3.18.54 \(\int \frac {(a^2+2 a b x+b^2 x^2)^p}{(d+e x)^{3/2}} \, dx\) [1754]

Optimal. Leaf size=81 \[ -\frac {2 \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (-\frac {1}{2},-2 p;\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{e \sqrt {d+e x}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {660, 72, 71} \begin {gather*} -\frac {2 \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (-\frac {1}{2},-2 p;\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 72

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c -
a*d)), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 660

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{(d+e x)^{3/2}} \, dx &=\left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \frac {\left (a b+b^2 x\right )^{2 p}}{(d+e x)^{3/2}} \, dx\\ &=\left (\left (\frac {e \left (a b+b^2 x\right )}{-b^2 d+a b e}\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \frac {\left (-\frac {a e}{b d-a e}-\frac {b e x}{b d-a e}\right )^{2 p}}{(d+e x)^{3/2}} \, dx\\ &=-\frac {2 \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (-\frac {1}{2},-2 p;\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{e \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 71, normalized size = 0.88 \begin {gather*} -\frac {2 \left (\frac {e (a+b x)}{-b d+a e}\right )^{-2 p} \left ((a+b x)^2\right )^p \, _2F_1\left (-\frac {1}{2},-2 p;\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \frac {\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{\left (e x +d \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x\right )^{2}\right )^{p}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p}{{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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