3.2841 \(\int (c+d x)^4 \left (a+b (c+d x)^2\right )^p \, dx\)

Optimal. Leaf size=55 \[ \frac{(c+d x)^5 \left (a+b (c+d x)^2\right )^{p+1} \, _2F_1\left (1,p+\frac{7}{2};\frac{7}{2};-\frac{b (c+d x)^2}{a}\right )}{5 a d} \]

[Out]

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

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Rubi [A]  time = 0.122669, antiderivative size = 68, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{(c+d x)^5 \left (a+b (c+d x)^2\right )^p \left (\frac{b (c+d x)^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b (c+d x)^2}{a}\right )}{5 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.0626, size = 53, normalized size = 0.96 \[ \frac{\left (1 + \frac{b \left (c + d x\right )^{2}}{a}\right )^{- p} \left (a + b \left (c + d x\right )^{2}\right )^{p} \left (c + d x\right )^{5}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{- \frac{b \left (c + d x\right )^{2}}{a}} \right )}}{5 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0584199, size = 68, normalized size = 1.24 \[ \frac{(c+d x)^5 \left (a+b (c+d x)^2\right )^p \left (\frac{b (c+d x)^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b (c+d x)^2}{a}\right )}{5 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.224, size = 0, normalized size = 0. \[ \int \left ( dx+c \right ) ^{4} \left ( a+b \left ( dx+c \right ) ^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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