3.21.69 \(\int \frac {1}{\sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [2069]

Optimal. Leaf size=196 \[ \frac {1}{\left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 c d \sqrt {d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 c d \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \]

[Out]

-3*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))*e^(1/2)/(-a*
e^2+c*d^2)^(5/2)+1/(-a*e^2+c*d^2)/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-3*c*d*(e*x+d)^(1/2)/(-
a*e^2+c*d^2)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {686, 680, 674, 211} \begin {gather*} -\frac {3 c d \sqrt {e} \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 )}{\left (c d^2-a e^2\right )^{5/2}}-\frac {3 c d \sqrt {d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 680

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

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

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Mathematica [A]
time = 0.32, size = 141, normalized size = 0.72 \begin {gather*} \frac {-\sqrt {c d^2-a e^2} \left (a e^2+c d (2 d+3 e x)\right )-3 c d \sqrt {e} \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 )}{\left (c d^2-a e^2\right )^{5/2} \sqrt {d+e x} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[c*d^2 - a*e^2]*(a*e^2 + c*d*(2*d + 3*e*x))) - 3*c*d*Sqrt[e]*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]])/((c*d^2 - a*e^2)^(5/2)*Sqrt[d + e*x]*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]
time = 0.72, size = 225, normalized size = 1.15

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c d \,e^{2} x +3 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c \,d^{2} e -3 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c d e x -\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,e^{2}-2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c \,d^{2}\right )}{\left (e x +d \right )^{\frac {3}{2}} \left (c d x +a e \right ) \left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(225\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (176) = 352\).
time = 2.45, size = 746, normalized size = 3.81 \begin {gather*} \left [\frac {3 \, {\left (c^{2} d^{4} x + a c d x^{2} e^{3} + {\left (c^{2} d^{2} x^{3} + 2 \, a c d^{2} x\right )} e^{2} + {\left (2 \, c^{2} d^{3} x^{2} + a c d^{3}\right )} e\right )} \sqrt {-\frac {e}{c d^{2} - a e^{2}}} \log \left (\frac {c d^{3} - 2 \, a x e^{3} + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {-\frac {e}{c d^{2} - a e^{2}}} - {\left (c d x^{2} + 2 \, a d\right )} e^{2}}{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 (3 \, c d x e + 2 \, c d^{2} + a e^{2}\right )} \sqrt {x e + d}}{2 \, {\left (c^{3} d^{5} x^{3} e^{2} + c^{3} d^{7} x + a^{3} x^{2} e^{7} + a^{3} d^{2} e^{5} + {\left (a^{2} c d x^{3} + 2 \, a^{3} d x\right )} e^{6} - {\left (2 \, a c^{2} d^{3} x^{3} + 3 \, a^{2} c d^{3} x\right )} e^{4} - {\left (3 \, a c^{2} d^{4} x^{2} + 2 \, a^{2} c d^{4}\right )} e^{3} + {\left (2 \, c^{3} d^{6} x^{2} + a c^{2} d^{6}\right )} e\right )}}, -\frac {\frac {3 \, {\left (c^{2} d^{4} x + a c d x^{2} e^{3} + {\left (c^{2} d^{2} x^{3} + 2 \, a c d^{2} x\right )} e^{2} + {\left (2 \, c^{2} d^{3} x^{2} + a c d^{3}\right )} e\right )} \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} - a e^{2}} \sqrt {x e + d} e^{\frac {1}{2}}}{c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}}\right ) e^{\frac {1}{2}}}{\sqrt {c d^{2} - a e^{2}}} + \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (3 \, c d x e + 2 \, c d^{2} + a e^{2}\right )} \sqrt {x e + d}}{c^{3} d^{5} x^{3} e^{2} + c^{3} d^{7} x + a^{3} x^{2} e^{7} + a^{3} d^{2} e^{5} + {\left (a^{2} c d x^{3} + 2 \, a^{3} d x\right )} e^{6} - {\left (2 \, a c^{2} d^{3} x^{3} + 3 \, a^{2} c d^{3} x\right )} e^{4} - {\left (3 \, a c^{2} d^{4} x^{2} + 2 \, a^{2} c d^{4}\right )} e^{3} + {\left (2 \, c^{3} d^{6} x^{2} + a c^{2} d^{6}\right )} e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.93, size = 258, normalized size = 1.32 \begin {gather*} -\frac {3 \, c d \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}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {2 \, c^{2} d^{3} e^{2} - 2 \, a c d e^{4} + 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d e}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c d^{2} e - \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a e^{3} + {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*c*d*arctan(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))*e/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2
*e^4)*sqrt(c*d^2*e - a*e^3)) - (2*c^2*d^3*e^2 - 2*a*c*d*e^4 + 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c*d*e)/((c
^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c*d^2*e - sqrt((x*e + d)*c*d*e - c*
d^2*e + a*e^3)*a*e^3 + ((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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