3.18.99 \(\int \frac {a+b x}{(d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx\)
Optimal. Leaf size=42 \[ \frac {-a-b x}{4 e \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4} \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 0.93,
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} \begin {gather*} -\frac {a+b x}{4 e \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x)/((d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
-(a + b*x)/(4*e*(d + e*x)^4*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 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]
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)^5 \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}{\left (a b+b^2 x\right ) (d+e x)^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{(d+e x)^5} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {a+b x}{4 e (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.71 \begin {gather*} -\frac {a+b x}{4 e \sqrt {(a+b x)^2} (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)/((d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
-1/4*(a + b*x)/(e*Sqrt[(a + b*x)^2]*(d + e*x)^4)
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IntegrateAlgebraic [B] time = 87.25, size = 4872, normalized size = 116.00 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(a + b*x)/((d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
((a*b^5*x^4)/d^4 - (a*b^4*Sqrt[b^2]*x^4)/d^4 + (b^6*x^5)/d^4 - (b^5*Sqrt[b^2]*x^5)/d^4 + (b^5*x^4*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/d^4 - (b^4*Sqrt[b^2]*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^4)/((-a - Sqrt[b^2]*x + Sqrt[a^2
+ 2*a*b*x + b^2*x^2])*(-(a*b) - b^2*x + Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])*(2*b*d - a*e - Sqrt[b^2]*e*x
+ e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])*(-a^2 - a*b*x - a*Sqrt[b^2]*x - b^2*x^2 + a*Sqrt[a^2 + 2*a*b*x + b^2*x^2] +
Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + ((2*a*b^6*x^4)/d^3 - (2*a*b^5*Sqrt[b^2]*x^4)/d^3 + (2*b^7*x^5)/
d^3 - (2*b^6*Sqrt[b^2]*x^5)/d^3 + (2*b^6*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3 - (2*b^5*Sqrt[b^2]*x^4*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/d^3)/((-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2])*(-(a*b) - b^2*x + Sqrt[b^2]*S
qrt[a^2 + 2*a*b*x + b^2*x^2])*(2*b*d - a*e - Sqrt[b^2]*e*x + e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])^2*(-a^2 - a*b*x
- a*Sqrt[b^2]*x - b^2*x^2 + a*Sqrt[a^2 + 2*a*b*x + b^2*x^2] + Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + ((
4*a*b^7*x^4)/d^2 - (4*a*b^6*Sqrt[b^2]*x^4)/d^2 + (4*b^8*x^5)/d^2 - (4*b^7*Sqrt[b^2]*x^5)/d^2 + (4*b^7*x^4*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/d^2 - (4*b^6*Sqrt[b^2]*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^2)/((-a - Sqrt[b^2]*x +
Sqrt[a^2 + 2*a*b*x + b^2*x^2])*(-(a*b) - b^2*x + Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])*(2*b*d - a*e - Sqrt
[b^2]*e*x + e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])^3*(-a^2 - a*b*x - a*Sqrt[b^2]*x - b^2*x^2 + a*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2] + Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + ((8*a*b^8*x^4)/d - (8*a*b^7*Sqrt[b^2]*x^4)/d + (8*b
^9*x^5)/d - (8*b^8*Sqrt[b^2]*x^5)/d + (8*b^8*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d - (8*b^7*Sqrt[b^2]*x^4*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/d)/((-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2])*(-(a*b) - b^2*x + Sqrt[b^2]*S
qrt[a^2 + 2*a*b*x + b^2*x^2])*(2*b*d - a*e - Sqrt[b^2]*e*x + e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])^4*(-a^2 - a*b*x
- a*Sqrt[b^2]*x - b^2*x^2 + a*Sqrt[a^2 + 2*a*b*x + b^2*x^2] + Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + ((
2*a^7)/d^4 + (2*a^7*Sqrt[b^2])/(b*d^4) - (2*b^7*x^7)/d^4 - (2*b^6*Sqrt[b^2]*x^7)/d^4 + (2*a^6*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/d^4 + (2*a^6*Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(b*d^4) - (2*a^5*b*x*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/d^4 - (2*a^5*Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^4 + (2*a^4*b^2*x^2*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/d^4 + (2*a^4*(b^2)^(3/2)*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(b*d^4) - (2*a^3*b^3*x^3*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/d^4 - (2*a^3*(b^2)^(3/2)*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^4 + (2*a^2*b^4*x^4*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/d^4 + (2*a^2*b^3*Sqrt[b^2]*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^4 - (2*a*b^5*x^5*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/d^4 - (2*a*b^4*Sqrt[b^2]*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^4 + (2*b^6*x^6*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/d^4 + (2*b^5*Sqrt[b^2]*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^4)/((a - Sqrt[b^2]*x + Sqrt[a^2 +
2*a*b*x + b^2*x^2])*(2*b*d - a*e + Sqrt[b^2]*e*x - e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])*(a^2 + a*b*x - a*Sqrt[b^2
]*x + b^2*x^2 + a*Sqrt[a^2 + 2*a*b*x + b^2*x^2] - Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])*(2*a^3 + 3*a^2*b*
x - 3*a^2*Sqrt[b^2]*x + 3*a*b^2*x^2 - 3*a*b*Sqrt[b^2]*x^2 - 2*(b^2)^(3/2)*x^3 + 2*a^2*Sqrt[a^2 + 2*a*b*x + b^2
*x^2] + a*b*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2] - 3*a*Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2] + 2*b^2*x^2*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])) + ((4*a^7*b)/d^3 + (4*a^7*Sqrt[b^2])/d^3 - (4*b^8*x^7)/d^3 - (4*b^7*Sqrt[b^2]*x^7)/d
^3 + (4*a^6*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3 + (4*a^6*Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3 - (4*a^
5*b^2*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3 - (4*a^5*b*Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3 + (4*a^4*
b^3*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3 + (4*a^4*(b^2)^(3/2)*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3 - (4*a^
3*b^4*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3 - (4*a^3*b^3*Sqrt[b^2]*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3 + (
4*a^2*b^5*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3 + (4*a^2*b^4*Sqrt[b^2]*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3
- (4*a*b^6*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3 - (4*a*b^5*Sqrt[b^2]*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3
+ (4*b^7*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3 + (4*b^6*Sqrt[b^2]*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^3)/((
a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2])*(2*b*d - a*e + Sqrt[b^2]*e*x - e*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])^2*(a^2 + a*b*x - a*Sqrt[b^2]*x + b^2*x^2 + a*Sqrt[a^2 + 2*a*b*x + b^2*x^2] - Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])*(2*a^3 + 3*a^2*b*x - 3*a^2*Sqrt[b^2]*x + 3*a*b^2*x^2 - 3*a*b*Sqrt[b^2]*x^2 - 2*(b^2)^(3/2)*x^3 +
2*a^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2] + a*b*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2] - 3*a*Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2] + 2*b^2*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + ((8*a^7*b^2)/d^2 + (8*a^7*b*Sqrt[b^2])/d^2 - (8*b^9*
x^7)/d^2 - (8*b^8*Sqrt[b^2]*x^7)/d^2 + (8*a^6*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^2 + (8*a^6*b*Sqrt[b^2]*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/d^2 - (8*a^5*b^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^2 - (8*a^5*(b^2)^(3/2)*x*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])/d^2 + (8*a^4*b^4*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^2 + (8*a^4*b^3*Sqrt[b^2]*x^2*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/d^2 - (8*a^3*b^5*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^2 - (8*a^3*b^4*Sqrt[b^2]*x^
3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^2 + (8*a^2*b^6*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^2 + (8*a^2*b^5*Sqrt[b^2
]*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^2 - (8*a*b^7*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^2 - (8*a*b^6*Sqrt[b^2
]*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^2 + (8*b^8*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^2 + (8*b^7*Sqrt[b^2]*x^
6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d^2)/((a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2])*(2*b*d - a*e + Sqrt[b
^2]*e*x - e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])^3*(a^2 + a*b*x - a*Sqrt[b^2]*x + b^2*x^2 + a*Sqrt[a^2 + 2*a*b*x + b
^2*x^2] - Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])*(2*a^3 + 3*a^2*b*x - 3*a^2*Sqrt[b^2]*x + 3*a*b^2*x^2 - 3*
a*b*Sqrt[b^2]*x^2 - 2*(b^2)^(3/2)*x^3 + 2*a^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2] + a*b*x*Sqrt[a^2 + 2*a*b*x + b^2*x
^2] - 3*a*Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2] + 2*b^2*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + ((16*a^7*b^3
)/d + (16*a^7*(b^2)^(3/2))/d - (16*b^10*x^7)/d - (16*b^9*Sqrt[b^2]*x^7)/d + (16*a^6*b^3*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/d + (16*a^6*(b^2)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d - (16*a^5*b^4*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/d - (16*a^5*b^3*Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d + (16*a^4*b^5*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/d + (16*a^4*b^4*Sqrt[b^2]*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d - (16*a^3*b^6*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/d - (16*a^3*b^5*Sqrt[b^2]*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d + (16*a^2*b^7*x^4*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/d + (16*a^2*b^6*Sqrt[b^2]*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d - (16*a*b^8*x^5*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/d - (16*a*b^7*Sqrt[b^2]*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d + (16*b^9*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/d + (16*b^8*Sqrt[b^2]*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/d)/((a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^
2])*(2*b*d - a*e + Sqrt[b^2]*e*x - e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])^4*(a^2 + a*b*x - a*Sqrt[b^2]*x + b^2*x^2 +
a*Sqrt[a^2 + 2*a*b*x + b^2*x^2] - Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])*(2*a^3 + 3*a^2*b*x - 3*a^2*Sqrt[
b^2]*x + 3*a*b^2*x^2 - 3*a*b*Sqrt[b^2]*x^2 - 2*(b^2)^(3/2)*x^3 + 2*a^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2] + a*b*x*S
qrt[a^2 + 2*a*b*x + b^2*x^2] - 3*a*Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2] + 2*b^2*x^2*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]))
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fricas [A] time = 0.42, size = 46, normalized size = 1.10 \begin {gather*} -\frac {1}{4 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)/(e*x+d)^5/((b*x+a)^2)^(1/2),x, algorithm="fricas")
[Out]
-1/4/(e^5*x^4 + 4*d*e^4*x^3 + 6*d^2*e^3*x^2 + 4*d^3*e^2*x + d^4*e)
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giac [A] time = 0.15, size = 18, normalized size = 0.43 \begin {gather*} -\frac {e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right )}{4 \, {\left (x e + d\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)/(e*x+d)^5/((b*x+a)^2)^(1/2),x, algorithm="giac")
[Out]
-1/4*e^(-1)*sgn(b*x + a)/(x*e + d)^4
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maple [A] time = 0.05, size = 27, normalized size = 0.64 \begin {gather*} -\frac {b x +a}{4 \left (e x +d \right )^{4} \sqrt {\left (b x +a \right )^{2}}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)/(e*x+d)^5/((b*x+a)^2)^(1/2),x)
[Out]
-1/4/(e*x+d)^4/e*(b*x+a)/((b*x+a)^2)^(1/2)
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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)/(e*x+d)^5/((b*x+a)^2)^(1/2),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 = 2.17, size = 28, normalized size = 0.67 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}}{4\,e\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)/(((a + b*x)^2)^(1/2)*(d + e*x)^5),x)
[Out]
-((a + b*x)^2)^(1/2)/(4*e*(a + b*x)*(d + e*x)^4)
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sympy [A] time = 0.34, size = 49, normalized size = 1.17 \begin {gather*} - \frac {1}{4 d^{4} e + 16 d^{3} e^{2} x + 24 d^{2} e^{3} x^{2} + 16 d e^{4} x^{3} + 4 e^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)/(e*x+d)**5/((b*x+a)**2)**(1/2),x)
[Out]
-1/(4*d**4*e + 16*d**3*e**2*x + 24*d**2*e**3*x**2 + 16*d*e**4*x**3 + 4*e**5*x**4)
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