \[ \frac{(e x)^{m+1} \left (a^2 B d^2-a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{b^3 e (m+1)}+\frac{d (e x)^{m+3} (-a B d+A b d+2 b B c)}{b^2 e^3 (m+3)}+\frac{(e x)^{m+1} (A b-a B) (b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b^3 e (m+1)}+\frac{B d^2 (e x)^{m+5}}{b e^5 (m+5)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.189075, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {570, 364} \[ \frac{(e x)^{m+1} \left (a^2 B d^2-a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{b^3 e (m+1)}+\frac{d (e x)^{m+3} (-a B d+A b d+2 b B c)}{b^2 e^3 (m+3)}+\frac{(e x)^{m+1} (A b-a B) (b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b^3 e (m+1)}+\frac{B d^2 (e x)^{m+5}}{b e^5 (m+5)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 570
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2}{a+b x^2} \, dx &=\int \left (\frac{\left (a^2 B d^2-a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) (e x)^m}{b^3}+\frac{d (2 b B c+A b d-a B d) (e x)^{2+m}}{b^2 e^2}+\frac{B d^2 (e x)^{4+m}}{b e^4}+\frac{\left (A b^3 c^2-a b^2 B c^2-2 a A b^2 c d+2 a^2 b B c d+a^2 A b d^2-a^3 B d^2\right ) (e x)^m}{b^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{\left (a^2 B d^2-a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) (e x)^{1+m}}{b^3 e (1+m)}+\frac{d (2 b B c+A b d-a B d) (e x)^{3+m}}{b^2 e^3 (3+m)}+\frac{B d^2 (e x)^{5+m}}{b e^5 (5+m)}+\frac{\left ((A b-a B) (b c-a d)^2\right ) \int \frac{(e x)^m}{a+b x^2} \, dx}{b^3}\\ &=\frac{\left (a^2 B d^2-a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) (e x)^{1+m}}{b^3 e (1+m)}+\frac{d (2 b B c+A b d-a B d) (e x)^{3+m}}{b^2 e^3 (3+m)}+\frac{B d^2 (e x)^{5+m}}{b e^5 (5+m)}+\frac{(A b-a B) (b c-a d)^2 (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{b x^2}{a}\right )}{a b^3 e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.199976, size = 146, normalized size = 0.82 \[ \frac{x (e x)^m \left (\frac{a^2 B d^2-a b d (A d+2 B c)+b^2 c (2 A d+B c)}{m+1}+\frac{(A b-a B) (b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a (m+1)}+\frac{b d x^2 (-a B d+A b d+2 b B c)}{m+3}+\frac{b^2 B d^2 x^4}{m+5}\right )}{b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( B{x}^{2}+A \right ) \left ( d{x}^{2}+c \right ) ^{2}}{b{x}^{2}+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )}{\left (d x^{2} + c\right )}^{2} \left (e x\right )^{m}}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B d^{2} x^{6} +{\left (2 \, B c d + A d^{2}\right )} x^{4} + A c^{2} +{\left (B c^{2} + 2 \, A c d\right )} x^{2}\right )} \left (e x\right )^{m}}{b x^{2} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 32.2816, size = 666, normalized size = 3.74 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )}{\left (d x^{2} + c\right )}^{2} \left (e x\right )^{m}}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]