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

Optimal. Leaf size=260 \[ \frac{b (e x)^{m+3} \left (3 a^2 B d^2-3 a b d (B c-A d)+b^2 c (B c-A d)\right )}{d^3 e^3 (m+3)}+\frac{(e x)^{m+1} \left (a^3 B d^3-3 a^2 b d^2 (B c-A d)+3 a b^2 c d (B c-A d)+b^3 \left (-c^2\right ) (B c-A d)\right )}{d^4 e (m+1)}-\frac{b^2 (e x)^{m+5} (-3 a B d-A b d+b B c)}{d^2 e^5 (m+5)}+\frac{(e x)^{m+1} (b c-a d)^3 (B c-A d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c d^4 e (m+1)}+\frac{b^3 B (e x)^{m+7}}{d e^7 (m+7)} \]

[Out]

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

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Rubi [A]  time = 0.644206, antiderivative size = 260, 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 \[ \frac{b (e x)^{m+3} \left (3 a^2 B d^2-3 a b d (B c-A d)+b^2 c (B c-A d)\right )}{d^3 e^3 (m+3)}+\frac{(e x)^{m+1} \left (a^3 B d^3-3 a^2 b d^2 (B c-A d)+3 a b^2 c d (B c-A d)+b^3 \left (-c^2\right ) (B c-A d)\right )}{d^4 e (m+1)}-\frac{b^2 (e x)^{m+5} (-3 a B d-A b d+b B c)}{d^2 e^5 (m+5)}+\frac{(e x)^{m+1} (b c-a d)^3 (B c-A d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c d^4 e (m+1)}+\frac{b^3 B (e x)^{m+7}}{d e^7 (m+7)} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 0.7998, size = 218, normalized size = 0.84 \[ \frac{x (e x)^m \left (\frac{a^3 A \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{m+1}+\frac{a^2 x^2 (a B+3 A b) \, _2F_1\left (1,\frac{m+3}{2};\frac{m+5}{2};-\frac{d x^2}{c}\right )}{m+3}+b x^4 \left (b x^2 \left (\frac{(3 a B+A b) \, _2F_1\left (1,\frac{m+7}{2};\frac{m+9}{2};-\frac{d x^2}{c}\right )}{m+7}+\frac{b B x^2 \, _2F_1\left (1,\frac{m+9}{2};\frac{m+11}{2};-\frac{d x^2}{c}\right )}{m+9}\right )+\frac{3 a (a B+A b) \, _2F_1\left (1,\frac{m+5}{2};\frac{m+7}{2};-\frac{d x^2}{c}\right )}{m+5}\right )\right )}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( b{x}^{2}+a \right ) ^{3} \left ( B{x}^{2}+A \right ) }{d{x}^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 97.4542, size = 911, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**3*e**m*m*x*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2
+ 1/2)/(4*c*gamma(m/2 + 3/2)) + A*a**3*e**m*x*x**m*lerchphi(d*x**2*exp_polar(I*p
i)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(4*c*gamma(m/2 + 3/2)) + 3*A*a**2*b*e**m*m*
x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*c
*gamma(m/2 + 5/2)) + 9*A*a**2*b*e**m*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c
, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*c*gamma(m/2 + 5/2)) + 3*A*a*b**2*e**m*m*x**5
*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 5/2)*gamma(m/2 + 5/2)/(4*c*gam
ma(m/2 + 7/2)) + 15*A*a*b**2*e**m*x**5*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1
, m/2 + 5/2)*gamma(m/2 + 5/2)/(4*c*gamma(m/2 + 7/2)) + A*b**3*e**m*m*x**7*x**m*l
erchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 7/2)*gamma(m/2 + 7/2)/(4*c*gamma(m/2
+ 9/2)) + 7*A*b**3*e**m*x**7*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 7/
2)*gamma(m/2 + 7/2)/(4*c*gamma(m/2 + 9/2)) + B*a**3*e**m*m*x**3*x**m*lerchphi(d*
x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*c*gamma(m/2 + 5/2)) +
3*B*a**3*e**m*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m
/2 + 3/2)/(4*c*gamma(m/2 + 5/2)) + 3*B*a**2*b*e**m*m*x**5*x**m*lerchphi(d*x**2*e
xp_polar(I*pi)/c, 1, m/2 + 5/2)*gamma(m/2 + 5/2)/(4*c*gamma(m/2 + 7/2)) + 15*B*a
**2*b*e**m*x**5*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 5/2)*gamma(m/2
+ 5/2)/(4*c*gamma(m/2 + 7/2)) + 3*B*a*b**2*e**m*m*x**7*x**m*lerchphi(d*x**2*exp_
polar(I*pi)/c, 1, m/2 + 7/2)*gamma(m/2 + 7/2)/(4*c*gamma(m/2 + 9/2)) + 21*B*a*b*
*2*e**m*x**7*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 7/2)*gamma(m/2 + 7
/2)/(4*c*gamma(m/2 + 9/2)) + B*b**3*e**m*m*x**9*x**m*lerchphi(d*x**2*exp_polar(I
*pi)/c, 1, m/2 + 9/2)*gamma(m/2 + 9/2)/(4*c*gamma(m/2 + 11/2)) + 9*B*b**3*e**m*x
**9*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 9/2)*gamma(m/2 + 9/2)/(4*c*
gamma(m/2 + 11/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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