Optimal. Leaf size=340 \[ -\frac{b (e x)^{m+1} \left (3 a^2 d^2 (A d (m+1)-B c (m+3))-3 a b c d (A d (m+3)-B c (m+5))+b^2 c^2 (A d (m+5)-B c (m+7))\right )}{2 c d^4 e (m+1)}-\frac{b^2 (e x)^{m+3} (3 a d (A d (m+3)-B c (m+5))-b c (A d (m+5)-B c (m+7)))}{2 c d^3 e^3 (m+3)}+\frac{(e x)^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) (a d (A d (1-m)+B c (m+1))+b c (A d (m+5)-B c (m+7)))}{2 c^2 d^4 e (m+1)}-\frac{\left (a+b x^2\right )^3 (e x)^{m+1} (B c-A d)}{2 c d e \left (c+d x^2\right )}-\frac{b^3 (e x)^{m+5} (A d (m+5)-B c (m+7))}{2 c d^2 e^5 (m+5)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.715592, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {577, 570, 364} \[ -\frac{b (e x)^{m+1} \left (3 a^2 d^2 (A d (m+1)-B c (m+3))-3 a b c d (A d (m+3)-B c (m+5))+b^2 c^2 (A d (m+5)-B c (m+7))\right )}{2 c d^4 e (m+1)}-\frac{b^2 (e x)^{m+3} (3 a d (A d (m+3)-B c (m+5))-b c (A d (m+5)-B c (m+7)))}{2 c d^3 e^3 (m+3)}+\frac{(e x)^{m+1} (b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) (a d (A d (1-m)+B c (m+1))+b c (A d (m+5)-B c (m+7)))}{2 c^2 d^4 e (m+1)}-\frac{\left (a+b x^2\right )^3 (e x)^{m+1} (B c-A d)}{2 c d e \left (c+d x^2\right )}-\frac{b^3 (e x)^{m+5} (A d (m+5)-B c (m+7))}{2 c d^2 e^5 (m+5)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 577
Rule 570
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right )}{\left (c+d x^2\right )^2} \, dx &=-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^3}{2 c d e \left (c+d x^2\right )}-\frac{\int \frac{(e x)^m \left (a+b x^2\right )^2 \left (-a (A d (1-m)+B c (1+m))+b (A d (5+m)-B c (7+m)) x^2\right )}{c+d x^2} \, dx}{2 c d}\\ &=-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^3}{2 c d e \left (c+d x^2\right )}-\frac{\int \left (\frac{b \left (3 a^2 d^2 (A d (1+m)-B c (3+m))-3 a b c d (A d (3+m)-B c (5+m))+b^2 c^2 (A d (5+m)-B c (7+m))\right ) (e x)^m}{d^3}+\frac{b^2 (3 a d (A d (3+m)-B c (5+m))-b c (A d (5+m)-B c (7+m))) (e x)^{2+m}}{d^2 e^2}+\frac{b^3 (A d (5+m)-B c (7+m)) (e x)^{4+m}}{d e^4}+\frac{\left (7 b^3 B c^4-5 A b^3 c^3 d-15 a b^2 B c^3 d+9 a A b^2 c^2 d^2+9 a^2 b B c^2 d^2-3 a^2 A b c d^3-a^3 B c d^3-a^3 A d^4+b^3 B c^4 m-A b^3 c^3 d m-3 a b^2 B c^3 d m+3 a A b^2 c^2 d^2 m+3 a^2 b B c^2 d^2 m-3 a^2 A b c d^3 m-a^3 B c d^3 m+a^3 A d^4 m\right ) (e x)^m}{d^3 \left (c+d x^2\right )}\right ) \, dx}{2 c d}\\ &=-\frac{b \left (3 a^2 d^2 (A d (1+m)-B c (3+m))-3 a b c d (A d (3+m)-B c (5+m))+b^2 c^2 (A d (5+m)-B c (7+m))\right ) (e x)^{1+m}}{2 c d^4 e (1+m)}-\frac{b^2 (3 a d (A d (3+m)-B c (5+m))-b c (A d (5+m)-B c (7+m))) (e x)^{3+m}}{2 c d^3 e^3 (3+m)}-\frac{b^3 (A d (5+m)-B c (7+m)) (e x)^{5+m}}{2 c d^2 e^5 (5+m)}-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^3}{2 c d e \left (c+d x^2\right )}+\frac{\left ((b c-a d)^2 (a d (A d (1-m)+B c (1+m))+b c (A d (5+m)-B c (7+m)))\right ) \int \frac{(e x)^m}{c+d x^2} \, dx}{2 c d^4}\\ &=-\frac{b \left (3 a^2 d^2 (A d (1+m)-B c (3+m))-3 a b c d (A d (3+m)-B c (5+m))+b^2 c^2 (A d (5+m)-B c (7+m))\right ) (e x)^{1+m}}{2 c d^4 e (1+m)}-\frac{b^2 (3 a d (A d (3+m)-B c (5+m))-b c (A d (5+m)-B c (7+m))) (e x)^{3+m}}{2 c d^3 e^3 (3+m)}-\frac{b^3 (A d (5+m)-B c (7+m)) (e x)^{5+m}}{2 c d^2 e^5 (5+m)}-\frac{(B c-A d) (e x)^{1+m} \left (a+b x^2\right )^3}{2 c d e \left (c+d x^2\right )}+\frac{(b c-a d)^2 (a d (A d (1-m)+B c (1+m))+b c (A d (5+m)-B c (7+m))) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{d x^2}{c}\right )}{2 c^2 d^4 e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.34182, size = 212, normalized size = 0.62 \[ \frac{x (e x)^m \left (\frac{b \left (3 a^2 B d^2+3 a b d (A d-2 B c)+b^2 c (3 B c-2 A d)\right )}{m+1}+\frac{b^2 d x^2 (3 a B d+A b d-2 b B c)}{m+3}+\frac{(b c-a d)^3 (B c-A d) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c^2 (m+1)}-\frac{(b c-a d)^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) (-a B d-3 A b d+4 b B c)}{c (m+1)}+\frac{b^3 B d^2 x^4}{m+5}\right )}{d^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( B{x}^{2}+A \right ) \left ( b{x}^{2}+a \right ) ^{3} \left ( ex \right ) ^{m}}{ \left ( d{x}^{2}+c \right ) ^{2}}}\, 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 (b x^{2} + a\right )}^{3} \left (e x\right )^{m}}{{\left (d x^{2} + c\right )}^{2}}\,{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 b^{3} x^{8} +{\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} x^{4} + A a^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )} \left (e x\right )^{m}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (b x^{2} + a\right )}^{3} \left (e x\right )^{m}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]