3.20.24 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^2} \, dx\) [1924]

Optimal. Leaf size=187 \[ \frac {3}{4} \left (a-\frac {c d^2}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e (d+e x)}+\frac {3 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \sqrt {c} \sqrt {d} e^{5/2}} \]

[Out]

1/2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e/(e*x+d)+3/8*(-a*e^2+c*d^2)^2*arctanh(1/2*(2*c*d*e*x+a*e^2+c*d^2)
/c^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/e^(5/2)/c^(1/2)/d^(1/2)+3/4*(a-c*d^2/e^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 = 187, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {678, 635, 212} \begin {gather*} \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e (d+e x)}+\frac {3}{4} \left (a-\frac {c d^2}{e^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}+\frac {3 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 \sqrt {c} \sqrt {d} e^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a - (c*d^2)/e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/4 + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^
(3/2)/(2*e*(d + e*x)) + (3*(c*d^2 - a*e^2)^2*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(8*Sqrt[c]*Sqrt[d]*e^(5/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

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

Rule 678

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

Rubi steps

\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^2} \, dx &=\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e (d+e x)}-\frac {\left (3 \left (2 c d^2 e-e \left (c d^2+a e^2\right )\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx}{4 e^2}\\ &=\frac {3}{4} \left (a-\frac {c d^2}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e (d+e x)}+\frac {\left (3 \left (c d^2-a e^2\right )^2\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 e^2}\\ &=\frac {3}{4} \left (a-\frac {c d^2}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e (d+e x)}+\frac {\left (3 \left (c d^2-a e^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 e^2}\\ &=\frac {3}{4} \left (a-\frac {c d^2}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e (d+e x)}+\frac {3 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \sqrt {c} \sqrt {d} e^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 172, normalized size = 0.92 \begin {gather*} \frac {\sqrt {c} \sqrt {d} \sqrt {e} (d+e x) \left (5 a^2 e^3+c^2 d^2 x (-3 d+2 e x)+a c d e (-3 d+7 e x)\right )+3 \left (c d^2-a e^2\right )^2 \sqrt {a e+c d x} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{4 \sqrt {c} \sqrt {d} e^{5/2} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c]*Sqrt[d]*Sqrt[e]*(d + e*x)*(5*a^2*e^3 + c^2*d^2*x*(-3*d + 2*e*x) + a*c*d*e*(-3*d + 7*e*x)) + 3*(c*d^2
- a*e^2)^2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])
])/(4*Sqrt[c]*Sqrt[d]*e^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]
time = 0.70, size = 311, normalized size = 1.66

method	result	size
default	\(\frac {\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {6 c d e \left (\frac {\left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (e^{2} a -c \,d^{2}\right ) \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 c d e}-\frac {\left (e^{2} a -c \,d^{2}\right )^{2} \ln \left (\frac {\frac {e^{2} a}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 c d e \sqrt {c d e}}\right )}{2}\right )}{e^{2} a -c \,d^{2}}}{e^{2}}\)	\(311\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^2*(2/(a*e^2-c*d^2)/(x+d/e)^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)-6*c*d*e/(a*e^2-c*d^2)*(1/3*(c*d
*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+1/2*(a*e^2-c*d^2)*(1/4*(2*c*d*e*(x+d/e)+e^2*a-c*d^2)/c/d/e*(c*d*e*(x
+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/8*(a*e^2-c*d^2)^2/c/d/e*ln((1/2*e^2*a-1/2*c*d^2+c*d*e*(x+d/e))/(c*d*e)^
(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^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 [A]
time = 2.63, size = 409, normalized size = 2.19 \begin {gather*} \left [\frac {{\left (3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d} e^{\frac {1}{2}} \log \left (8 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + a^{2} e^{4} + 4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {c d} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) + 4 \, {\left (2 \, c^{2} d^{2} x e^{2} - 3 \, c^{2} d^{3} e + 5 \, a c d e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-3\right )}}{16 \, c d}, -\frac {{\left (3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) - 2 \, {\left (2 \, c^{2} d^{2} x e^{2} - 3 \, c^{2} d^{3} e + 5 \, a c d e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-3\right )}}{8 \, c d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(3*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(c*d)*e^(1/2)*log(8*c^2*d^3*x*e + c^2*d^4 + 8*a*c*d*x*e^3 + a
^2*e^4 + 4*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqrt(c*d)*e^(1/2) + 2*(4*c^
2*d^2*x^2 + 3*a*c*d^2)*e^2) + 4*(2*c^2*d^2*x*e^2 - 3*c^2*d^3*e + 5*a*c*d*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^
2 + a*d)*e))*e^(-3)/(c*d), -1/8*(3*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c*d*e)*arctan(1/2*sqrt(c*d^2*x +
a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^3*x*e + a*c*d*x*e^3 + (c^2*d^2*x^
2 + a*c*d^2)*e^2)) - 2*(2*c^2*d^2*x*e^2 - 3*c^2*d^3*e + 5*a*c*d*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*
e))*e^(-3)/(c*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (161) = 322\).
time = 0.79, size = 472, normalized size = 2.52 \begin {gather*} -\frac {1}{4} \, {\left (\frac {3 \, {\left (c^{2} d^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 2 \, a c d^{2} e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + a^{2} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} \arctan \left (\frac {\sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}}}{\sqrt {-c d e}}\right ) e^{\left (-3\right )}}{\sqrt {-c d e}} + \frac {{\left (3 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} c^{3} d^{5} e \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 5 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} c^{2} d^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 6 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} a c^{2} d^{3} e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 10 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} a c d^{2} e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, \sqrt {c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}} a^{2} c d e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 5 \, {\left (c d e - \frac {c d^{2} e}{x e + d} + \frac {a e^{3}}{x e + d}\right )}^{\frac {3}{2}} a^{2} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} e^{\left (-3\right )}}{{\left (\frac {c d^{2} e}{x e + d} - \frac {a e^{3}}{x e + d}\right )}^{2}}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(3*(c^2*d^4*sgn(1/(x*e + d)) - 2*a*c*d^2*e^2*sgn(1/(x*e + d)) + a^2*e^4*sgn(1/(x*e + d)))*arctan(sqrt(c*d
*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))/sqrt(-c*d*e))*e^(-3)/sqrt(-c*d*e) + (3*sqrt(c*d*e - c*d^2*e/(x*e + d
) + a*e^3/(x*e + d))*c^3*d^5*e*sgn(1/(x*e + d)) - 5*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*c^2*d^
4*sgn(1/(x*e + d)) - 6*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a*c^2*d^3*e^3*sgn(1/(x*e + d)) + 10*(
c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a*c*d^2*e^2*sgn(1/(x*e + d)) + 3*sqrt(c*d*e - c*d^2*e/(x*e
+ d) + a*e^3/(x*e + d))*a^2*c*d*e^5*sgn(1/(x*e + d)) - 5*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a
^2*e^4*sgn(1/(x*e + d)))*e^(-3)/(c*d^2*e/(x*e + d) - a*e^3/(x*e + d))^2)*e

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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