3.381 \(\int x^5 (d+e x) \left (a+b x^2\right )^p \, dx\)

Optimal. Leaf size=125 \[ \frac{a^2 d \left (a+b x^2\right )^{p+1}}{2 b^3 (p+1)}-\frac{a d \left (a+b x^2\right )^{p+2}}{b^3 (p+2)}+\frac{d \left (a+b x^2\right )^{p+3}}{2 b^3 (p+3)}+\frac{1}{7} e x^7 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right ) \]

[Out]

(a^2*d*(a + b*x^2)^(1 + p))/(2*b^3*(1 + p)) - (a*d*(a + b*x^2)^(2 + p))/(b^3*(2
+ p)) + (d*(a + b*x^2)^(3 + p))/(2*b^3*(3 + p)) + (e*x^7*(a + b*x^2)^p*Hypergeom
etric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(7*(1 + (b*x^2)/a)^p)

_______________________________________________________________________________________

Rubi [A]  time = 0.20005, 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 \[ \frac{a^2 d \left (a+b x^2\right )^{p+1}}{2 b^3 (p+1)}-\frac{a d \left (a+b x^2\right )^{p+2}}{b^3 (p+2)}+\frac{d \left (a+b x^2\right )^{p+3}}{2 b^3 (p+3)}+\frac{1}{7} e x^7 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^5*(d + e*x)*(a + b*x^2)^p,x]

[Out]

(a^2*d*(a + b*x^2)^(1 + p))/(2*b^3*(1 + p)) - (a*d*(a + b*x^2)^(2 + p))/(b^3*(2
+ p)) + (d*(a + b*x^2)^(3 + p))/(2*b^3*(3 + p)) + (e*x^7*(a + b*x^2)^p*Hypergeom
etric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(7*(1 + (b*x^2)/a)^p)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.9166, size = 104, normalized size = 0.83 \[ \frac{a^{2} d \left (a + b x^{2}\right )^{p + 1}}{2 b^{3} \left (p + 1\right )} - \frac{a d \left (a + b x^{2}\right )^{p + 2}}{b^{3} \left (p + 2\right )} + \frac{e x^{7} \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{7} + \frac{d \left (a + b x^{2}\right )^{p + 3}}{2 b^{3} \left (p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(e*x+d)*(b*x**2+a)**p,x)

[Out]

a**2*d*(a + b*x**2)**(p + 1)/(2*b**3*(p + 1)) - a*d*(a + b*x**2)**(p + 2)/(b**3*
(p + 2)) + e*x**7*(1 + b*x**2/a)**(-p)*(a + b*x**2)**p*hyper((-p, 7/2), (9/2,),
-b*x**2/a)/7 + d*(a + b*x**2)**(p + 3)/(2*b**3*(p + 3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.240914, size = 183, normalized size = 1.46 \[ \frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (7 d \left (2 a^3 \left (\left (\frac{b x^2}{a}+1\right )^p-1\right )-2 a^2 b p x^2 \left (\frac{b x^2}{a}+1\right )^p+b^3 \left (p^2+3 p+2\right ) x^6 \left (\frac{b x^2}{a}+1\right )^p+a b^2 p (p+1) x^4 \left (\frac{b x^2}{a}+1\right )^p\right )+2 b^3 e \left (p^3+6 p^2+11 p+6\right ) x^7 \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right )\right )}{14 b^3 (p+1) (p+2) (p+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(d + e*x)*(a + b*x^2)^p,x]

[Out]

((a + b*x^2)^p*(7*d*(-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*e*(6 + 11*p + 6*p^2 + p^3)*x^7*Hypergeometric2F1[7/2, -p, 9/
2, -((b*x^2)/a)]))/(14*b^3*(1 + p)*(2 + p)*(3 + p)*(1 + (b*x^2)/a)^p)

_______________________________________________________________________________________

Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{x}^{5} \left ( ex+d \right ) \left ( b{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(e*x+d)*(b*x^2+a)^p,x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ e \int{\left (b x^{2} + a\right )}^{p} x^{6}\,{d x} + \frac{{\left ({\left (p^{2} + 3 \, p + 2\right )} b^{3} x^{6} +{\left (p^{2} + p\right )} a b^{2} x^{4} - 2 \, a^{2} b p x^{2} + 2 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{p} d}{2 \,{\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(b*x^2 + a)^p*x^5,x, algorithm="maxima")

[Out]

e*integrate((b*x^2 + a)^p*x^6, x) + 1/2*((p^2 + 3*p + 2)*b^3*x^6 + (p^2 + p)*a*b
^2*x^4 - 2*a^2*b*p*x^2 + 2*a^3)*(b*x^2 + a)^p*d/((p^3 + 6*p^2 + 11*p + 6)*b^3)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x^{6} + d x^{5}\right )}{\left (b x^{2} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(b*x^2 + a)^p*x^5,x, algorithm="fricas")

[Out]

integral((e*x^6 + d*x^5)*(b*x^2 + a)^p, x)

_______________________________________________________________________________________

Sympy [A]  time = 117.074, 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**5*(e*x+d)*(b*x**2+a)**p,x)

[Out]

a**p*e*x**7*hyper((7/2, -p), (9/2,), b*x**2*exp_polar(I*pi)/a)/7 + d*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))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}{\left (b x^{2} + a\right )}^{p} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(b*x^2 + a)^p*x^5,x, algorithm="giac")

[Out]

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