3.424 \(\int \frac{x^3 \left (a+b x^2\right )^p}{(d+e x)^3} \, dx\)

Optimal. Leaf size=416 \[ -\frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (3 a^2 e^4+a b d^2 e^2 (7 p+6)+b^2 d^4 \left (2 p^2+5 p+3\right )\right ) F_1\left (\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{e^3 \left (a e^2+b d^2\right )^2}+\frac{d \left (a+b x^2\right )^{p+1} \left (3 a^2 e^4+a b d^2 e^2 (7 p+6)+b^2 d^4 \left (2 p^2+5 p+3\right )\right ) \, _2F_1\left (1,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e^2 (p+1) \left (a e^2+b d^2\right )^3}+\frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (a^2 e^4+a b d^2 e^2 (6 p+5)+b^2 d^4 \left (2 p^2+5 p+3\right )\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{e^3 \left (a e^2+b d^2\right )^2}-\frac{d^2 \left (a+b x^2\right )^{p+1} \left (3 a e^2+b d^2 (p+2)\right )}{e^2 (d+e x) \left (a e^2+b d^2\right )^2}+\frac{d^3 \left (a+b x^2\right )^{p+1}}{2 e^2 (d+e x)^2 \left (a e^2+b d^2\right )} \]

[Out]

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

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Rubi [A]  time = 1.08218, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ -\frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (3 a^2 e^4+a b d^2 e^2 (7 p+6)+b^2 d^4 \left (2 p^2+5 p+3\right )\right ) F_1\left (\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{e^3 \left (a e^2+b d^2\right )^2}+\frac{d \left (a+b x^2\right )^{p+1} \left (3 a^2 e^4+a b d^2 e^2 (7 p+6)+b^2 d^4 \left (2 p^2+5 p+3\right )\right ) \, _2F_1\left (1,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e^2 (p+1) \left (a e^2+b d^2\right )^3}+\frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (a^2 e^4+a b d^2 e^2 (6 p+5)+b^2 d^4 \left (2 p^2+5 p+3\right )\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{e^3 \left (a e^2+b d^2\right )^2}-\frac{d^2 \left (a+b x^2\right )^{p+1} \left (3 a e^2+b d^2 (p+2)\right )}{e^2 (d+e x) \left (a e^2+b d^2\right )^2}+\frac{d^3 \left (a+b x^2\right )^{p+1}}{2 e^2 (d+e x)^2 \left (a e^2+b d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 86.0886, size = 464, normalized size = 1.12 \[ \frac{d^{3} \left (\frac{e \left (\sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (- \frac{e \left (- \sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (a + b x^{2}\right )^{p} \left (\frac{1}{d + e x}\right )^{2 p} \left (\frac{1}{d + e x}\right )^{- 2 p + 2} \operatorname{appellf_{1}}{\left (- 2 p + 2,- p,- p,- 2 p + 3,\frac{d - \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x},\frac{d + \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x} \right )}}{2 e^{4} \left (- p + 1\right )} - \frac{3 d^{2} \left (\frac{e \left (\sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (- \frac{e \left (- \sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (a + b x^{2}\right )^{p} \left (\frac{1}{d + e x}\right )^{2 p} \left (\frac{1}{d + e x}\right )^{- 2 p + 1} \operatorname{appellf_{1}}{\left (- 2 p + 1,- p,- p,- 2 p + 2,\frac{d - \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x},\frac{d + \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x} \right )}}{e^{4} \left (- 2 p + 1\right )} - \frac{3 d \left (\frac{e \left (\sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (- \frac{e \left (- \sqrt{b} x + \sqrt{- a}\right )}{\sqrt{b} \left (d + e x\right )}\right )^{- p} \left (a + b x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (- 2 p,- p,- p,- 2 p + 1,\frac{d - \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x},\frac{d + \frac{e \sqrt{- a}}{\sqrt{b}}}{d + e x} \right )}}{2 e^{4} p} + \frac{x \left (1 + \frac{b x^{2}}{a}\right )^{- p} \left (a + b x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*(e*(sqrt(b)*x + sqrt(-a))/(sqrt(b)*(d + e*x)))**(-p)*(-e*(-sqrt(b)*x + sqrt
(-a))/(sqrt(b)*(d + e*x)))**(-p)*(a + b*x**2)**p*(1/(d + e*x))**(2*p)*(1/(d + e*
x))**(-2*p + 2)*appellf1(-2*p + 2, -p, -p, -2*p + 3, (d - e*sqrt(-a)/sqrt(b))/(d
+ e*x), (d + e*sqrt(-a)/sqrt(b))/(d + e*x))/(2*e**4*(-p + 1)) - 3*d**2*(e*(sqrt
(b)*x + sqrt(-a))/(sqrt(b)*(d + e*x)))**(-p)*(-e*(-sqrt(b)*x + sqrt(-a))/(sqrt(b
)*(d + e*x)))**(-p)*(a + b*x**2)**p*(1/(d + e*x))**(2*p)*(1/(d + e*x))**(-2*p +
1)*appellf1(-2*p + 1, -p, -p, -2*p + 2, (d - e*sqrt(-a)/sqrt(b))/(d + e*x), (d +
e*sqrt(-a)/sqrt(b))/(d + e*x))/(e**4*(-2*p + 1)) - 3*d*(e*(sqrt(b)*x + sqrt(-a)
)/(sqrt(b)*(d + e*x)))**(-p)*(-e*(-sqrt(b)*x + sqrt(-a))/(sqrt(b)*(d + e*x)))**(
-p)*(a + b*x**2)**p*appellf1(-2*p, -p, -p, -2*p + 1, (d - e*sqrt(-a)/sqrt(b))/(d
+ e*x), (d + e*sqrt(-a)/sqrt(b))/(d + e*x))/(2*e**4*p) + x*(1 + b*x**2/a)**(-p)
*(a + b*x**2)**p*hyper((-p, 1/2), (3/2,), -b*x**2/a)/e**3

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Mathematica [A]  time = 1.06905, size = 0, normalized size = 0. \[ \int \frac{x^3 \left (a+b x^2\right )^p}{(d+e x)^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(x^3*(a + b*x^2)^p)/(d + e*x)^3, x]

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Maple [F]  time = 0.117, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( b{x}^{2}+a \right ) ^{p}}{ \left ( ex+d \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^3*(b*x^2+a)^p/(e*x+d)^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^p*x^3/(e*x + d)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^p*x^3/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), 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(x**3*(b*x**2+a)**p/(e*x+d)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^p*x^3/(e*x + d)^3, x)