3.17.52 \(\int \frac {1}{(d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)} \, dx\) [1652]

Optimal. Leaf size=124 \[ -\frac {5 e}{3 (b d-a e)^2 (d+e x)^{3/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac {5 b e}{(b d-a e)^3 \sqrt {d+e x}}+\frac {5 b^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 44, 53, 65, 214} \begin {gather*} \frac {5 b^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}}-\frac {5 b e}{\sqrt {d+e x} (b d-a e)^3}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}-\frac {5 e}{3 (d+e x)^{3/2} (b d-a e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

(-5*e)/(3*(b*d - a*e)^2*(d + e*x)^(3/2)) - 1/((b*d - a*e)*(a + b*x)*(d + e*x)^(3/2)) - (5*b*e)/((b*d - a*e)^3*
Sqrt[d + e*x]) + (5*b^(3/2)*e*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(7/2)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 44

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] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {1}{(a+b x)^2 (d+e x)^{5/2}} \, dx\\ &=-\frac {1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac {(5 e) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{2 (b d-a e)}\\ &=-\frac {5 e}{3 (b d-a e)^2 (d+e x)^{3/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac {(5 b e) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 (b d-a e)^2}\\ &=-\frac {5 e}{3 (b d-a e)^2 (d+e x)^{3/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac {5 b e}{(b d-a e)^3 \sqrt {d+e x}}-\frac {\left (5 b^2 e\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 (b d-a e)^3}\\ &=-\frac {5 e}{3 (b d-a e)^2 (d+e x)^{3/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac {5 b e}{(b d-a e)^3 \sqrt {d+e x}}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{(b d-a e)^3}\\ &=-\frac {5 e}{3 (b d-a e)^2 (d+e x)^{3/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac {5 b e}{(b d-a e)^3 \sqrt {d+e x}}+\frac {5 b^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.31, size = 125, normalized size = 1.01 \begin {gather*} \frac {2 a^2 e^2-2 a b e (7 d+5 e x)-b^2 \left (3 d^2+20 d e x+15 e^2 x^2\right )}{3 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}+\frac {5 b^{3/2} e \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

(2*a^2*e^2 - 2*a*b*e*(7*d + 5*e*x) - b^2*(3*d^2 + 20*d*e*x + 15*e^2*x^2))/(3*(b*d - a*e)^3*(a + b*x)*(d + e*x)
^(3/2)) + (5*b^(3/2)*e*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(7/2)

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Maple [A]
time = 0.76, size = 121, normalized size = 0.98

method	result	size
derivativedivides	\(2 e \left (\frac {b^{2} \left (\frac {\sqrt {e x +d}}{2 \left (e x +d \right ) b +2 a e -2 b d}+\frac {5 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{3}}-\frac {1}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 b}{\left (a e -b d \right )^{3} \sqrt {e x +d}}\right )\)	\(121\)
default	\(2 e \left (\frac {b^{2} \left (\frac {\sqrt {e x +d}}{2 \left (e x +d \right ) b +2 a e -2 b d}+\frac {5 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{3}}-\frac {1}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 b}{\left (a e -b d \right )^{3} \sqrt {e x +d}}\right )\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e*(1/(a*e-b*d)^3*b^2*(1/2*(e*x+d)^(1/2)/((e*x+d)*b+a*e-b*d)+5/2/(b*(a*e-b*d))^(1/2)*arctan(b*(e*x+d)^(1/2)/(
b*(a*e-b*d))^(1/2)))-1/3/(a*e-b*d)^2/(e*x+d)^(3/2)+2/(a*e-b*d)^3*b/(e*x+d)^(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(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (118) = 236\).
time = 2.29, size = 775, normalized size = 6.25 \begin {gather*} \left [-\frac {15 \, {\left ({\left (b^{2} x^{3} + a b x^{2}\right )} e^{3} + 2 \, {\left (b^{2} d x^{2} + a b d x\right )} e^{2} + {\left (b^{2} d^{2} x + a b d^{2}\right )} e\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {2 \, b d - 2 \, {\left (b d - a e\right )} \sqrt {x e + d} \sqrt {\frac {b}{b d - a e}} + {\left (b x - a\right )} e}{b x + a}\right ) + 2 \, {\left (3 \, b^{2} d^{2} + {\left (15 \, b^{2} x^{2} + 10 \, a b x - 2 \, a^{2}\right )} e^{2} + 2 \, {\left (10 \, b^{2} d x + 7 \, a b d\right )} e\right )} \sqrt {x e + d}}{6 \, {\left (b^{4} d^{5} x + a b^{3} d^{5} - {\left (a^{3} b x^{3} + a^{4} x^{2}\right )} e^{5} + {\left (3 \, a^{2} b^{2} d x^{3} + a^{3} b d x^{2} - 2 \, a^{4} d x\right )} e^{4} - {\left (3 \, a b^{3} d^{2} x^{3} - 3 \, a^{2} b^{2} d^{2} x^{2} - 5 \, a^{3} b d^{2} x + a^{4} d^{2}\right )} e^{3} + {\left (b^{4} d^{3} x^{3} - 5 \, a b^{3} d^{3} x^{2} - 3 \, a^{2} b^{2} d^{3} x + 3 \, a^{3} b d^{3}\right )} e^{2} + {\left (2 \, b^{4} d^{4} x^{2} - a b^{3} d^{4} x - 3 \, a^{2} b^{2} d^{4}\right )} e\right )}}, \frac {15 \, {\left ({\left (b^{2} x^{3} + a b x^{2}\right )} e^{3} + 2 \, {\left (b^{2} d x^{2} + a b d x\right )} e^{2} + {\left (b^{2} d^{2} x + a b d^{2}\right )} e\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {x e + d} \sqrt {-\frac {b}{b d - a e}}}{b x e + b d}\right ) - {\left (3 \, b^{2} d^{2} + {\left (15 \, b^{2} x^{2} + 10 \, a b x - 2 \, a^{2}\right )} e^{2} + 2 \, {\left (10 \, b^{2} d x + 7 \, a b d\right )} e\right )} \sqrt {x e + d}}{3 \, {\left (b^{4} d^{5} x + a b^{3} d^{5} - {\left (a^{3} b x^{3} + a^{4} x^{2}\right )} e^{5} + {\left (3 \, a^{2} b^{2} d x^{3} + a^{3} b d x^{2} - 2 \, a^{4} d x\right )} e^{4} - {\left (3 \, a b^{3} d^{2} x^{3} - 3 \, a^{2} b^{2} d^{2} x^{2} - 5 \, a^{3} b d^{2} x + a^{4} d^{2}\right )} e^{3} + {\left (b^{4} d^{3} x^{3} - 5 \, a b^{3} d^{3} x^{2} - 3 \, a^{2} b^{2} d^{3} x + 3 \, a^{3} b d^{3}\right )} e^{2} + {\left (2 \, b^{4} d^{4} x^{2} - a b^{3} d^{4} x - 3 \, a^{2} b^{2} d^{4}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(15*((b^2*x^3 + a*b*x^2)*e^3 + 2*(b^2*d*x^2 + a*b*d*x)*e^2 + (b^2*d^2*x + a*b*d^2)*e)*sqrt(b/(b*d - a*e)
)*log((2*b*d - 2*(b*d - a*e)*sqrt(x*e + d)*sqrt(b/(b*d - a*e)) + (b*x - a)*e)/(b*x + a)) + 2*(3*b^2*d^2 + (15*
b^2*x^2 + 10*a*b*x - 2*a^2)*e^2 + 2*(10*b^2*d*x + 7*a*b*d)*e)*sqrt(x*e + d))/(b^4*d^5*x + a*b^3*d^5 - (a^3*b*x
^3 + a^4*x^2)*e^5 + (3*a^2*b^2*d*x^3 + a^3*b*d*x^2 - 2*a^4*d*x)*e^4 - (3*a*b^3*d^2*x^3 - 3*a^2*b^2*d^2*x^2 - 5
*a^3*b*d^2*x + a^4*d^2)*e^3 + (b^4*d^3*x^3 - 5*a*b^3*d^3*x^2 - 3*a^2*b^2*d^3*x + 3*a^3*b*d^3)*e^2 + (2*b^4*d^4
*x^2 - a*b^3*d^4*x - 3*a^2*b^2*d^4)*e), 1/3*(15*((b^2*x^3 + a*b*x^2)*e^3 + 2*(b^2*d*x^2 + a*b*d*x)*e^2 + (b^2*
d^2*x + a*b*d^2)*e)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(x*e + d)*sqrt(-b/(b*d - a*e))/(b*x*e + b*d))
- (3*b^2*d^2 + (15*b^2*x^2 + 10*a*b*x - 2*a^2)*e^2 + 2*(10*b^2*d*x + 7*a*b*d)*e)*sqrt(x*e + d))/(b^4*d^5*x +
a*b^3*d^5 - (a^3*b*x^3 + a^4*x^2)*e^5 + (3*a^2*b^2*d*x^3 + a^3*b*d*x^2 - 2*a^4*d*x)*e^4 - (3*a*b^3*d^2*x^3 - 3
*a^2*b^2*d^2*x^2 - 5*a^3*b*d^2*x + a^4*d^2)*e^3 + (b^4*d^3*x^3 - 5*a*b^3*d^3*x^2 - 3*a^2*b^2*d^3*x + 3*a^3*b*d
^3)*e^2 + (2*b^4*d^4*x^2 - a*b^3*d^4*x - 3*a^2*b^2*d^4)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{2} \left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.39, size = 224, normalized size = 1.81 \begin {gather*} -\frac {5 \, b^{2} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e}} - \frac {\sqrt {x e + d} b^{2} e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}} - \frac {2 \, {\left (6 \, {\left (x e + d\right )} b e + b d e - a e^{2}\right )}}{3 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*b^2*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e/((b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*sqr
t(-b^2*d + a*b*e)) - sqrt(x*e + d)*b^2*e/((b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*((x*e + d)*b - b
*d + a*e)) - 2/3*(6*(x*e + d)*b*e + b*d*e - a*e^2)/((b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*(x*e +
d)^(3/2))

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Mupad [B]
time = 0.67, size = 161, normalized size = 1.30 \begin {gather*} \frac {\frac {10\,b\,e\,\left (d+e\,x\right )}{3\,{\left (a\,e-b\,d\right )}^2}-\frac {2\,e}{3\,\left (a\,e-b\,d\right )}+\frac {5\,b^2\,e\,{\left (d+e\,x\right )}^2}{{\left (a\,e-b\,d\right )}^3}}{b\,{\left (d+e\,x\right )}^{5/2}+\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{3/2}}+\frac {5\,b^{3/2}\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^{7/2}}\right )}{{\left (a\,e-b\,d\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((10*b*e*(d + e*x))/(3*(a*e - b*d)^2) - (2*e)/(3*(a*e - b*d)) + (5*b^2*e*(d + e*x)^2)/(a*e - b*d)^3)/(b*(d + e
*x)^(5/2) + (a*e - b*d)*(d + e*x)^(3/2)) + (5*b^(3/2)*e*atan((b^(1/2)*(d + e*x)^(1/2)*(a^3*e^3 - b^3*d^3 + 3*a
*b^2*d^2*e - 3*a^2*b*d*e^2))/(a*e - b*d)^(7/2)))/(a*e - b*d)^(7/2)

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