3.21.63 \(\int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [2063]

Optimal. Leaf size=207 \[ \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {3 c^2 d^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{4 \sqrt {e} \left (c d^2-a e^2\right )^{5/2}} \]

[Out]

3/4*c^2*d^2*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)^(5/2)/e^(1/2)+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)^(5/2)+3/4*c*d*(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^2/(e*x+d)^(3/2)

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Rubi [A]
time = 0.08, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {3 c^2 d^2 \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 )}{4 \sqrt {e} \left (c d^2-a e^2\right )^{5/2}}+\frac {3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*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]/(2*(c*d^2 - a*e^2)*(d + e*x)^(5/2)) + (3*c*d*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(4*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) + (3*c^2*d^2*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])])/(4*Sqrt[e]*(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 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)^{5/2} \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}}{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {(3 c d) \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}{4 \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}}{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {\left (3 c^2 d^2\right ) \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}{8 \left (c d^2-a e^2\right )^2}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {\left (3 c^2 d^2 e\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 )}{4 \left (c d^2-a e^2\right )^2}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {3 c^2 d^2 \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 )}{4 \sqrt {e} \left (c d^2-a e^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 174, normalized size = 0.84 \begin {gather*} \frac {\sqrt {e} \sqrt {c d^2-a e^2} \left (-2 a^2 e^3+a c d e (5 d+e x)+c^2 d^2 x (5 d+3 e x)\right )+3 c^2 d^2 \sqrt {a e+c d x} (d+e x)^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{4 \sqrt {e} \left (c d^2-a e^2\right )^{5/2} (d+e x)^{3/2} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*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]*(-2*a^2*e^3 + a*c*d*e*(5*d + e*x) + c^2*d^2*x*(5*d + 3*e*x)) + 3*c^2*d^2*Sqrt[a*e
+ c*d*x]*(d + e*x)^2*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(4*Sqrt[e]*(c*d^2 - a*e^2)^(5/2
)*(d + e*x)^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]
time = 0.72, size = 282, normalized size = 1.36

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{2} e^{2} x^{2}+6 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{3} e x +3 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{4}-3 c d e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+2 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,e^{2}-5 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c \,d^{2}\right )}{4 \left (e x +d \right )^{\frac {5}{2}} \sqrt {c d x +a e}\, \left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(282\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(e*x+d)^(5/2)*((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^2*d^2
*e^2*x^2+6*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^2*d^3*e*x+3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*
e^2-c*d^2)*e)^(1/2))*c^2*d^4-3*c*d*e*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+2*(c*d*x+a*e)^(1/2)*((a*e^2-c
*d^2)*e)^(1/2)*a*e^2-5*(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)^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)^(5/2)/(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)^(5/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (181) = 362\).
time = 3.27, size = 861, normalized size = 4.16 \begin {gather*} \left [-\frac {3 \, {\left (c^{2} d^{2} x^{3} e^{3} + 3 \, c^{2} d^{3} x^{2} e^{2} + 3 \, c^{2} d^{4} x e + c^{2} d^{5}\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 \, {\left (3 \, c^{2} d^{3} x e^{2} + 5 \, c^{2} d^{4} e - 3 \, a c d x e^{4} - 7 \, a c d^{2} e^{3} + 2 \, a^{2} e^{5}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{8 \, {\left (3 \, c^{3} d^{8} x e^{2} + c^{3} d^{9} e - a^{3} x^{3} e^{10} - 3 \, a^{3} d x^{2} e^{9} + 3 \, {\left (a^{2} c d^{2} x^{3} - a^{3} d^{2} x\right )} e^{8} + {\left (9 \, a^{2} c d^{3} x^{2} - a^{3} d^{3}\right )} e^{7} - 3 \, {\left (a c^{2} d^{4} x^{3} - 3 \, a^{2} c d^{4} x\right )} e^{6} - 3 \, {\left (3 \, a c^{2} d^{5} x^{2} - a^{2} c d^{5}\right )} e^{5} + {\left (c^{3} d^{6} x^{3} - 9 \, a c^{2} d^{6} x\right )} e^{4} + 3 \, {\left (c^{3} d^{7} x^{2} - a c^{2} d^{7}\right )} e^{3}\right )}}, -\frac {3 \, {\left (c^{2} d^{2} x^{3} e^{3} + 3 \, c^{2} d^{3} x^{2} e^{2} + 3 \, c^{2} d^{4} x e + c^{2} d^{5}\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 ) - {\left (3 \, c^{2} d^{3} x e^{2} + 5 \, c^{2} d^{4} e - 3 \, a c d x e^{4} - 7 \, a c d^{2} e^{3} + 2 \, a^{2} e^{5}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{4 \, {\left (3 \, c^{3} d^{8} x e^{2} + c^{3} d^{9} e - a^{3} x^{3} e^{10} - 3 \, a^{3} d x^{2} e^{9} + 3 \, {\left (a^{2} c d^{2} x^{3} - a^{3} d^{2} x\right )} e^{8} + {\left (9 \, a^{2} c d^{3} x^{2} - a^{3} d^{3}\right )} e^{7} - 3 \, {\left (a c^{2} d^{4} x^{3} - 3 \, a^{2} c d^{4} x\right )} e^{6} - 3 \, {\left (3 \, a c^{2} d^{5} x^{2} - a^{2} c d^{5}\right )} e^{5} + {\left (c^{3} d^{6} x^{3} - 9 \, a c^{2} d^{6} x\right )} e^{4} + 3 \, {\left (c^{3} d^{7} x^{2} - a c^{2} d^{7}\right )} e^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*(3*(c^2*d^2*x^3*e^3 + 3*c^2*d^3*x^2*e^2 + 3*c^2*d^4*x*e + c^2*d^5)*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*(3*c^2*d^3*x*e^2 + 5*c^2*d^4*e - 3*a*c*d*x*e^4 - 7*a*c*d^2*e^3 + 2*a^2
*e^5)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(3*c^3*d^8*x*e^2 + c^3*d^9*e - a^3*x^3*e^10 -
3*a^3*d*x^2*e^9 + 3*(a^2*c*d^2*x^3 - a^3*d^2*x)*e^8 + (9*a^2*c*d^3*x^2 - a^3*d^3)*e^7 - 3*(a*c^2*d^4*x^3 - 3*
a^2*c*d^4*x)*e^6 - 3*(3*a*c^2*d^5*x^2 - a^2*c*d^5)*e^5 + (c^3*d^6*x^3 - 9*a*c^2*d^6*x)*e^4 + 3*(c^3*d^7*x^2 -
a*c^2*d^7)*e^3), -1/4*(3*(c^2*d^2*x^3*e^3 + 3*c^2*d^3*x^2*e^2 + 3*c^2*d^4*x*e + c^2*d^5)*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)) - (3*c^2*d^3*x*e^2 + 5*c^2*d^4*e - 3*a*c*d*x*e^4 - 7*a*c*d^2*e^3 + 2*a^2*e^5)*sqrt(c*d^
2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(3*c^3*d^8*x*e^2 + c^3*d^9*e - a^3*x^3*e^10 - 3*a^3*d*x^2*e^
9 + 3*(a^2*c*d^2*x^3 - a^3*d^2*x)*e^8 + (9*a^2*c*d^3*x^2 - a^3*d^3)*e^7 - 3*(a*c^2*d^4*x^3 - 3*a^2*c*d^4*x)*e^
6 - 3*(3*a*c^2*d^5*x^2 - a^2*c*d^5)*e^5 + (c^3*d^6*x^3 - 9*a*c^2*d^6*x)*e^4 + 3*(c^3*d^7*x^2 - a*c^2*d^7)*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 {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [A]
time = 1.72, size = 256, normalized size = 1.24 \begin {gather*} \frac {{\left (\frac {3 \, c^{3} d^{3} \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 {{\left (5 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{4} d^{5} e^{2} - 5 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{3} d^{3} e^{4} + 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{3} d^{3} e\right )} e^{\left (-2\right )}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (x e + d\right )}^{2} c^{2} d^{2}}\right )} e^{\left (-1\right )}}{4 \, c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3*c^3*d^3*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)) + (5*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^4*d^5*e^2 - 5*sqrt((x*e +
d)*c*d*e - c*d^2*e + a*e^3)*a*c^3*d^3*e^4 + 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^3*d^3*e)*e^(-2)/((c^
2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(x*e + d)^2*c^2*d^2))*e^(-1)/(c*d)

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

[Out]

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

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