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

Optimal. Leaf size=201 \[ \frac{x^{m+1} (b c-a d)^2 (a d (m+5)+b (c-c m)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{2 a^2 b^3 (m+1)}-\frac{d x^{m+1} \left (a^2 d^2 (m+5)-3 a b c d (m+3)+2 b^2 c^2 (m+1)\right )}{2 a b^3 (m+1)}-\frac{d^2 x^{m+3} (b c (m+3)-a d (m+5))}{2 a b^2 (m+3)}+\frac{x^{m+1} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.55184, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{x^{m+1} (b c-a d)^2 (a d (m+5)+b (c-c m)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{2 a^2 b^3 (m+1)}-\frac{d x^{m+1} \left (a^2 d^2 (m+5)-3 a b c d (m+3)+2 b^2 c^2 (m+1)\right )}{2 a b^3 (m+1)}-\frac{d^2 x^{m+3} (b c (m+3)-a d (m+5))}{2 a b^2 (m+3)}+\frac{x^{m+1} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 139.365, size = 206, normalized size = 1.02 \[ - \frac{x^{m + 1} \left (c + d x^{2}\right )^{2} \left (a d - b c\right )}{2 a b \left (a + b x^{2}\right )} + \frac{d^{2} x^{m + 3} \left (a d m + 5 a d - b c m - 3 b c\right )}{2 a b^{2} \left (m + 3\right )} - \frac{d x^{m + 1} \left (a^{2} d^{2} m + 5 a^{2} d^{2} - 3 a b c d m - 9 a b c d + 2 b^{2} c^{2} m + 2 b^{2} c^{2}\right )}{2 a b^{3} \left (m + 1\right )} + \frac{x^{m + 1} \left (a d - b c\right )^{2} \left (a d m + 5 a d - b c m + b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{2 a^{2} b^{3} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(d*x**2+c)**3/(b*x**2+a)**2,x)

[Out]

-x**(m + 1)*(c + d*x**2)**2*(a*d - b*c)/(2*a*b*(a + b*x**2)) + d**2*x**(m + 3)*(
a*d*m + 5*a*d - b*c*m - 3*b*c)/(2*a*b**2*(m + 3)) - d*x**(m + 1)*(a**2*d**2*m +
5*a**2*d**2 - 3*a*b*c*d*m - 9*a*b*c*d + 2*b**2*c**2*m + 2*b**2*c**2)/(2*a*b**3*(
m + 1)) + x**(m + 1)*(a*d - b*c)**2*(a*d*m + 5*a*d - b*c*m + b*c)*hyper((1, m/2
+ 1/2), (m/2 + 3/2,), -b*x**2/a)/(2*a**2*b**3*(m + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.267561, size = 159, normalized size = 0.79 \[ \frac{x^{m+1} \left (\frac{c^3 \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{m+1}+d x^2 \left (\frac{3 c^2 \, _2F_1\left (2,\frac{m+3}{2};\frac{m+5}{2};-\frac{b x^2}{a}\right )}{m+3}+d x^2 \left (\frac{3 c \, _2F_1\left (2,\frac{m+5}{2};\frac{m+7}{2};-\frac{b x^2}{a}\right )}{m+5}+\frac{d x^2 \, _2F_1\left (2,\frac{m+7}{2};\frac{m+9}{2};-\frac{b x^2}{a}\right )}{m+7}\right )\right )\right )}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( d{x}^{2}+c \right ) ^{3}}{ \left ( b{x}^{2}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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(x**m*(d*x**2+c)**3/(b*x**2+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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