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

Optimal. Leaf size=133 \[ \frac{d x^{m+1} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3 (m+1)}+\frac{x^{m+1} (b c-a d)^3 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b^3 (m+1)}+\frac{d^2 x^{m+3} (3 b c-a d)}{b^2 (m+3)}+\frac{d^3 x^{m+5}}{b (m+5)} \]

[Out]

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

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Rubi [A]  time = 0.209192, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{d x^{m+1} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3 (m+1)}+\frac{x^{m+1} (b c-a d)^3 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b^3 (m+1)}+\frac{d^2 x^{m+3} (3 b c-a d)}{b^2 (m+3)}+\frac{d^3 x^{m+5}}{b (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.481, size = 116, normalized size = 0.87 \[ \frac{d^{3} x^{m + 5}}{b \left (m + 5\right )} - \frac{d^{2} x^{m + 3} \left (a d - 3 b c\right )}{b^{2} \left (m + 3\right )} + \frac{d x^{m + 1} \left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right )}{b^{3} \left (m + 1\right )} - \frac{x^{m + 1} \left (a d - b c\right )^{3}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a b^{3} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.266775, size = 159, normalized size = 1.2 \[ \frac{x^{m+1} \left (\frac{c^3 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{m+1}+d x^2 \left (\frac{3 c^2 \, _2F_1\left (1,\frac{m+3}{2};\frac{m+5}{2};-\frac{b x^2}{a}\right )}{m+3}+d x^2 \left (\frac{3 c \, _2F_1\left (1,\frac{m+5}{2};\frac{m+7}{2};-\frac{b x^2}{a}\right )}{m+5}+\frac{d x^2 \, _2F_1\left (1,\frac{m+7}{2};\frac{m+9}{2};-\frac{b x^2}{a}\right )}{m+7}\right )\right )\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( d{x}^{2}+c \right ) ^{3}}{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 64.6448, size = 411, normalized size = 3.09 \[ \frac{c^{3} m x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{c^{3} x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{3 c^{2} d m x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{9 c^{2} d x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{3 c d^{2} m x^{5} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{5}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )} + \frac{15 c d^{2} x^{5} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{5}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )} + \frac{d^{3} m x^{7} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{7}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{9}{2}\right )} + \frac{7 d^{3} x^{7} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{7}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{9}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**3*m*x*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/
(4*a*gamma(m/2 + 3/2)) + c**3*x*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 +
1/2)*gamma(m/2 + 1/2)/(4*a*gamma(m/2 + 3/2)) + 3*c**2*d*m*x**3*x**m*lerchphi(b*
x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*a*gamma(m/2 + 5/2)) +
9*c**2*d*x**3*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 +
3/2)/(4*a*gamma(m/2 + 5/2)) + 3*c*d**2*m*x**5*x**m*lerchphi(b*x**2*exp_polar(I*p
i)/a, 1, m/2 + 5/2)*gamma(m/2 + 5/2)/(4*a*gamma(m/2 + 7/2)) + 15*c*d**2*x**5*x**
m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 5/2)*gamma(m/2 + 5/2)/(4*a*gamma(m
/2 + 7/2)) + d**3*m*x**7*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 7/2)*g
amma(m/2 + 7/2)/(4*a*gamma(m/2 + 9/2)) + 7*d**3*x**7*x**m*lerchphi(b*x**2*exp_po
lar(I*pi)/a, 1, m/2 + 7/2)*gamma(m/2 + 7/2)/(4*a*gamma(m/2 + 9/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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