3.18.93 \(\int \frac {(a+b x) (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=62 \[ \frac {d x (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e x^2 (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {770, 21} \begin {gather*} \frac {d x (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e x^2 (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x*(a + b*x))/Sqrt[a^2 + 2*a*b*x + b^2*x^2] + (e*x^2*(a + b*x))/(2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(a+b x) (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(a+b x) (d+e x)}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int (d+e x) \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {d x (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e x^2 (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.45 \begin {gather*} \frac {x (a+b x) (2 d+e x)}{2 \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*x)*(2*d + e*x))/(2*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [F]  time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(d + e*x))/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

Defer[IntegrateAlgebraic][((a + b*x)*(d + e*x))/Sqrt[a^2 + 2*a*b*x + b^2*x^2], x]

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fricas [A]  time = 0.42, size = 10, normalized size = 0.16 \begin {gather*} \frac {1}{2} \, e x^{2} + d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/2*e*x^2 + d*x

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giac [A]  time = 0.15, size = 19, normalized size = 0.31 \begin {gather*} \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

1/2*(x^2*e + 2*d*x)*sgn(b*x + a)

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maple [A]  time = 0.05, size = 25, normalized size = 0.40 \begin {gather*} \frac {\left (e x +2 d \right ) \left (b x +a \right ) x}{2 \sqrt {\left (b x +a \right )^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*(e*x+2*d)*(b*x+a)/((b*x+a)^2)^(1/2)

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maxima [B]  time = 0.76, size = 95, normalized size = 1.53 \begin {gather*} \frac {1}{2} \, e x^{2} - \frac {a e x}{b} + \frac {a d \log \left (x + \frac {a}{b}\right )}{b} + \frac {a^{2} e \log \left (x + \frac {a}{b}\right )}{b^{2}} - \frac {{\left (b d + a e\right )} a \log \left (x + \frac {a}{b}\right )}{b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (b d + a e\right )}}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

1/2*e*x^2 - a*e*x/b + a*d*log(x + a/b)/b + a^2*e*log(x + a/b)/b^2 - (b*d + a*e)*a*log(x + a/b)/b^2 + sqrt(b^2*
x^2 + 2*a*b*x + a^2)*(b*d + a*e)/b^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (a+b\,x\right )\,\left (d+e\,x\right )}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.09, size = 8, normalized size = 0.13 \begin {gather*} d x + \frac {e x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/((b*x+a)**2)**(1/2),x)

[Out]

d*x + e*x**2/2

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