3.382 \(\int x^4 (d+e x) \left (a+b x^2\right )^p \, dx\)

Optimal. Leaf size=125 \[ \frac{a^2 e \left (a+b x^2\right )^{p+1}}{2 b^3 (p+1)}-\frac{a e \left (a+b x^2\right )^{p+2}}{b^3 (p+2)}+\frac{e \left (a+b x^2\right )^{p+3}}{2 b^3 (p+3)}+\frac{1}{5} d x^5 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right ) \]

[Out]

(a^2*e*(a + b*x^2)^(1 + p))/(2*b^3*(1 + p)) - (a*e*(a + b*x^2)^(2 + p))/(b^3*(2
+ p)) + (e*(a + b*x^2)^(3 + p))/(2*b^3*(3 + p)) + (d*x^5*(a + b*x^2)^p*Hypergeom
etric2F1[5/2, -p, 7/2, -((b*x^2)/a)])/(5*(1 + (b*x^2)/a)^p)

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Rubi [A]  time = 0.1958, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{a^2 e \left (a+b x^2\right )^{p+1}}{2 b^3 (p+1)}-\frac{a e \left (a+b x^2\right )^{p+2}}{b^3 (p+2)}+\frac{e \left (a+b x^2\right )^{p+3}}{2 b^3 (p+3)}+\frac{1}{5} d x^5 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^4*(d + e*x)*(a + b*x^2)^p,x]

[Out]

(a^2*e*(a + b*x^2)^(1 + p))/(2*b^3*(1 + p)) - (a*e*(a + b*x^2)^(2 + p))/(b^3*(2
+ p)) + (e*(a + b*x^2)^(3 + p))/(2*b^3*(3 + p)) + (d*x^5*(a + b*x^2)^p*Hypergeom
etric2F1[5/2, -p, 7/2, -((b*x^2)/a)])/(5*(1 + (b*x^2)/a)^p)

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Rubi in Sympy [A]  time = 28.7686, size = 104, normalized size = 0.83 \[ \frac{a^{2} e \left (a + b x^{2}\right )^{p + 1}}{2 b^{3} \left (p + 1\right )} - \frac{a e \left (a + b x^{2}\right )^{p + 2}}{b^{3} \left (p + 2\right )} + \frac{d x^{5} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{5} + \frac{e \left (a + b x^{2}\right )^{p + 3}}{2 b^{3} \left (p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(e*x+d)*(b*x**2+a)**p,x)

[Out]

a**2*e*(a + b*x**2)**(p + 1)/(2*b**3*(p + 1)) - a*e*(a + b*x**2)**(p + 2)/(b**3*
(p + 2)) + d*x**5*(1 + b*x**2/a)**(-p)*(a + b*x**2)**p*hyper((-p, 5/2), (7/2,),
-b*x**2/a)/5 + e*(a + b*x**2)**(p + 3)/(2*b**3*(p + 3))

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Mathematica [A]  time = 0.203704, size = 183, normalized size = 1.46 \[ \frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (5 e \left (2 a^3 \left (\left (\frac{b x^2}{a}+1\right )^p-1\right )-2 a^2 b p x^2 \left (\frac{b x^2}{a}+1\right )^p+b^3 \left (p^2+3 p+2\right ) x^6 \left (\frac{b x^2}{a}+1\right )^p+a b^2 p (p+1) x^4 \left (\frac{b x^2}{a}+1\right )^p\right )+2 b^3 d \left (p^3+6 p^2+11 p+6\right ) x^5 \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right )\right )}{10 b^3 (p+1) (p+2) (p+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(d + e*x)*(a + b*x^2)^p,x]

[Out]

((a + b*x^2)^p*(5*e*(-2*a^2*b*p*x^2*(1 + (b*x^2)/a)^p + a*b^2*p*(1 + p)*x^4*(1 +
(b*x^2)/a)^p + b^3*(2 + 3*p + p^2)*x^6*(1 + (b*x^2)/a)^p + 2*a^3*(-1 + (1 + (b*
x^2)/a)^p)) + 2*b^3*d*(6 + 11*p + 6*p^2 + p^3)*x^5*Hypergeometric2F1[5/2, -p, 7/
2, -((b*x^2)/a)]))/(10*b^3*(1 + p)*(2 + p)*(3 + p)*(1 + (b*x^2)/a)^p)

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{x}^{4} \left ( ex+d \right ) \left ( b{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(e*x+d)*(b*x^2+a)^p,x)

[Out]

int(x^4*(e*x+d)*(b*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}{\left (b x^{2} + a\right )}^{p} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(b*x^2 + a)^p*x^4,x, algorithm="maxima")

[Out]

integrate((e*x + d)*(b*x^2 + a)^p*x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x^{5} + d x^{4}\right )}{\left (b x^{2} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(b*x^2 + a)^p*x^4,x, algorithm="fricas")

[Out]

integral((e*x^5 + d*x^4)*(b*x^2 + a)^p, x)

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Sympy [A]  time = 83.1233, size = 1012, normalized size = 8.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(e*x+d)*(b*x**2+a)**p,x)

[Out]

a**p*d*x**5*hyper((5/2, -p), (7/2,), b*x**2*exp_polar(I*pi)/a)/5 + e*Piecewise((
a**p*x**6/6, Eq(b, 0)), (2*a**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**3 + 8*a
*b**4*x**2 + 4*b**5*x**4) + 2*a**2*log(I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**3 + 8
*a*b**4*x**2 + 4*b**5*x**4) + 3*a**2/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4)
+ 4*a*b*x**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**
5*x**4) + 4*a*b*x**2*log(I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**3 + 8*a*b**4*x**2 +
4*b**5*x**4) + 4*a*b*x**2/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 2*b**2*
x**4*log(-I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) +
2*b**2*x**4*log(I*sqrt(a)*sqrt(1/b) + x)/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*
x**4), Eq(p, -3)), (-2*a**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**3 + 2*b**4*x**
2) - 2*a**2*log(I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**3 + 2*b**4*x**2) - 2*a**2/(2*a*
b**3 + 2*b**4*x**2) - 2*a*b*x**2*log(-I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**3 + 2*b**
4*x**2) - 2*a*b*x**2*log(I*sqrt(a)*sqrt(1/b) + x)/(2*a*b**3 + 2*b**4*x**2) + b**
2*x**4/(2*a*b**3 + 2*b**4*x**2), Eq(p, -2)), (a**2*log(-I*sqrt(a)*sqrt(1/b) + x)
/(2*b**3) + a**2*log(I*sqrt(a)*sqrt(1/b) + x)/(2*b**3) - a*x**2/(2*b**2) + x**4/
(4*b), Eq(p, -1)), (2*a**3*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**2 + 22*b**3
*p + 12*b**3) - 2*a**2*b*p*x**2*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**2 + 22
*b**3*p + 12*b**3) + a*b**2*p**2*x**4*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**
2 + 22*b**3*p + 12*b**3) + a*b**2*p*x**4*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*
p**2 + 22*b**3*p + 12*b**3) + b**3*p**2*x**6*(a + b*x**2)**p/(2*b**3*p**3 + 12*b
**3*p**2 + 22*b**3*p + 12*b**3) + 3*b**3*p*x**6*(a + b*x**2)**p/(2*b**3*p**3 + 1
2*b**3*p**2 + 22*b**3*p + 12*b**3) + 2*b**3*x**6*(a + b*x**2)**p/(2*b**3*p**3 +
12*b**3*p**2 + 22*b**3*p + 12*b**3), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}{\left (b x^{2} + a\right )}^{p} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(b*x^2 + a)^p*x^4,x, algorithm="giac")

[Out]

integrate((e*x + d)*(b*x^2 + a)^p*x^4, x)