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

Optimal. Leaf size=480 \[ -\frac{d^2 (e x)^{m+3} \left (A b (m+3) (a d (m+5)+b c (3-m))+a B \left (b c \left (m^2+4 m+3\right )-a d \left (m^2+12 m+35\right )\right )\right )}{8 a^2 b^3 e^3 (m+3)}+\frac{\left (c+d x^2\right )^2 (e x)^{m+1} (A b (a d (m+3)+b c (3-m))+a B (b c (m+1)-a d (m+7)))}{8 a^2 b^2 e \left (a+b x^2\right )}-\frac{d (e x)^{m+1} \left (A b \left (-a^2 d^2 \left (m^2+8 m+15\right )+3 a b c d \left (m^2+4 m+3\right )+2 b^2 c^2 \left (-m^2+2 m+3\right )\right )+a B \left (a^2 d^2 \left (m^2+12 m+35\right )-3 a b c d \left (m^2+8 m+15\right )+2 b^2 c^2 (m+1)^2\right )\right )}{8 a^2 b^4 e (m+1)}+\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2+8 m+15\right )+2 a b c d \left (-m^2-2 m+3\right )+b^2 c^2 \left (m^2-4 m+3\right )\right )+a B \left (-a^2 d^2 \left (m^2+12 m+35\right )+2 a b c d \left (m^2+6 m+5\right )+b^2 c^2 \left (1-m^2\right )\right )\right )}{8 a^3 b^4 e (m+1)}+\frac{\left (c+d x^2\right )^3 (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2} \]

[Out]

-(d*(A*b*(2*b^2*c^2*(3 + 2*m - m^2) + 3*a*b*c*d*(3 + 4*m + m^2) - a^2*d^2*(15 +
8*m + m^2)) + a*B*(2*b^2*c^2*(1 + m)^2 - 3*a*b*c*d*(15 + 8*m + m^2) + a^2*d^2*(3
5 + 12*m + m^2)))*(e*x)^(1 + m))/(8*a^2*b^4*e*(1 + m)) - (d^2*(A*b*(3 + m)*(b*c*
(3 - m) + a*d*(5 + m)) + a*B*(b*c*(3 + 4*m + m^2) - a*d*(35 + 12*m + m^2)))*(e*x
)^(3 + m))/(8*a^2*b^3*e^3*(3 + m)) + ((A*b*(b*c*(3 - m) + a*d*(3 + m)) + a*B*(b*
c*(1 + m) - a*d*(7 + m)))*(e*x)^(1 + m)*(c + d*x^2)^2)/(8*a^2*b^2*e*(a + b*x^2))
+ ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^2)^3)/(4*a*b*e*(a + b*x^2)^2) + ((b*c - a
*d)*(A*b*(2*a*b*c*d*(3 - 2*m - m^2) + b^2*c^2*(3 - 4*m + m^2) + a^2*d^2*(15 + 8*
m + m^2)) + a*B*(b^2*c^2*(1 - m^2) + 2*a*b*c*d*(5 + 6*m + m^2) - a^2*d^2*(35 + 1
2*m + m^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/
a)])/(8*a^3*b^4*e*(1 + m))

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Rubi [A]  time = 2.91074, antiderivative size = 480, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ -\frac{d^2 (e x)^{m+3} \left (A b (m+3) (a d (m+5)+b c (3-m))+a B \left (b c \left (m^2+4 m+3\right )-a d \left (m^2+12 m+35\right )\right )\right )}{8 a^2 b^3 e^3 (m+3)}+\frac{\left (c+d x^2\right )^2 (e x)^{m+1} (A b (a d (m+3)+b c (3-m))+a B (b c (m+1)-a d (m+7)))}{8 a^2 b^2 e \left (a+b x^2\right )}-\frac{d (e x)^{m+1} \left (A b \left (-a^2 d^2 \left (m^2+8 m+15\right )+3 a b c d \left (m^2+4 m+3\right )+2 b^2 c^2 \left (-m^2+2 m+3\right )\right )+a B \left (a^2 d^2 \left (m^2+12 m+35\right )-3 a b c d \left (m^2+8 m+15\right )+2 b^2 c^2 (m+1)^2\right )\right )}{8 a^2 b^4 e (m+1)}+\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2+8 m+15\right )+2 a b c d \left (-m^2-2 m+3\right )+b^2 c^2 \left (m^2-4 m+3\right )\right )+a B \left (-a^2 d^2 \left (m^2+12 m+35\right )+2 a b c d \left (m^2+6 m+5\right )+b^2 c^2 \left (1-m^2\right )\right )\right )}{8 a^3 b^4 e (m+1)}+\frac{\left (c+d x^2\right )^3 (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(d*(A*b*(2*b^2*c^2*(3 + 2*m - m^2) + 3*a*b*c*d*(3 + 4*m + m^2) - a^2*d^2*(15 +
8*m + m^2)) + a*B*(2*b^2*c^2*(1 + m)^2 - 3*a*b*c*d*(15 + 8*m + m^2) + a^2*d^2*(3
5 + 12*m + m^2)))*(e*x)^(1 + m))/(8*a^2*b^4*e*(1 + m)) - (d^2*(A*b*(3 + m)*(b*c*
(3 - m) + a*d*(5 + m)) + a*B*(b*c*(3 + 4*m + m^2) - a*d*(35 + 12*m + m^2)))*(e*x
)^(3 + m))/(8*a^2*b^3*e^3*(3 + m)) + ((A*b*(b*c*(3 - m) + a*d*(3 + m)) + a*B*(b*
c*(1 + m) - a*d*(7 + m)))*(e*x)^(1 + m)*(c + d*x^2)^2)/(8*a^2*b^2*e*(a + b*x^2))
+ ((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^2)^3)/(4*a*b*e*(a + b*x^2)^2) + ((b*c - a
*d)*(A*b*(2*a*b*c*d*(3 - 2*m - m^2) + b^2*c^2*(3 - 4*m + m^2) + a^2*d^2*(15 + 8*
m + m^2)) + a*B*(b^2*c^2*(1 - m^2) + 2*a*b*c*d*(5 + 6*m + m^2) - a^2*d^2*(35 + 1
2*m + m^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/
a)])/(8*a^3*b^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)*(d*x**2+c)**3/(b*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.855526, size = 218, normalized size = 0.45 \[ \frac{x (e x)^m \left (\frac{c^2 x^2 (3 A d+B c) \, _2F_1\left (3,\frac{m+3}{2};\frac{m+5}{2};-\frac{b x^2}{a}\right )}{m+3}+d x^4 \left (d x^2 \left (\frac{(A d+3 B c) \, _2F_1\left (3,\frac{m+7}{2};\frac{m+9}{2};-\frac{b x^2}{a}\right )}{m+7}+\frac{B d x^2 \, _2F_1\left (3,\frac{m+9}{2};\frac{m+11}{2};-\frac{b x^2}{a}\right )}{m+9}\right )+\frac{3 c (A d+B c) \, _2F_1\left (3,\frac{m+5}{2};\frac{m+7}{2};-\frac{b x^2}{a}\right )}{m+5}\right )+\frac{A c^3 \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{m+1}\right )}{a^3} \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*((A*c^3*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(1
+ m) + (c^2*(B*c + 3*A*d)*x^2*Hypergeometric2F1[3, (3 + m)/2, (5 + m)/2, -((b*x^
2)/a)])/(3 + m) + d*x^4*((3*c*(B*c + A*d)*Hypergeometric2F1[3, (5 + m)/2, (7 + m
)/2, -((b*x^2)/a)])/(5 + m) + d*x^2*(((3*B*c + A*d)*Hypergeometric2F1[3, (7 + m)
/2, (9 + m)/2, -((b*x^2)/a)])/(7 + m) + (B*d*x^2*Hypergeometric2F1[3, (9 + m)/2,
(11 + m)/2, -((b*x^2)/a)])/(9 + m)))))/a^3

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( B{x}^{2}+A \right ) \left ( d{x}^{2}+c \right ) ^{3}}{ \left ( b{x}^{2}+a \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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