### 3.2075 $$\int \frac{(d+e x)^{3/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx$$

Optimal. Leaf size=251 $\frac{2 e \sqrt{d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 e}{3 c d \sqrt{d+e x} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}}$

[Out]

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

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Rubi [A]  time = 0.17421, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 39, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.128, Rules used = {668, 672, 666, 660, 205} $\frac{2 e \sqrt{d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 e}{3 c d \sqrt{d+e x} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 668

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

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*(m + 2*p + 2))/((m + p + 1)*(2*c*d - b*e)), I
nt[(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] && EqQ
[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rule 666

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 660

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 205

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

Rubi steps

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

Mathematica [C]  time = 0.0245399, size = 77, normalized size = 0.31 $\frac{2 (d+e x)^{3/2} \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{3 \left (a e^2-c d^2\right ) ((d+e x) (a e+c d x))^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*Hypergeometric2F1[-3/2, 1, -1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*(-(c*d^2) + a*e^
2)*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [A]  time = 0.208, size = 234, normalized size = 0.9 \begin{align*} -{\frac{2}{3\, \left ( cdx+ae \right ) ^{2} \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}xcd{e}^{2}+3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) a{e}^{3}\sqrt{cdx+ae}-3\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}xcde-4\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}a{e}^{2}+\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}c{d}^{2} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

-2/3*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(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)*x*c*d*e^2+3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*e^3*(c*d*x+a*e)^(1/2)-3*((a*e^2-c*
d^2)*e)^(1/2)*x*c*d*e-4*((a*e^2-c*d^2)*e)^(1/2)*a*e^2+((a*e^2-c*d^2)*e)^(1/2)*c*d^2)/(e*x+d)^(1/2)/(c*d*x+a*e)
^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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2), x)

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Fricas [A]  time = 2.29472, size = 1592, normalized size = 6.34 \begin{align*} \left [\frac{3 \,{\left (c^{2} d^{2} e^{2} x^{3} + a^{2} d e^{3} +{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x^{2} +{\left (2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x\right )} \sqrt{-\frac{e}{c d^{2} - a e^{2}}} \log \left (-\frac{c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{-\frac{e}{c d^{2} - a e^{2}}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (3 \, c d e x - c d^{2} + 4 \, a e^{2}\right )} \sqrt{e x + d}}{3 \,{\left (a^{2} c^{2} d^{5} e^{2} - 2 \, a^{3} c d^{3} e^{4} + a^{4} d e^{6} +{\left (c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}\right )} x^{3} +{\left (c^{4} d^{7} - 3 \, a^{2} c^{2} d^{3} e^{4} + 2 \, a^{3} c d e^{6}\right )} x^{2} +{\left (2 \, a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + a^{4} e^{7}\right )} x\right )}}, \frac{2 \,{\left (3 \,{\left (c^{2} d^{2} e^{2} x^{3} + a^{2} d e^{3} +{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x^{2} +{\left (2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x\right )} \sqrt{\frac{e}{c d^{2} - a e^{2}}} \arctan \left (-\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{e}{c d^{2} - a e^{2}}}}{c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x}\right ) + \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (3 \, c d e x - c d^{2} + 4 \, a e^{2}\right )} \sqrt{e x + d}\right )}}{3 \,{\left (a^{2} c^{2} d^{5} e^{2} - 2 \, a^{3} c d^{3} e^{4} + a^{4} d e^{6} +{\left (c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}\right )} x^{3} +{\left (c^{4} d^{7} - 3 \, a^{2} c^{2} d^{3} e^{4} + 2 \, a^{3} c d e^{6}\right )} x^{2} +{\left (2 \, a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + a^{4} e^{7}\right )} x\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x