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

Optimal. Leaf size=120 \[ \frac{(e x)^{m+1} (b c-a d) (B c-A d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c d^2 e (m+1)}-\frac{(e x)^{m+1} (-a B d-A b d+b B c)}{d^2 e (m+1)}+\frac{b B (e x)^{m+3}}{d e^3 (m+3)} \]

[Out]

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

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Rubi [A]  time = 0.257458, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{(e x)^{m+1} (b c-a d) (B c-A d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c d^2 e (m+1)}-\frac{(e x)^{m+1} (-a B d-A b d+b B c)}{d^2 e (m+1)}+\frac{b B (e x)^{m+3}}{d e^3 (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.5286, size = 99, normalized size = 0.82 \[ \frac{B b \left (e x\right )^{m + 3}}{d e^{3} \left (m + 3\right )} + \frac{\left (e x\right )^{m + 1} \left (A b d + B a d - B b c\right )}{d^{2} e \left (m + 1\right )} + \frac{\left (e x\right )^{m + 1} \left (A d - B c\right ) \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{d x^{2}}{c}} \right )}}{c d^{2} e \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b*(e*x)**(m + 3)/(d*e**3*(m + 3)) + (e*x)**(m + 1)*(A*b*d + B*a*d - B*b*c)/(d*
*2*e*(m + 1)) + (e*x)**(m + 1)*(A*d - B*c)*(a*d - b*c)*hyper((1, m/2 + 1/2), (m/
2 + 3/2,), -d*x**2/c)/(c*d**2*e*(m + 1))

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Mathematica [A]  time = 0.202129, size = 121, normalized size = 1.01 \[ \frac{x (e x)^m \left (\frac{x^2 (a B+A b) \, _2F_1\left (1,\frac{m+3}{2};\frac{m+5}{2};-\frac{d x^2}{c}\right )}{m+3}+\frac{a A \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{m+1}+\frac{b B x^4 \, _2F_1\left (1,\frac{m+5}{2};\frac{m+7}{2};-\frac{d x^2}{c}\right )}{m+5}\right )}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( b{x}^{2}+a \right ) \left ( B{x}^{2}+A \right ) }{d{x}^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 22.0524, size = 428, normalized size = 3.57 \[ \frac{A a e^{m} m x x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{A a e^{m} x x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{A b e^{m} m x^{3} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{3 A b e^{m} x^{3} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{B a e^{m} m x^{3} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{3 B a e^{m} x^{3} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{B b e^{m} m x^{5} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{5}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )} + \frac{5 B b e^{m} x^{5} x^{m} \Phi \left (\frac{d x^{2} e^{i \pi }}{c}, 1, \frac{m}{2} + \frac{5}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )}{4 c \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*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*e**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*b*e**m*m*x**3*x**m*ler
chphi(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*b*e**m*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*ga
mma(m/2 + 3/2)/(4*c*gamma(m/2 + 5/2)) + B*a*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*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)) + B*b*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*gamma(m/2 + 7/2)) + 5*B*b*e**m*x**5*x**m*ler
chphi(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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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