3.45 \(\int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx\)

Optimal. Leaf size=495 \[ \frac{(e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b^3 e (m+2 p+3) (m+2 p+5) (m+2 p+7)}-\frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right ) (b c (m+2 p+3) ((m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 b c (p+2) (a B (m+1)-A b (m+2 p+7)))-a (m+1) (2 b c d (p+2) (a B (m+1)-A b (m+2 p+7))+d (m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 (b c-a d) (a B d (m+5)-b (A d (m+2 p+7)+4 B c))))}{b^3 e (m+1) (m+2 p+3) (m+2 p+5) (m+2 p+7)}-\frac{\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (a B d (m+5)-b (A d (m+2 p+7)+4 B c))}{b^2 e (m+2 p+5) (m+2 p+7)}+\frac{B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)} \]

[Out]

((a^2*B*d^2*(15 + 8*m + m^2) + b^2*c*(8*B*c + A*d*(7 + m + 2*p)^2) - a*b*d*(A*d*
(3 + m)*(7 + m + 2*p) + B*c*(27 + m^2 + 2*p + 2*m*(6 + p))))*(e*x)^(1 + m)*(a +
b*x^2)^(1 + p))/(b^3*e*(3 + m + 2*p)*(5 + m + 2*p)*(7 + m + 2*p)) - ((a*B*d*(5 +
m) - b*(4*B*c + A*d*(7 + m + 2*p)))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^
2))/(b^2*e*(5 + m + 2*p)*(7 + m + 2*p)) + (B*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(
c + d*x^2)^2)/(b*e*(7 + m + 2*p)) - ((b*c*(3 + m + 2*p)*(2*b*c*(2 + p)*(a*B*(1 +
m) - A*b*(7 + m + 2*p)) + (b*c - a*d)*(1 + m)*(a*B*(5 + m) - A*b*(7 + m + 2*p))
) - a*(1 + m)*(2*b*c*d*(2 + p)*(a*B*(1 + m) - A*b*(7 + m + 2*p)) + d*(b*c - a*d)
*(1 + m)*(a*B*(5 + m) - A*b*(7 + m + 2*p)) + 2*(b*c - a*d)*(a*B*d*(5 + m) - b*(4
*B*c + A*d*(7 + m + 2*p)))))*(e*x)^(1 + m)*(a + b*x^2)^p*Hypergeometric2F1[(1 +
m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(b^3*e*(1 + m)*(3 + m + 2*p)*(5 + m + 2*p)*(
7 + m + 2*p)*(1 + (b*x^2)/a)^p)

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Rubi [A]  time = 1.93388, antiderivative size = 464, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ -\frac{(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right ) \left (\frac{a \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b (m+2 p+3)}+c \left ((b c-a d) (a B (m+5)-A b (m+2 p+7))+\frac{2 b c (p+2) (a B (m+1)-A b (m+2 p+7))}{m+1}\right )\right )}{b^2 e (m+2 p+5) (m+2 p+7)}+\frac{(e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b^3 e (m+2 p+3) (m+2 p+5) (m+2 p+7)}+\frac{\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+5)+A b d (m+2 p+7)+4 b B c)}{b^2 e (m+2 p+5) (m+2 p+7)}+\frac{B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)} \]

Antiderivative was successfully verified.

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

[Out]

((a^2*B*d^2*(15 + 8*m + m^2) + b^2*c*(8*B*c + A*d*(7 + m + 2*p)^2) - a*b*d*(A*d*
(3 + m)*(7 + m + 2*p) + B*c*(27 + m^2 + 2*p + 2*m*(6 + p))))*(e*x)^(1 + m)*(a +
b*x^2)^(1 + p))/(b^3*e*(3 + m + 2*p)*(5 + m + 2*p)*(7 + m + 2*p)) + ((4*b*B*c -
a*B*d*(5 + m) + A*b*d*(7 + m + 2*p))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^
2))/(b^2*e*(5 + m + 2*p)*(7 + m + 2*p)) + (B*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(
c + d*x^2)^2)/(b*e*(7 + m + 2*p)) - ((c*((2*b*c*(2 + p)*(a*B*(1 + m) - A*b*(7 +
m + 2*p)))/(1 + m) + (b*c - a*d)*(a*B*(5 + m) - A*b*(7 + m + 2*p))) + (a*(a^2*B*
d^2*(15 + 8*m + m^2) + b^2*c*(8*B*c + A*d*(7 + m + 2*p)^2) - a*b*d*(A*d*(3 + m)*
(7 + m + 2*p) + B*c*(27 + m^2 + 2*p + 2*m*(6 + p)))))/(b*(3 + m + 2*p)))*(e*x)^(
1 + m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/
(b^2*e*(5 + m + 2*p)*(7 + m + 2*p)*(1 + (b*x^2)/a)^p)

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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)**p*(B*x**2+A)*(d*x**2+c)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.547255, size = 198, normalized size = 0.4 \[ x (e x)^m \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (\frac{c x^2 (2 A d+B c) \, _2F_1\left (\frac{m+3}{2},-p;\frac{m+5}{2};-\frac{b x^2}{a}\right )}{m+3}+d x^4 \left (\frac{(A d+2 B c) \, _2F_1\left (\frac{m+5}{2},-p;\frac{m+7}{2};-\frac{b x^2}{a}\right )}{m+5}+\frac{B d x^2 \, _2F_1\left (\frac{m+7}{2},-p;\frac{m+9}{2};-\frac{b x^2}{a}\right )}{m+7}\right )+\frac{A c^2 \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right )}{m+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.094, size = 0, normalized size = 0. \[ \int \left ( ex \right ) ^{m} \left ( b{x}^{2}+a \right ) ^{p} \left ( B{x}^{2}+A \right ) \left ( d{x}^{2}+c \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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