3.2845 \(\int \frac{\left (a+b (c+d x)^2\right )^p}{c+d x} \, dx\)

Optimal. Leaf size=52 \[ -\frac{\left (a+b (c+d x)^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b (c+d x)^2}{a}+1\right )}{2 a d (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.117322, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{\left (a+b (c+d x)^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b (c+d x)^2}{a}+1\right )}{2 a d (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.1708, size = 39, normalized size = 0.75 \[ - \frac{\left (a + b \left (c + d x\right )^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{1 + \frac{b \left (c + d x\right )^{2}}{a}} \right )}}{2 a d \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0477715, size = 66, normalized size = 1.27 \[ \frac{\left (\frac{a}{b (c+d x)^2}+1\right )^{-p} \left (a+b (c+d x)^2\right )^p \, _2F_1\left (-p,-p;1-p;-\frac{a}{b (c+d x)^2}\right )}{2 d p} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.154, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b \left ( dx+c \right ) ^{2} \right ) ^{p}}{dx+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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