3.15.33 \(\int \frac {(b d+2 c d x)^m}{(a+b x+c x^2)^{3/2}} \, dx\) [1433]

Optimal. Leaf size=94 \[ -\frac {4 (b d+2 c d x)^{1+m} \, _2F_1\left (1,\frac {m}{2};\frac {3+m}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right ) d (1+m) \sqrt {4 a-\frac {b^2}{c}+\frac {(b+2 c x)^2}{c}}} \]

[Out]

-4*(2*c*d*x+b*d)^(1+m)*hypergeom([1, 1/2*m],[3/2+1/2*m],(2*c*x+b)^2/(-4*a*c+b^2))/(-4*a*c+b^2)/d/(1+m)/(4*a-b^
2/c+(2*c*x+b)^2/c)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 109, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {708, 372, 371} \begin {gather*} -\frac {2 \sqrt {1-\frac {(b+2 c x)^2}{b^2-4 a c}} (d (b+2 c x))^{m+1} \, _2F_1\left (\frac {3}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(d*(b + 2*c*x))^(1 + m)*Sqrt[1 - (b + 2*c*x)^2/(b^2 - 4*a*c)]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2,
(b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)*d*(1 + m)*Sqrt[a + b*x + c*x^2])

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/
(1 + b*(x^n/a))^FracPart[p]), Int[(c*x)^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 708

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[x^m*(
a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int \frac {(b d+2 c d x)^m}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {x^m}{\left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )^{3/2}} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=\frac {\sqrt {4+\frac {(b d+2 c d x)^2}{\left (a-\frac {b^2}{4 c}\right ) c d^2}} \text {Subst}\left (\int \frac {x^m}{\left (1+\frac {x^2}{4 \left (a-\frac {b^2}{4 c}\right ) c d^2}\right )^{3/2}} \, dx,x,b d+2 c d x\right )}{4 \left (a-\frac {b^2}{4 c}\right ) c d \sqrt {a+b x+c x^2}}\\ &=-\frac {2 (d (b+2 c x))^{1+m} \sqrt {1-\frac {(b+2 c x)^2}{b^2-4 a c}} \, _2F_1\left (\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right ) d (1+m) \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.54, size = 110, normalized size = 1.17 \begin {gather*} -\frac {4 (b+2 c x) (d (b+2 c x))^m \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \, _2F_1\left (\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right ) (1+m) \sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(b + 2*c*x)*(d*(b + 2*c*x))^m*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[3/2, (1 + m)/2,
(3 + m)/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/((b^2 - 4*a*c)*(1 + m)*Sqrt[a + x*(b + c*x)])

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Maple [F]
time = 0.33, size = 0, normalized size = 0.00 \[\int \frac {\left (2 c d x +b d \right )^{m}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral(sqrt(c*x^2 + b*x + a)*(2*c*d*x + b*d)^m/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d*(b + 2*c*x))**m/(a + b*x + c*x**2)**(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((2*c*d*x+b*d)^m/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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