3.42 \(\int \frac{(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=452 \[ \frac{d (e x)^{m+1} \left (A \left (-a^2 d^2 (3-m)+a b c d (11-m)+4 b^2 c^2\right )-a B c (a d (m+1)+b c (11-m))\right )}{8 a c^2 e \left (c+d x^2\right ) (b c-a d)^3}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) \left (-a^2 d^2 (1-m) (A d (3-m)+B c (m+1))+2 a b c d \left (A d \left (m^2-8 m+7\right )+B c \left (-m^2+4 m+5\right )\right )+b^2 c^2 (5-m) (B c (3-m)-A d (7-m))\right )}{8 c^3 e (m+1) (b c-a d)^4}+\frac{b^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (A b (b c (1-m)-a d (7-m))+a B (a d (5-m)+b c (m+1)))}{2 a^2 e (m+1) (b c-a d)^4}+\frac{d (e x)^{m+1} (a A d-3 a B c+2 A b c)}{4 a c e \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{(e x)^{m+1} (A b-a B)}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

(d*(2*A*b*c - 3*a*B*c + a*A*d)*(e*x)^(1 + m))/(4*a*c*(b*c - a*d)^2*e*(c + d*x^2)
^2) + ((A*b - a*B)*(e*x)^(1 + m))/(2*a*(b*c - a*d)*e*(a + b*x^2)*(c + d*x^2)^2)
+ (d*(A*(4*b^2*c^2 - a^2*d^2*(3 - m) + a*b*c*d*(11 - m)) - a*B*c*(b*c*(11 - m) +
a*d*(1 + m)))*(e*x)^(1 + m))/(8*a*c^2*(b*c - a*d)^3*e*(c + d*x^2)) + (b^2*(A*b*
(b*c*(1 - m) - a*d*(7 - m)) + a*B*(a*d*(5 - m) + b*c*(1 + m)))*(e*x)^(1 + m)*Hyp
ergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*(b*c - a*d)^4*e*(1
+ m)) - (d*(b^2*c^2*(B*c*(3 - m) - A*d*(7 - m))*(5 - m) - a^2*d^2*(1 - m)*(A*d*
(3 - m) + B*c*(1 + m)) + 2*a*b*c*d*(B*c*(5 + 4*m - m^2) + A*d*(7 - 8*m + m^2)))*
(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(8*c^3*(
b*c - a*d)^4*e*(1 + m))

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Rubi [A]  time = 3.59139, antiderivative size = 452, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{d (e x)^{m+1} \left (A \left (-a^2 d^2 (3-m)+a b c d (11-m)+4 b^2 c^2\right )-a B c (a d (m+1)+b c (11-m))\right )}{8 a c^2 e \left (c+d x^2\right ) (b c-a d)^3}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) \left (-a^2 d^2 (1-m) (A d (3-m)+B c (m+1))+2 a b c d \left (A d \left (m^2-8 m+7\right )+B c \left (-m^2+4 m+5\right )\right )+b^2 c^2 (5-m) (B c (3-m)-A d (7-m))\right )}{8 c^3 e (m+1) (b c-a d)^4}+\frac{b^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (A b (b c (1-m)-a d (7-m))+a B (a d (5-m)+b c (m+1)))}{2 a^2 e (m+1) (b c-a d)^4}+\frac{d (e x)^{m+1} (a A d-3 a B c+2 A b c)}{4 a c e \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{(e x)^{m+1} (A b-a B)}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^m*(A + B*x^2))/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(d*(2*A*b*c - 3*a*B*c + a*A*d)*(e*x)^(1 + m))/(4*a*c*(b*c - a*d)^2*e*(c + d*x^2)
^2) + ((A*b - a*B)*(e*x)^(1 + m))/(2*a*(b*c - a*d)*e*(a + b*x^2)*(c + d*x^2)^2)
+ (d*(A*(4*b^2*c^2 - a^2*d^2*(3 - m) + a*b*c*d*(11 - m)) - a*B*c*(b*c*(11 - m) +
a*d*(1 + m)))*(e*x)^(1 + m))/(8*a*c^2*(b*c - a*d)^3*e*(c + d*x^2)) + (b^2*(A*b*
(b*c*(1 - m) - a*d*(7 - m)) + a*B*(a*d*(5 - m) + b*c*(1 + m)))*(e*x)^(1 + m)*Hyp
ergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*(b*c - a*d)^4*e*(1
+ m)) - (d*(b^2*c^2*(B*c*(3 - m) - A*d*(7 - m))*(5 - m) - a^2*d^2*(1 - m)*(A*d*
(3 - m) + B*c*(1 + m)) + 2*a*b*c*d*(B*c*(5 + 4*m - m^2) + A*d*(7 - 8*m + m^2)))*
(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(8*c^3*(
b*c - a*d)^4*e*(1 + m))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(B*x**2+A)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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Mathematica [C]  time = 1.13317, size = 379, normalized size = 0.84 \[ \frac{a c x (e x)^m \left (\frac{A (m+3)^2 F_1\left (\frac{m+1}{2};2,3;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{(m+1) \left (a c (m+3) F_1\left (\frac{m+1}{2};2,3;\frac{m+3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-2 x^2 \left (3 a d F_1\left (\frac{m+3}{2};2,4;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+2 b c F_1\left (\frac{m+3}{2};3,3;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )}+\frac{B (m+5) x^2 F_1\left (\frac{m+3}{2};2,3;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{a c (m+5) F_1\left (\frac{m+3}{2};2,3;\frac{m+5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-2 x^2 \left (3 a d F_1\left (\frac{m+5}{2};2,4;\frac{m+7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+2 b c F_1\left (\frac{m+5}{2};3,3;\frac{m+7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}\right )}{(m+3) \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((e*x)^m*(A + B*x^2))/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(a*c*x*(e*x)^m*((A*(3 + m)^2*AppellF1[(1 + m)/2, 2, 3, (3 + m)/2, -((b*x^2)/a),
-((d*x^2)/c)])/((1 + m)*(a*c*(3 + m)*AppellF1[(1 + m)/2, 2, 3, (3 + m)/2, -((b*x
^2)/a), -((d*x^2)/c)] - 2*x^2*(3*a*d*AppellF1[(3 + m)/2, 2, 4, (5 + m)/2, -((b*x
^2)/a), -((d*x^2)/c)] + 2*b*c*AppellF1[(3 + m)/2, 3, 3, (5 + m)/2, -((b*x^2)/a),
-((d*x^2)/c)]))) + (B*(5 + m)*x^2*AppellF1[(3 + m)/2, 2, 3, (5 + m)/2, -((b*x^2
)/a), -((d*x^2)/c)])/(a*c*(5 + m)*AppellF1[(3 + m)/2, 2, 3, (5 + m)/2, -((b*x^2)
/a), -((d*x^2)/c)] - 2*x^2*(3*a*d*AppellF1[(5 + m)/2, 2, 4, (7 + m)/2, -((b*x^2)
/a), -((d*x^2)/c)] + 2*b*c*AppellF1[(5 + m)/2, 3, 3, (7 + m)/2, -((b*x^2)/a), -(
(d*x^2)/c)]))))/((3 + m)*(a + b*x^2)^2*(c + d*x^2)^3)

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Maple [F]  time = 0.114, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( B{x}^{2}+A \right ) }{ \left ( b{x}^{2}+a \right ) ^{2} \left ( d{x}^{2}+c \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(B*x^2+A)/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

int((e*x)^m*(B*x^2+A)/(b*x^2+a)^2/(d*x^2+c)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(e*x)^m/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*(e*x)^m/((b*x^2 + a)^2*(d*x^2 + c)^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{b^{2} d^{3} x^{10} +{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} +{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} + a^{2} c^{3} +{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} +{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(e*x)^m/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="fricas")

[Out]

integral((B*x^2 + A)*(e*x)^m/(b^2*d^3*x^10 + (3*b^2*c*d^2 + 2*a*b*d^3)*x^8 + (3*
b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^6 + a^2*c^3 + (b^2*c^3 + 6*a*b*c^2*d + 3*a^
2*c*d^2)*x^4 + (2*a*b*c^3 + 3*a^2*c^2*d)*x^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(B*x**2+A)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(e*x)^m/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*(e*x)^m/((b*x^2 + a)^2*(d*x^2 + c)^3), x)