3.15.96 \(\int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=104 \[ \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2} (B d-A e)}{3 (d+e x)^3 (b d-a e)^2}+\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{2 (d+e x)^2 (b d-a e)^2} \]

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Rubi [A]  time = 0.06, antiderivative size = 104, normalized size of antiderivative = 1.00, 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 = {769, 646, 37} \begin {gather*} \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2} (B d-A e)}{3 (d+e x)^3 (b d-a e)^2}+\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{2 (d+e x)^2 (b d-a e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^4,x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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]

Rule 769

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(-2*c*(e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)^2), x] + Dist[(2*c*f -
b*g)/(2*c*d - b*e), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x]
&& EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && NeQ[2*c*f - b*g, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 81, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a e (2 A e+B (d+3 e x))+b \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )\right )}{6 e^3 (a+b x) (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^4,x]

[Out]

-1/6*(Sqrt[(a + b*x)^2]*(a*e*(2*A*e + B*(d + 3*e*x)) + b*(A*e*(d + 3*e*x) + 2*B*(d^2 + 3*d*e*x + 3*e^2*x^2))))
/(e^3*(a + b*x)*(d + e*x)^3)

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IntegrateAlgebraic [F]  time = 2.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^4,x]

[Out]

Defer[IntegrateAlgebraic][((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^4, x]

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fricas [A]  time = 0.41, size = 93, normalized size = 0.89 \begin {gather*} -\frac {6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} + {\left (B a + A b\right )} d e + 3 \, {\left (2 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x}{6 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 117, normalized size = 1.12 \begin {gather*} -\frac {{\left (6 \, B b x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, B b d x e \mathrm {sgn}\left (b x + a\right ) + 2 \, B b d^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a x e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, A b x e^{2} \mathrm {sgn}\left (b x + a\right ) + B a d e \mathrm {sgn}\left (b x + a\right ) + A b d e \mathrm {sgn}\left (b x + a\right ) + 2 \, A a e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

-1/6*(6*B*b*x^2*e^2*sgn(b*x + a) + 6*B*b*d*x*e*sgn(b*x + a) + 2*B*b*d^2*sgn(b*x + a) + 3*B*a*x*e^2*sgn(b*x + a
) + 3*A*b*x*e^2*sgn(b*x + a) + B*a*d*e*sgn(b*x + a) + A*b*d*e*sgn(b*x + a) + 2*A*a*e^2*sgn(b*x + a))*e^(-3)/(x
*e + d)^3

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maple [A]  time = 0.05, size = 87, normalized size = 0.84 \begin {gather*} -\frac {\left (6 B b \,e^{2} x^{2}+3 A b \,e^{2} x +3 B a \,e^{2} x +6 B b d e x +2 A a \,e^{2}+A b d e +B a d e +2 B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{6 \left (e x +d \right )^{3} \left (b x +a \right ) e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^4,x)

[Out]

-1/6*(6*B*b*e^2*x^2+3*A*b*e^2*x+3*B*a*e^2*x+6*B*b*d*e*x+2*A*a*e^2+A*b*d*e+B*a*d*e+2*B*b*d^2)*((b*x+a)^2)^(1/2)
/(e*x+d)^3/e^3/(b*x+a)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [B]  time = 2.13, size = 86, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (2\,A\,a\,e^2+2\,B\,b\,d^2+3\,A\,b\,e^2\,x+3\,B\,a\,e^2\,x+6\,B\,b\,e^2\,x^2+A\,b\,d\,e+B\,a\,d\,e+6\,B\,b\,d\,e\,x\right )}{6\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((a + b*x)^2)^(1/2)*(A + B*x))/(d + e*x)^4,x)

[Out]

-(((a + b*x)^2)^(1/2)*(2*A*a*e^2 + 2*B*b*d^2 + 3*A*b*e^2*x + 3*B*a*e^2*x + 6*B*b*e^2*x^2 + A*b*d*e + B*a*d*e +
6*B*b*d*e*x))/(6*e^3*(a + b*x)*(d + e*x)^3)

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sympy [A]  time = 1.51, size = 107, normalized size = 1.03 \begin {gather*} \frac {- 2 A a e^{2} - A b d e - B a d e - 2 B b d^{2} - 6 B b e^{2} x^{2} + x \left (- 3 A b e^{2} - 3 B a e^{2} - 6 B b d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)**2)**(1/2)/(e*x+d)**4,x)

[Out]

(-2*A*a*e**2 - A*b*d*e - B*a*d*e - 2*B*b*d**2 - 6*B*b*e**2*x**2 + x*(-3*A*b*e**2 - 3*B*a*e**2 - 6*B*b*d*e))/(6
*d**3*e**3 + 18*d**2*e**4*x + 18*d*e**5*x**2 + 6*e**6*x**3)

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