3.5.82 \(\int \frac {1}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [482]

Optimal. Leaf size=121 \[ \frac {2}{3 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {8 c d \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

2/3/(-a*e^2+c*d^2)/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-8/3*c*d*(2*c*d*e*x+a*e^2+c*d^2)/(-a*e^2+c*d
^2)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 627} \begin {gather*} \frac {2}{3 (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}}-\frac {8 c d \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \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/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

2/(3*(c*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (8*c*d*(c*d^2 + a*e^2 + 2*c*d*e*
x))/(3*(c*d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 672

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*(Simplify[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, m, p}, x] && NeQ[b^2 - 4*a
*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rubi steps

\begin {align*} \int \frac {1}{(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 {2}{3 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(4 c d) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 \left (c d^2-a e^2\right )}\\ &=\frac {2}{3 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {8 c d \left (c d^2+a e^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 97, normalized size = 0.80 \begin {gather*} -\frac {2 (a e+c d x)^3 \left (-e^2+\frac {6 c d e (d+e x)}{a e+c d x}+\frac {3 c^2 d^2 (d+e x)^2}{(a e+c d x)^2}\right )}{3 \left (c d^2-a e^2\right )^3 ((a e+c d x) (d+e x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a*e + c*d*x)^3*(-e^2 + (6*c*d*e*(d + e*x))/(a*e + c*d*x) + (3*c^2*d^2*(d + e*x)^2)/(a*e + c*d*x)^2))/(3*(
c*d^2 - a*e^2)^3*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [A]
time = 0.09, size = 146, normalized size = 1.21

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-8 c^{2} d^{2} e^{2} x^{2}-4 a c d \,e^{3} x -12 c^{2} d^{3} e x +a^{2} e^{4}-6 a c \,d^{2} e^{2}-3 c^{2} d^{4}\right )}{3 \left (a^{3} e^{6}-3 a^{2} c \,d^{2} e^{4}+3 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(138\)
default	\(\frac {-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}}{e}\)	\(146\)
trager	\(-\frac {2 \left (-8 c^{2} d^{2} e^{2} x^{2}-4 a c d \,e^{3} x -12 c^{2} d^{3} e x +a^{2} e^{4}-6 a c \,d^{2} e^{2}-3 c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{2} \left (a \,e^{2}-c \,d^{2}\right ) \left (c d x +a e \right )}\)	\(146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e*(-2/3/(a*e^2-c*d^2)/(x+d/e)/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+8/3*c*d*e/(a*e^2-c*d^2)^3*(2*c*d
*e*(x+d/e)+a*e^2-c*d^2)/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(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)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),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(c*d^2-%e^2*a>0)', see `assume?
` for more d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (114) = 228\).
time = 8.47, size = 308, normalized size = 2.55 \begin {gather*} -\frac {2 \, {\left (12 \, c^{2} d^{3} x e + 3 \, c^{2} d^{4} + 4 \, a c d x e^{3} - a^{2} e^{4} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{3 \, {\left (c^{4} d^{9} x - a^{4} x^{2} e^{9} - {\left (a^{3} c d x^{3} + 2 \, a^{4} d x\right )} e^{8} + {\left (a^{3} c d^{2} x^{2} - a^{4} d^{2}\right )} e^{7} + {\left (3 \, a^{2} c^{2} d^{3} x^{3} + 5 \, a^{3} c d^{3} x\right )} e^{6} + 3 \, {\left (a^{2} c^{2} d^{4} x^{2} + a^{3} c d^{4}\right )} e^{5} - 3 \, {\left (a c^{3} d^{5} x^{3} + a^{2} c^{2} d^{5} x\right )} e^{4} - {\left (5 \, a c^{3} d^{6} x^{2} + 3 \, a^{2} c^{2} d^{6}\right )} e^{3} + {\left (c^{4} d^{7} x^{3} - a c^{3} d^{7} x\right )} e^{2} + {\left (2 \, c^{4} d^{8} x^{2} + a c^{3} d^{8}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(12*c^2*d^3*x*e + 3*c^2*d^4 + 4*a*c*d*x*e^3 - a^2*e^4 + 2*(4*c^2*d^2*x^2 + 3*a*c*d^2)*e^2)*sqrt(c*d^2*x +
a*x*e^2 + (c*d*x^2 + a*d)*e)/(c^4*d^9*x - a^4*x^2*e^9 - (a^3*c*d*x^3 + 2*a^4*d*x)*e^8 + (a^3*c*d^2*x^2 - a^4*
d^2)*e^7 + (3*a^2*c^2*d^3*x^3 + 5*a^3*c*d^3*x)*e^6 + 3*(a^2*c^2*d^4*x^2 + a^3*c*d^4)*e^5 - 3*(a*c^3*d^5*x^3 +
a^2*c^2*d^5*x)*e^4 - (5*a*c^3*d^6*x^2 + 3*a^2*c^2*d^6)*e^3 + (c^4*d^7*x^3 - a*c^3*d^7*x)*e^2 + (2*c^4*d^8*x^2
+ a*c^3*d^8)*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}} \left (d + e x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(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)*(d + e*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 2.88, size = 120, normalized size = 0.99 \begin {gather*} \frac {2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (-a^2\,e^4+6\,a\,c\,d^2\,e^2+4\,a\,c\,d\,e^3\,x+3\,c^2\,d^4+12\,c^2\,d^3\,e\,x+8\,c^2\,d^2\,e^2\,x^2\right )}{3\,\left (a\,e+c\,d\,x\right )\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(3*c^2*d^4 - a^2*e^4 + 8*c^2*d^2*e^2*x^2 + 6*a*c*d^2*e^2 + 12
*c^2*d^3*e*x + 4*a*c*d*e^3*x))/(3*(a*e + c*d*x)*(a*e^2 - c*d^2)^3*(d + e*x)^2)

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