3.1432 \(\int \frac {(b d+2 c d x)^m}{\sqrt {a+b x+c x^2}} \, dx\)

Optimal. Leaf size=96 \[ -\frac {\sqrt {4 a-\frac {b^2}{c}+\frac {(b+2 c x)^2}{c}} (b d+2 c d x)^{m+1} \, _2F_1\left (1,\frac {m+2}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 104, normalized size of antiderivative = 1.08, 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 = {694, 365, 364} \[ \frac {\sqrt {1-\frac {(b+2 c x)^2}{b^2-4 a c}} (d (b+2 c x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{2 c d (m+1) \sqrt {a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 365

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 694

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

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Mathematica [A]  time = 0.04, size = 102, normalized size = 1.06 \[ \frac {(b+2 c x) \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^m \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c (m+1) \sqrt {a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (2 \, c d x + b d\right )}^{m}}{\sqrt {c x^{2} + b x + a}}, x\right ) \]

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)^(1/2),x, algorithm="fricas")

[Out]

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

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

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)^(1/2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 1.26, size = 0, normalized size = 0.00 \[ \int \frac {\left (2 c d x +b d \right )^{m}}{\sqrt {c \,x^{2}+b x +a}}\, dx \]

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)^(1/2),x)

[Out]

int((2*c*d*x+b*d)^m/(c*x^2+b*x+a)^(1/2),x)

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

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)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,d+2\,c\,d\,x\right )}^m}{\sqrt {c\,x^2+b\,x+a}} \,d x \]

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)^(1/2),x)

[Out]

int((b*d + 2*c*d*x)^m/(a + b*x + c*x^2)^(1/2), x)

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

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)**(1/2),x)

[Out]

Integral((d*(b + 2*c*x))**m/sqrt(a + b*x + c*x**2), x)

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