3.1964 \(\int \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=146 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^3 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^3 (a+b x) (d+e x)^4} \]

[Out]

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

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Rubi [A]  time = 0.0822581, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^3 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^3 (a+b x) (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^5,x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^5} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{(a+b x) \left (a b+b^2 x\right )}{(d+e x)^5} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{(a+b x)^2}{(d+e x)^5} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^2}{e^2 (d+e x)^5}-\frac{2 b (b d-a e)}{e^2 (d+e x)^4}+\frac{b^2}{e^2 (d+e x)^3}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}+\frac{2 b (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^3}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0311172, size = 73, normalized size = 0.5 \[ -\frac{\sqrt{(a+b x)^2} \left (3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )\right )}{12 e^3 (a+b x) (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^5,x]

[Out]

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

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Maple [A]  time = 0.005, size = 78, normalized size = 0.5 \begin{align*} -{\frac{6\,{x}^{2}{b}^{2}{e}^{2}+8\,xab{e}^{2}+4\,x{b}^{2}de+3\,{a}^{2}{e}^{2}+2\,abde+{b}^{2}{d}^{2}}{12\,{e}^{3} \left ( ex+d \right ) ^{4} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^5,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54662, size = 201, normalized size = 1.38 \begin{align*} -\frac{6 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 2 \, a b d e + 3 \, a^{2} e^{2} + 4 \,{\left (b^{2} d e + 2 \, a b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

-1/12*(6*b^2*e^2*x^2 + b^2*d^2 + 2*a*b*d*e + 3*a^2*e^2 + 4*(b^2*d*e + 2*a*b*e^2)*x)/(e^7*x^4 + 4*d*e^6*x^3 + 6
*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

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Sympy [A]  time = 0.932119, size = 104, normalized size = 0.71 \begin{align*} - \frac{3 a^{2} e^{2} + 2 a b d e + b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (8 a b e^{2} + 4 b^{2} d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*((b*x+a)**2)**(1/2)/(e*x+d)**5,x)

[Out]

-(3*a**2*e**2 + 2*a*b*d*e + b**2*d**2 + 6*b**2*e**2*x**2 + x*(8*a*b*e**2 + 4*b**2*d*e))/(12*d**4*e**3 + 48*d**
3*e**4*x + 72*d**2*e**5*x**2 + 48*d*e**6*x**3 + 12*e**7*x**4)

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Giac [A]  time = 1.14124, size = 130, normalized size = 0.89 \begin{align*} -\frac{{\left (6 \, b^{2} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 4 \, b^{2} d x e \mathrm{sgn}\left (b x + a\right ) + b^{2} d^{2} \mathrm{sgn}\left (b x + a\right ) + 8 \, a b x e^{2} \mathrm{sgn}\left (b x + a\right ) + 2 \, a b d e \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} e^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{12 \,{\left (x e + d\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^5,x, algorithm="giac")

[Out]

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