3.18.27 \(\int \frac {1}{\sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1727]

Optimal. Leaf size=278 \[ \frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} (b d-a e)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

-35/64*e^4*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(9/2)/b^(1/2)/((b*x+a)^2)^(1/2)+
35/64*e^3*(e*x+d)^(1/2)/(-a*e+b*d)^4/((b*x+a)^2)^(1/2)-1/4*(e*x+d)^(1/2)/(-a*e+b*d)/(b*x+a)^3/((b*x+a)^2)^(1/2
)+7/24*e*(e*x+d)^(1/2)/(-a*e+b*d)^2/(b*x+a)^2/((b*x+a)^2)^(1/2)-35/96*e^2*(e*x+d)^(1/2)/(-a*e+b*d)^3/(b*x+a)/(
(b*x+a)^2)^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {660, 44, 65, 214} \begin {gather*} -\frac {35 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac {35 e^3 \sqrt {d+e x}}{64 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {35 e^2 \sqrt {d+e x}}{96 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac {7 e \sqrt {d+e x}}{24 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {\sqrt {d+e x}}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(35*e^3*Sqrt[d + e*x])/(64*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt[d + e*x]/(4*(b*d - a*e)*(a + b*
x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*Sqrt[d + e*x])/(24*(b*d - a*e)^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b
^2*x^2]) - (35*e^2*Sqrt[d + e*x])/(96*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^4*(a + b*
x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*Sqrt[b]*(b*d - a*e)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])

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 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]

Rule 660

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x} \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 {1}{\left (a b+b^2 x\right )^5 \sqrt {d+e x}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 b^3 e \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^4 \sqrt {d+e x}} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 b^2 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 \sqrt {d+e x}} \, dx}{48 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 b e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 \sqrt {d+e x}} \, dx}{64 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^3 \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} (b d-a e)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.51, size = 187, normalized size = 0.67 \begin {gather*} \frac {e^4 (a+b x)^5 \left (\frac {\sqrt {d+e x} \left (279 a^3 e^3+a^2 b e^2 (-326 d+511 e x)+a b^2 e \left (200 d^2-252 d e x+385 e^2 x^2\right )+b^3 \left (-48 d^3+56 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )}{e^4 (b d-a e)^4 (a+b x)^4}+\frac {105 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{\sqrt {b} (-b d+a e)^{9/2}}\right )}{192 \left ((a+b x)^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*(a + b*x)^5*((Sqrt[d + e*x]*(279*a^3*e^3 + a^2*b*e^2*(-326*d + 511*e*x) + a*b^2*e*(200*d^2 - 252*d*e*x +
385*e^2*x^2) + b^3*(-48*d^3 + 56*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3)))/(e^4*(b*d - a*e)^4*(a + b*x)^4) + (10
5*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(Sqrt[b]*(-(b*d) + a*e)^(9/2))))/(192*((a + b*x)^2)^(5/2
))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(496\) vs. \(2(195)=390\).
time = 0.68, size = 497, normalized size = 1.79

method	result	size
default	\(\frac {\left (105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{4} e^{4} x^{4}+420 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{3} e^{4} x^{3}+105 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, b^{3} e^{3} x^{3}+630 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{2} e^{4} x^{2}+385 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a \,b^{2} e^{3} x^{2}-70 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, b^{3} d \,e^{2} x^{2}+420 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3} b \,e^{4} x +511 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{2} b \,e^{3} x -252 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a \,b^{2} d \,e^{2} x +56 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, b^{3} d^{2} e x +105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{4} e^{4}+279 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{3} e^{3}-326 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{2} b d \,e^{2}+200 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a \,b^{2} d^{2} e -48 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, b^{3} d^{3}\right ) \left (b x +a \right )}{192 \sqrt {b \left (a e -b d \right )}\, \left (a e -b d \right )^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(497\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(105*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*b^4*e^4*x^4+420*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1
/2))*a*b^3*e^4*x^3+105*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^3*e^3*x^3+630*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^
(1/2))*a^2*b^2*e^4*x^2+385*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^2*e^3*x^2-70*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2
)*b^3*d*e^2*x^2+420*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^3*b*e^4*x+511*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1
/2)*a^2*b*e^3*x-252*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^2*d*e^2*x+56*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^3*d
^2*e*x+105*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^4*e^4+279*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*e^3-3
26*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b*d*e^2+200*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^2*d^2*e-48*(b*(a*e-
b*d))^(1/2)*(e*x+d)^(1/2)*b^3*d^3)*(b*x+a)/(b*(a*e-b*d))^(1/2)/(a*e-b*d)^4/((b*x+a)^2)^(5/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (204) = 408\).
time = 3.38, size = 1268, normalized size = 4.56 \begin {gather*} \left [\frac {105 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {b^{2} d - a b e} e^{4} \log \left (\frac {2 \, b d + {\left (b x - a\right )} e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {x e + d}}{b x + a}\right ) - 2 \, {\left (48 \, b^{5} d^{4} + {\left (105 \, a b^{4} x^{3} + 385 \, a^{2} b^{3} x^{2} + 511 \, a^{3} b^{2} x + 279 \, a^{4} b\right )} e^{4} - {\left (105 \, b^{5} d x^{3} + 455 \, a b^{4} d x^{2} + 763 \, a^{2} b^{3} d x + 605 \, a^{3} b^{2} d\right )} e^{3} + 2 \, {\left (35 \, b^{5} d^{2} x^{2} + 154 \, a b^{4} d^{2} x + 263 \, a^{2} b^{3} d^{2}\right )} e^{2} - 8 \, {\left (7 \, b^{5} d^{3} x + 31 \, a b^{4} d^{3}\right )} e\right )} \sqrt {x e + d}}{384 \, {\left (b^{10} d^{5} x^{4} + 4 \, a b^{9} d^{5} x^{3} + 6 \, a^{2} b^{8} d^{5} x^{2} + 4 \, a^{3} b^{7} d^{5} x + a^{4} b^{6} d^{5} - {\left (a^{5} b^{5} x^{4} + 4 \, a^{6} b^{4} x^{3} + 6 \, a^{7} b^{3} x^{2} + 4 \, a^{8} b^{2} x + a^{9} b\right )} e^{5} + 5 \, {\left (a^{4} b^{6} d x^{4} + 4 \, a^{5} b^{5} d x^{3} + 6 \, a^{6} b^{4} d x^{2} + 4 \, a^{7} b^{3} d x + a^{8} b^{2} d\right )} e^{4} - 10 \, {\left (a^{3} b^{7} d^{2} x^{4} + 4 \, a^{4} b^{6} d^{2} x^{3} + 6 \, a^{5} b^{5} d^{2} x^{2} + 4 \, a^{6} b^{4} d^{2} x + a^{7} b^{3} d^{2}\right )} e^{3} + 10 \, {\left (a^{2} b^{8} d^{3} x^{4} + 4 \, a^{3} b^{7} d^{3} x^{3} + 6 \, a^{4} b^{6} d^{3} x^{2} + 4 \, a^{5} b^{5} d^{3} x + a^{6} b^{4} d^{3}\right )} e^{2} - 5 \, {\left (a b^{9} d^{4} x^{4} + 4 \, a^{2} b^{8} d^{4} x^{3} + 6 \, a^{3} b^{7} d^{4} x^{2} + 4 \, a^{4} b^{6} d^{4} x + a^{5} b^{5} d^{4}\right )} e\right )}}, \frac {105 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {x e + d}}{b x e + b d}\right ) e^{4} - {\left (48 \, b^{5} d^{4} + {\left (105 \, a b^{4} x^{3} + 385 \, a^{2} b^{3} x^{2} + 511 \, a^{3} b^{2} x + 279 \, a^{4} b\right )} e^{4} - {\left (105 \, b^{5} d x^{3} + 455 \, a b^{4} d x^{2} + 763 \, a^{2} b^{3} d x + 605 \, a^{3} b^{2} d\right )} e^{3} + 2 \, {\left (35 \, b^{5} d^{2} x^{2} + 154 \, a b^{4} d^{2} x + 263 \, a^{2} b^{3} d^{2}\right )} e^{2} - 8 \, {\left (7 \, b^{5} d^{3} x + 31 \, a b^{4} d^{3}\right )} e\right )} \sqrt {x e + d}}{192 \, {\left (b^{10} d^{5} x^{4} + 4 \, a b^{9} d^{5} x^{3} + 6 \, a^{2} b^{8} d^{5} x^{2} + 4 \, a^{3} b^{7} d^{5} x + a^{4} b^{6} d^{5} - {\left (a^{5} b^{5} x^{4} + 4 \, a^{6} b^{4} x^{3} + 6 \, a^{7} b^{3} x^{2} + 4 \, a^{8} b^{2} x + a^{9} b\right )} e^{5} + 5 \, {\left (a^{4} b^{6} d x^{4} + 4 \, a^{5} b^{5} d x^{3} + 6 \, a^{6} b^{4} d x^{2} + 4 \, a^{7} b^{3} d x + a^{8} b^{2} d\right )} e^{4} - 10 \, {\left (a^{3} b^{7} d^{2} x^{4} + 4 \, a^{4} b^{6} d^{2} x^{3} + 6 \, a^{5} b^{5} d^{2} x^{2} + 4 \, a^{6} b^{4} d^{2} x + a^{7} b^{3} d^{2}\right )} e^{3} + 10 \, {\left (a^{2} b^{8} d^{3} x^{4} + 4 \, a^{3} b^{7} d^{3} x^{3} + 6 \, a^{4} b^{6} d^{3} x^{2} + 4 \, a^{5} b^{5} d^{3} x + a^{6} b^{4} d^{3}\right )} e^{2} - 5 \, {\left (a b^{9} d^{4} x^{4} + 4 \, a^{2} b^{8} d^{4} x^{3} + 6 \, a^{3} b^{7} d^{4} x^{2} + 4 \, a^{4} b^{6} d^{4} x + a^{5} b^{5} d^{4}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/384*(105*(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*sqrt(b^2*d - a*b*e)*e^4*log((2*b*d + (b*
x - a)*e - 2*sqrt(b^2*d - a*b*e)*sqrt(x*e + d))/(b*x + a)) - 2*(48*b^5*d^4 + (105*a*b^4*x^3 + 385*a^2*b^3*x^2
+ 511*a^3*b^2*x + 279*a^4*b)*e^4 - (105*b^5*d*x^3 + 455*a*b^4*d*x^2 + 763*a^2*b^3*d*x + 605*a^3*b^2*d)*e^3 + 2
*(35*b^5*d^2*x^2 + 154*a*b^4*d^2*x + 263*a^2*b^3*d^2)*e^2 - 8*(7*b^5*d^3*x + 31*a*b^4*d^3)*e)*sqrt(x*e + d))/(
b^10*d^5*x^4 + 4*a*b^9*d^5*x^3 + 6*a^2*b^8*d^5*x^2 + 4*a^3*b^7*d^5*x + a^4*b^6*d^5 - (a^5*b^5*x^4 + 4*a^6*b^4*
x^3 + 6*a^7*b^3*x^2 + 4*a^8*b^2*x + a^9*b)*e^5 + 5*(a^4*b^6*d*x^4 + 4*a^5*b^5*d*x^3 + 6*a^6*b^4*d*x^2 + 4*a^7*
b^3*d*x + a^8*b^2*d)*e^4 - 10*(a^3*b^7*d^2*x^4 + 4*a^4*b^6*d^2*x^3 + 6*a^5*b^5*d^2*x^2 + 4*a^6*b^4*d^2*x + a^7
*b^3*d^2)*e^3 + 10*(a^2*b^8*d^3*x^4 + 4*a^3*b^7*d^3*x^3 + 6*a^4*b^6*d^3*x^2 + 4*a^5*b^5*d^3*x + a^6*b^4*d^3)*e
^2 - 5*(a*b^9*d^4*x^4 + 4*a^2*b^8*d^4*x^3 + 6*a^3*b^7*d^4*x^2 + 4*a^4*b^6*d^4*x + a^5*b^5*d^4)*e), 1/192*(105*
(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqr
t(x*e + d)/(b*x*e + b*d))*e^4 - (48*b^5*d^4 + (105*a*b^4*x^3 + 385*a^2*b^3*x^2 + 511*a^3*b^2*x + 279*a^4*b)*e^
4 - (105*b^5*d*x^3 + 455*a*b^4*d*x^2 + 763*a^2*b^3*d*x + 605*a^3*b^2*d)*e^3 + 2*(35*b^5*d^2*x^2 + 154*a*b^4*d^
2*x + 263*a^2*b^3*d^2)*e^2 - 8*(7*b^5*d^3*x + 31*a*b^4*d^3)*e)*sqrt(x*e + d))/(b^10*d^5*x^4 + 4*a*b^9*d^5*x^3
+ 6*a^2*b^8*d^5*x^2 + 4*a^3*b^7*d^5*x + a^4*b^6*d^5 - (a^5*b^5*x^4 + 4*a^6*b^4*x^3 + 6*a^7*b^3*x^2 + 4*a^8*b^2
*x + a^9*b)*e^5 + 5*(a^4*b^6*d*x^4 + 4*a^5*b^5*d*x^3 + 6*a^6*b^4*d*x^2 + 4*a^7*b^3*d*x + a^8*b^2*d)*e^4 - 10*(
a^3*b^7*d^2*x^4 + 4*a^4*b^6*d^2*x^3 + 6*a^5*b^5*d^2*x^2 + 4*a^6*b^4*d^2*x + a^7*b^3*d^2)*e^3 + 10*(a^2*b^8*d^3
*x^4 + 4*a^3*b^7*d^3*x^3 + 6*a^4*b^6*d^3*x^2 + 4*a^5*b^5*d^3*x + a^6*b^4*d^3)*e^2 - 5*(a*b^9*d^4*x^4 + 4*a^2*b
^8*d^4*x^3 + 6*a^3*b^7*d^4*x^2 + 4*a^4*b^6*d^4*x + a^5*b^5*d^4)*e)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 1.31, size = 391, normalized size = 1.41 \begin {gather*} \frac {35 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{4}}{64 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {105 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} e^{4} - 385 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} d e^{4} + 511 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{4} - 279 \, \sqrt {x e + d} b^{3} d^{3} e^{4} + 385 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{2} e^{5} - 1022 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} d e^{5} + 837 \, \sqrt {x e + d} a b^{2} d^{2} e^{5} + 511 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b e^{6} - 837 \, \sqrt {x e + d} a^{2} b d e^{6} + 279 \, \sqrt {x e + d} a^{3} e^{7}}{192 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/64*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^4*d^4*sgn(b*x + a) - 4*a*b^3*d^3*e*sgn(b*x + a) + 6
*a^2*b^2*d^2*e^2*sgn(b*x + a) - 4*a^3*b*d*e^3*sgn(b*x + a) + a^4*e^4*sgn(b*x + a))*sqrt(-b^2*d + a*b*e)) + 1/1
92*(105*(x*e + d)^(7/2)*b^3*e^4 - 385*(x*e + d)^(5/2)*b^3*d*e^4 + 511*(x*e + d)^(3/2)*b^3*d^2*e^4 - 279*sqrt(x
*e + d)*b^3*d^3*e^4 + 385*(x*e + d)^(5/2)*a*b^2*e^5 - 1022*(x*e + d)^(3/2)*a*b^2*d*e^5 + 837*sqrt(x*e + d)*a*b
^2*d^2*e^5 + 511*(x*e + d)^(3/2)*a^2*b*e^6 - 837*sqrt(x*e + d)*a^2*b*d*e^6 + 279*sqrt(x*e + d)*a^3*e^7)/((b^4*
d^4*sgn(b*x + a) - 4*a*b^3*d^3*e*sgn(b*x + a) + 6*a^2*b^2*d^2*e^2*sgn(b*x + a) - 4*a^3*b*d*e^3*sgn(b*x + a) +
a^4*e^4*sgn(b*x + a))*((x*e + d)*b - b*d + a*e)^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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