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

Optimal. Leaf size=92 \[ \frac {b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^2 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{3 e^2 (a+b x)} \]

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Rubi [A]  time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {646, 43} \begin {gather*} \frac {b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^2 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{3 e^2 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*d - a*e)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^2*(a + b*x)) + (b*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(4*e^2*(a + b*x))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 646

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 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right ) (d+e x)^2 \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e) (d+e x)^2}{e}+\frac {b^2 (d+e x)^3}{e}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x)}+\frac {b (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^2 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 67, normalized size = 0.73 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (4 a \left (3 d^2+3 d e x+e^2 x^2\right )+b x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )}{12 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(4*a*(3*d^2 + 3*d*e*x + e^2*x^2) + b*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)))/(12*(a + b*x))

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IntegrateAlgebraic [F]  time = 0.83, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 48, normalized size = 0.52 \begin {gather*} \frac {1}{4} \, b e^{2} x^{4} + a d^{2} x + \frac {1}{3} \, {\left (2 \, b d e + a e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b d^{2} + 2 \, a d e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*e^2*x^4 + a*d^2*x + 1/3*(2*b*d*e + a*e^2)*x^3 + 1/2*(b*d^2 + 2*a*d*e)*x^2

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giac [A]  time = 0.18, size = 85, normalized size = 0.92 \begin {gather*} \frac {1}{4} \, b x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, b d x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + a d x^{2} e \mathrm {sgn}\left (b x + a\right ) + a d^{2} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*x^4*e^2*sgn(b*x + a) + 2/3*b*d*x^3*e*sgn(b*x + a) + 1/2*b*d^2*x^2*sgn(b*x + a) + 1/3*a*x^3*e^2*sgn(b*x +
a) + a*d*x^2*e*sgn(b*x + a) + a*d^2*x*sgn(b*x + a)

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maple [A]  time = 0.04, size = 66, normalized size = 0.72 \begin {gather*} \frac {\left (3 b \,e^{2} x^{3}+4 x^{2} a \,e^{2}+8 x^{2} b d e +12 a d e x +6 x b \,d^{2}+12 a \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}\, x}{12 b x +12 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x*(3*b*e^2*x^3+4*a*e^2*x^2+8*b*d*e*x^2+12*a*d*e*x+6*b*d^2*x+12*a*d^2)*((b*x+a)^2)^(1/2)/(b*x+a)

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maxima [B]  time = 1.15, size = 245, normalized size = 2.66 \begin {gather*} \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} d^{2} x - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a d e x}{b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} e^{2} x}{2 \, b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a d^{2}}{2 \, b} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d e}{b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e^{2}}{2 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} e^{2} x}{4 \, b^{2}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} d e}{3 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a e^{2}}{12 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*d^2*x - sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*d*e*x/b + 1/2*sqrt(b^2*x^2 + 2*a*b*x
+ a^2)*a^2*e^2*x/b^2 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*d^2/b - sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*d*e/b^2
+ 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^3*e^2/b^3 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*e^2*x/b^2 + 2/3*(b^2*x^2
+ 2*a*b*x + a^2)^(3/2)*d*e/b^2 - 5/12*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*e^2/b^3

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mupad [B]  time = 0.78, size = 215, normalized size = 2.34 \begin {gather*} d^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}+\frac {e^2\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b^2}+\frac {d\,e\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{12\,b^4}-\frac {a^2\,e^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b^2}-\frac {5\,a\,e^2\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{96\,b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^2*(x/2 + a/(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2) + (e^2*x*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(4*b^2) + (d*e*(
8*b^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(12*b^4) - (a^2*e^2*(x/2 + a/
(2*b))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*b^2) - (5*a*e^2*(8*b^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*(a
^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(96*b^5)

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sympy [A]  time = 0.11, size = 49, normalized size = 0.53 \begin {gather*} a d^{2} x + \frac {b e^{2} x^{4}}{4} + x^{3} \left (\frac {a e^{2}}{3} + \frac {2 b d e}{3}\right ) + x^{2} \left (a d e + \frac {b d^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**2*x + b*e**2*x**4/4 + x**3*(a*e**2/3 + 2*b*d*e/3) + x**2*(a*d*e + b*d**2/2)

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