3.21.62 \(\int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [2062]

Optimal. Leaf size=139 \[ \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {c d \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{\sqrt {e} \left (c d^2-a e^2\right )^{3/2}} \]

[Out]

c*d*arctan(e^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^(1/2)/(e*x+d)^(1/2))/(-a*e^2+c*d^2)^
(3/2)/e^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)/(e*x+d)^(3/2)

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Rubi [A]
time = 0.05, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {686, 674, 211} \begin {gather*} \frac {c d \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\sqrt {e} \left (c d^2-a e^2\right )^{3/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/((c*d^2 - a*e^2)*(d + e*x)^(3/2)) + (c*d*ArcTan[(Sqrt[e]*Sqrt[a*d*
e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(Sqrt[e]*(c*d^2 - a*e^2)^(3/2))

Rule 211

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

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a
+ b*x + c*x^2)^(p + 1)/((m + p + 1)*(2*c*d - b*e))), x] + Dist[c*((m + 2*p + 2)/((m + p + 1)*(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, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rubi steps

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

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Mathematica [A]
time = 0.20, size = 135, normalized size = 0.97 \begin {gather*} \frac {\sqrt {e} \sqrt {c d^2-a e^2} (a e+c d x)+c d \sqrt {a e+c d x} (d+e x) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {e} \left (c d^2-a e^2\right )^{3/2} \sqrt {d+e x} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(Sqrt[e]*Sqrt[c*d^2 - a*e^2]*(a*e + c*d*x) + c*d*Sqrt[a*e + c*d*x]*(d + e*x)*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x]
)/Sqrt[c*d^2 - a*e^2]])/(Sqrt[e]*(c*d^2 - a*e^2)^(3/2)*Sqrt[d + e*x]*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]
time = 0.72, size = 162, normalized size = 1.17

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (\arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c d e x +\arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c \,d^{2}-\sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\right )}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {c d x +a e}\, \left (e^{2} a -c \,d^{2}\right ) \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(162\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

((c*d*x+a*e)*(e*x+d))^(1/2)*(arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c*d*e*x+arctanh(e*(c*d*x+a*e
)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c*d^2-(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2))/(e*x+d)^(3/2)/(c*d*x+a*e)^(1
/2)/(a*e^2-c*d^2)/((a*e^2-c*d^2)*e)^(1/2)

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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/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)*(x*e + d)^(3/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (122) = 244\).
time = 2.95, size = 555, normalized size = 3.99 \begin {gather*} \left [\frac {{\left (c d x^{2} e^{2} + 2 \, c d^{2} x e + c d^{3}\right )} \sqrt {-c d^{2} e + a e^{3}} \log \left (\frac {c d^{3} - 2 \, a x e^{3} - {\left (c d x^{2} + 2 \, a d\right )} e^{2} - 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-c d^{2} e + a e^{3}} \sqrt {x e + d}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d^{2} e - a e^{3}\right )} \sqrt {x e + d}}{2 \, {\left (2 \, c^{2} d^{5} x e^{2} + c^{2} d^{6} e - 4 \, a c d^{3} x e^{4} + a^{2} x^{2} e^{7} + 2 \, a^{2} d x e^{6} - {\left (2 \, a c d^{2} x^{2} - a^{2} d^{2}\right )} e^{5} + {\left (c^{2} d^{4} x^{2} - 2 \, a c d^{4}\right )} e^{3}\right )}}, -\frac {{\left (c d x^{2} e^{2} + 2 \, c d^{2} x e + c d^{3}\right )} \sqrt {c d^{2} e - a e^{3}} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {c d^{2} e - a e^{3}} \sqrt {x e + d}}{c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}}\right ) - \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d^{2} e - a e^{3}\right )} \sqrt {x e + d}}{2 \, c^{2} d^{5} x e^{2} + c^{2} d^{6} e - 4 \, a c d^{3} x e^{4} + a^{2} x^{2} e^{7} + 2 \, a^{2} d x e^{6} - {\left (2 \, a c d^{2} x^{2} - a^{2} d^{2}\right )} e^{5} + {\left (c^{2} d^{4} x^{2} - 2 \, a c d^{4}\right )} e^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Integral(1/(sqrt((d + e*x)*(a*e + c*d*x))*(d + e*x)**(3/2)), x)

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Giac [A]
time = 1.52, size = 140, normalized size = 1.01 \begin {gather*} \frac {{\left (\frac {c^{2} d^{2} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{\sqrt {c d^{2} e - a e^{3}} {\left (c d^{2} - a e^{2}\right )}} + \frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c d}{{\left (c d^{2} - a e^{2}\right )} {\left (x e + d\right )}}\right )} e^{\left (-1\right )}}{c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

(c^2*d^2*arctan(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))*e/(sqrt(c*d^2*e - a*e^3)*(c*d^2
- a*e^2)) + sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c*d/((c*d^2 - a*e^2)*(x*e + d)))*e^(-1)/(c*d)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)),x)

[Out]

int(1/((d + e*x)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)), x)

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