3.2041 \(\int \frac{a+b x}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)
Optimal. Leaf size=323 \[ -\frac{4 b e^3 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}-\frac{e^3 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^4}-\frac{6 b^2 e^2}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{10 b^2 e^3 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac{10 b^2 e^3 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac{3 b^2 e}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^2}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]
[Out]
(-6*b^2*e^2)/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b^2/(3*(b*d - a*e)^3*(a + b*x)^2*Sqrt[a^2 + 2*a*b
*x + b^2*x^2]) + (3*b^2*e)/(2*(b*d - a*e)^4*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e^3*(a + b*x))/(2*(b*d
- a*e)^4*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (4*b*e^3*(a + b*x))/((b*d - a*e)^5*(d + e*x)*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) - (10*b^2*e^3*(a + b*x)*Log[a + b*x])/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (10
*b^2*e^3*(a + b*x)*Log[d + e*x])/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
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Rubi [A] time = 0.237623, antiderivative size = 323, 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, 44} \[ -\frac{4 b e^3 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}-\frac{e^3 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^4}-\frac{6 b^2 e^2}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{10 b^2 e^3 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac{10 b^2 e^3 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac{3 b^2 e}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^2}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(-6*b^2*e^2)/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b^2/(3*(b*d - a*e)^3*(a + b*x)^2*Sqrt[a^2 + 2*a*b
*x + b^2*x^2]) + (3*b^2*e)/(2*(b*d - a*e)^4*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e^3*(a + b*x))/(2*(b*d
- a*e)^4*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (4*b*e^3*(a + b*x))/((b*d - a*e)^5*(d + e*x)*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) - (10*b^2*e^3*(a + b*x)*Log[a + b*x])/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (10
*b^2*e^3*(a + b*x)*Log[d + e*x])/((b*d - a*e)^6*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 44
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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int \frac{a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{a+b x}{\left (a b+b^2 x\right )^5 (d+e x)^3} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \frac{1}{(a+b x)^4 (d+e x)^3} \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \left (\frac{b^3}{(b d-a e)^3 (a+b x)^4}-\frac{3 b^3 e}{(b d-a e)^4 (a+b x)^3}+\frac{6 b^3 e^2}{(b d-a e)^5 (a+b x)^2}-\frac{10 b^3 e^3}{(b d-a e)^6 (a+b x)}+\frac{e^4}{(b d-a e)^4 (d+e x)^3}+\frac{4 b e^4}{(b d-a e)^5 (d+e x)^2}+\frac{10 b^2 e^4}{(b d-a e)^6 (d+e x)}\right ) \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{6 b^2 e^2}{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2}{3 (b d-a e)^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b^2 e}{2 (b d-a e)^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e^3 (a+b x)}{2 (b d-a e)^4 (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 b e^3 (a+b x)}{(b d-a e)^5 (d+e x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{10 b^2 e^3 (a+b x) \log (a+b x)}{(b d-a e)^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{10 b^2 e^3 (a+b x) \log (d+e x)}{(b d-a e)^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.127413, size = 184, normalized size = 0.57 \[ \frac{-36 b^2 e^2 (a+b x)^2 (b d-a e)+60 b^2 e^3 (a+b x)^3 \log (d+e x)+9 b^2 e (a+b x) (b d-a e)^2-2 b^2 (b d-a e)^3-60 b^2 e^3 (a+b x)^3 \log (a+b x)-\frac{3 e^3 (a+b x)^3 (b d-a e)^2}{(d+e x)^2}-\frac{24 b e^3 (a+b x)^3 (b d-a e)}{d+e x}}{6 \left ((a+b x)^2\right )^{3/2} (b d-a e)^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(-2*b^2*(b*d - a*e)^3 + 9*b^2*e*(b*d - a*e)^2*(a + b*x) - 36*b^2*e^2*(b*d - a*e)*(a + b*x)^2 - (3*e^3*(b*d - a
*e)^2*(a + b*x)^3)/(d + e*x)^2 - (24*b*e^3*(b*d - a*e)*(a + b*x)^3)/(d + e*x) - 60*b^2*e^3*(a + b*x)^3*Log[a +
b*x] + 60*b^2*e^3*(a + b*x)^3*Log[d + e*x])/(6*(b*d - a*e)^6*((a + b*x)^2)^(3/2))
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Maple [B] time = 0.02, size = 753, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
1/6*(-180*ln(b*x+a)*x^2*a*b^4*d^2*e^3-3*a^5*e^5-2*b^5*d^5+120*ln(e*x+d)*x*a^3*b^2*d*e^4+180*ln(e*x+d)*x*a^2*b^
3*d^2*e^3+360*ln(e*x+d)*x^3*a*b^4*d*e^4+360*ln(e*x+d)*x^2*a^2*b^3*d*e^4+180*ln(e*x+d)*x^2*a*b^4*d^2*e^3-360*ln
(b*x+a)*x^2*a^2*b^3*d*e^4-360*ln(b*x+a)*x^3*a*b^4*d*e^4-120*ln(b*x+a)*x*a^3*b^2*d*e^4-180*ln(b*x+a)*x*a^2*b^3*
d^2*e^3-60*ln(b*x+a)*x^5*b^5*e^5-90*x^3*b^5*d^2*e^3+60*ln(e*x+d)*x^5*b^5*e^5+60*x^4*a*b^4*e^5-60*x^4*b^5*d*e^4
+110*x^2*a^3*b^2*e^5-20*x^2*b^5*d^3*e^2+15*x*a^4*b*e^5+5*x*b^5*d^4*e+150*x^3*a^2*b^3*e^5+15*a*d^4*b^4*e-60*a^2
*b^3*d^3*e^2+20*a^3*d^2*b^2*e^3+30*a^4*b*d*e^4-120*ln(b*x+a)*x^4*b^5*d*e^4+180*ln(e*x+d)*x^3*a^2*b^3*e^5+60*ln
(e*x+d)*x^3*b^5*d^2*e^3+60*ln(e*x+d)*x^2*a^3*b^2*e^5+60*ln(e*x+d)*a^3*b^2*d^2*e^3+180*ln(e*x+d)*x^4*a*b^4*e^5+
120*ln(e*x+d)*x^4*b^5*d*e^4-180*ln(b*x+a)*x^4*a*b^4*e^5-60*ln(b*x+a)*a^3*b^2*d^2*e^3-60*ln(b*x+a)*x^3*b^5*d^2*
e^3-60*ln(b*x+a)*x^2*a^3*b^2*e^5-180*ln(b*x+a)*x^3*a^2*b^3*e^5-210*x^2*a*b^4*d^2*e^3-120*x*a^2*b^3*d^2*e^3-60*
x*a*b^4*d^3*e^2+160*x*a^3*b^2*d*e^4-60*x^3*a*b^4*d*e^4+120*x^2*a^2*b^3*d*e^4)*(b*x+a)^2/(e*x+d)^2/(a*e-b*d)^6/
((b*x+a)^2)^(5/2)
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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)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.67361, size = 2313, normalized size = 7.16 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
[Out]
-1/6*(2*b^5*d^5 - 15*a*b^4*d^4*e + 60*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 30*a^4*b*d*e^4 + 3*a^5*e^5 + 60*(
b^5*d*e^4 - a*b^4*e^5)*x^4 + 30*(3*b^5*d^2*e^3 + 2*a*b^4*d*e^4 - 5*a^2*b^3*e^5)*x^3 + 10*(2*b^5*d^3*e^2 + 21*a
*b^4*d^2*e^3 - 12*a^2*b^3*d*e^4 - 11*a^3*b^2*e^5)*x^2 - 5*(b^5*d^4*e - 12*a*b^4*d^3*e^2 - 24*a^2*b^3*d^2*e^3 +
32*a^3*b^2*d*e^4 + 3*a^4*b*e^5)*x + 60*(b^5*e^5*x^5 + a^3*b^2*d^2*e^3 + (2*b^5*d*e^4 + 3*a*b^4*e^5)*x^4 + (b^
5*d^2*e^3 + 6*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + (3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 + (3*a^
2*b^3*d^2*e^3 + 2*a^3*b^2*d*e^4)*x)*log(b*x + a) - 60*(b^5*e^5*x^5 + a^3*b^2*d^2*e^3 + (2*b^5*d*e^4 + 3*a*b^4*
e^5)*x^4 + (b^5*d^2*e^3 + 6*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + (3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^
5)*x^2 + (3*a^2*b^3*d^2*e^3 + 2*a^3*b^2*d*e^4)*x)*log(e*x + d))/(a^3*b^6*d^8 - 6*a^4*b^5*d^7*e + 15*a^5*b^4*d^
6*e^2 - 20*a^6*b^3*d^5*e^3 + 15*a^7*b^2*d^4*e^4 - 6*a^8*b*d^3*e^5 + a^9*d^2*e^6 + (b^9*d^6*e^2 - 6*a*b^8*d^5*e
^3 + 15*a^2*b^7*d^4*e^4 - 20*a^3*b^6*d^3*e^5 + 15*a^4*b^5*d^2*e^6 - 6*a^5*b^4*d*e^7 + a^6*b^3*e^8)*x^5 + (2*b^
9*d^7*e - 9*a*b^8*d^6*e^2 + 12*a^2*b^7*d^5*e^3 + 5*a^3*b^6*d^4*e^4 - 30*a^4*b^5*d^3*e^5 + 33*a^5*b^4*d^2*e^6 -
16*a^6*b^3*d*e^7 + 3*a^7*b^2*e^8)*x^4 + (b^9*d^8 - 18*a^2*b^7*d^6*e^2 + 52*a^3*b^6*d^5*e^3 - 60*a^4*b^5*d^4*e
^4 + 24*a^5*b^4*d^3*e^5 + 10*a^6*b^3*d^2*e^6 - 12*a^7*b^2*d*e^7 + 3*a^8*b*e^8)*x^3 + (3*a*b^8*d^8 - 12*a^2*b^7
*d^7*e + 10*a^3*b^6*d^6*e^2 + 24*a^4*b^5*d^5*e^3 - 60*a^5*b^4*d^4*e^4 + 52*a^6*b^3*d^3*e^5 - 18*a^7*b^2*d^2*e^
6 + a^9*e^8)*x^2 + (3*a^2*b^7*d^8 - 16*a^3*b^6*d^7*e + 33*a^4*b^5*d^6*e^2 - 30*a^5*b^4*d^5*e^3 + 5*a^6*b^3*d^4
*e^4 + 12*a^7*b^2*d^3*e^5 - 9*a^8*b*d^2*e^6 + 2*a^9*d*e^7)*x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)/(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
[Out]
integrate((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^3), x)