3.2017 \(\int \frac{(a+b x) (d+e x)^2}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{(a+b x) (d+e x)^3}{3 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

((a + b*x)*(d + e*x)^3)/(3*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.0298452, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 32} \[ \frac{(a+b x) (d+e x)^3}{3 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)*(d + e*x)^3)/(3*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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]

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{(a+b x) (d+e x)^2}{\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)^2}{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)^2 \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(a+b x) (d+e x)^3}{3 e \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0089984, size = 30, normalized size = 0.77 \[ \frac{(a+b x) (d+e x)^3}{3 e \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)*(d + e*x)^3)/(3*e*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.003, size = 36, normalized size = 0.9 \begin{align*}{\frac{x \left ({e}^{2}{x}^{2}+3\,edx+3\,{d}^{2} \right ) \left ( bx+a \right ) }{3}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x*(e^2*x^2+3*d*e*x+3*d^2)*(b*x+a)/((b*x+a)^2)^(1/2)

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Maxima [B]  time = 0.998151, size = 392, normalized size = 10.05 \begin{align*} -\frac{5 \, a^{3} b^{2} e^{2} \log \left (x + \frac{a}{b}\right )}{3 \,{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{5 \, a^{2} b e^{2} x}{3 \,{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{5 \, a e^{2} x^{2}}{6 \, \sqrt{b^{2}}} + a \sqrt{\frac{1}{b^{2}}} d^{2} \log \left (x + \frac{a}{b}\right ) + \frac{2 \, a^{3} \sqrt{\frac{1}{b^{2}}} e^{2} \log \left (x + \frac{a}{b}\right )}{3 \, b^{2}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} e^{2} x^{2}}{3 \, b} + \frac{{\left (2 \, b d e + a e^{2}\right )} a^{2} b^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{{\left (2 \, b d e + a e^{2}\right )} a b x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{{\left (2 \, b d e + a e^{2}\right )} x^{2}}{2 \, \sqrt{b^{2}}} - \frac{{\left (b d^{2} + 2 \, a d e\right )} a \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{b} - \frac{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} e^{2}}{3 \, b^{3}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}{\left (b d^{2} + 2 \, a d e\right )}}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/3*a^3*b^2*e^2*log(x + a/b)/(b^2)^(5/2) + 5/3*a^2*b*e^2*x/(b^2)^(3/2) - 5/6*a*e^2*x^2/sqrt(b^2) + a*sqrt(b^(
-2))*d^2*log(x + a/b) + 2/3*a^3*sqrt(b^(-2))*e^2*log(x + a/b)/b^2 + 1/3*sqrt(b^2*x^2 + 2*a*b*x + a^2)*e^2*x^2/
b + (2*b*d*e + a*e^2)*a^2*b^2*log(x + a/b)/(b^2)^(5/2) - (2*b*d*e + a*e^2)*a*b*x/(b^2)^(3/2) + 1/2*(2*b*d*e +
a*e^2)*x^2/sqrt(b^2) - (b*d^2 + 2*a*d*e)*a*sqrt(b^(-2))*log(x + a/b)/b - 2/3*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2
*e^2/b^3 + sqrt(b^2*x^2 + 2*a*b*x + a^2)*(b*d^2 + 2*a*d*e)/b^2

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Fricas [A]  time = 1.45522, size = 42, normalized size = 1.08 \begin{align*} \frac{1}{3} \, e^{2} x^{3} + d e x^{2} + d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*e^2*x^3 + d*e*x^2 + d^2*x

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Sympy [A]  time = 0.097354, size = 19, normalized size = 0.49 \begin{align*} d^{2} x + d e x^{2} + \frac{e^{2} x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**2*x + d*e*x**2 + e**2*x**3/3

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Giac [A]  time = 1.09422, size = 24, normalized size = 0.62 \begin{align*} \frac{1}{3} \,{\left (x e + d\right )}^{3} e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x*e + d)^3*e^(-1)*sgn(b*x + a)