3.7.69 \(\int \frac {(d+e x)^{3/2}}{(f+g x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [669]

Optimal. Leaf size=133 \[ -\frac {2 \sqrt {d+e x}}{(c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {2 \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{(c d f-a e g)^{3/2}} \]

[Out]

-2*arctan(g^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^(1/2)/(e*x+d)^(1/2))*g^(1/2)/(-a*e*g+
c*d*f)^(3/2)-2*(e*x+d)^(1/2)/(-a*e*g+c*d*f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {882, 888, 211} \begin {gather*} -\frac {2 \sqrt {g} \text {ArcTan}\left (\frac {\sqrt {g} \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 f-a e g}}\right )}{(c d f-a e g)^{3/2}}-\frac {2 \sqrt {d+e x}}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 882

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(c*e*f + c*d*g - b*e*g))), x]
+ Dist[e^2*g*((m - n - 2)/((p + 1)*(c*e*f + c*d*g - b*e*g))), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*
x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[
c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && LtQ[p, -1] && RationalQ[n]

Rule 888

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[
2*e^2, Subst[Int[1/(c*(e*f + d*g) - b*e*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; Fre
eQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.11, size = 109, normalized size = 0.82 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (\sqrt {c d f-a e g}+\sqrt {g} \sqrt {a e+c d x} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{(c d f-a e g)^{3/2} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 118, normalized size = 0.89

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (g \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) \sqrt {c d x +a e}-\sqrt {\left (a e g -c d f \right ) g}\right )}{\sqrt {e x +d}\, \left (c d x +a e \right ) \left (a e g -c d f \right ) \sqrt {\left (a e g -c d f \right ) g}}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (124) = 248\).
time = 2.74, size = 555, normalized size = 4.17 \begin {gather*} \left [-\frac {{\left (c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e\right )} \sqrt {-\frac {g}{c d f - a g e}} \log \left (-\frac {c d^{2} g x - c d^{2} f + 2 \, a g x e^{2} + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d f - a g e\right )} \sqrt {x e + d} \sqrt {-\frac {g}{c d f - a g e}} + {\left (c d g x^{2} - c d f x + 2 \, a d g\right )} e}{d g x + d f + {\left (g x^{2} + f x\right )} e}\right ) + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{c^{2} d^{3} f x - a^{2} g x e^{3} - {\left (a c d g x^{2} - a c d f x + a^{2} d g\right )} e^{2} + {\left (c^{2} d^{2} f x^{2} - a c d^{2} g x + a c d^{2} f\right )} e}, -\frac {2 \, {\left ({\left (c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e\right )} \sqrt {\frac {g}{c d f - a g e}} \arctan \left (-\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d f - a g e\right )} \sqrt {x e + d} \sqrt {\frac {g}{c d f - a g e}}}{c d^{2} g x + a g x e^{2} + {\left (c d g x^{2} + a d g\right )} e}\right ) + \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}\right )}}{c^{2} d^{3} f x - a^{2} g x e^{3} - {\left (a c d g x^{2} - a c d f x + a^{2} d g\right )} e^{2} + {\left (c^{2} d^{2} f x^{2} - a c d^{2} g x + a c d^{2} f\right )} e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-((c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(-g/(c*d*f - a*g*e))*log(-(c*d^2*g*x - c*d^2*f + 2*a*g*x*e^2 +
2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(c*d*f - a*g*e)*sqrt(x*e + d)*sqrt(-g/(c*d*f - a*g*e)) + (c*d*g*
x^2 - c*d*f*x + 2*a*d*g)*e)/(d*g*x + d*f + (g*x^2 + f*x)*e)) + 2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*s
qrt(x*e + d))/(c^2*d^3*f*x - a^2*g*x*e^3 - (a*c*d*g*x^2 - a*c*d*f*x + a^2*d*g)*e^2 + (c^2*d^2*f*x^2 - a*c*d^2*
g*x + a*c*d^2*f)*e), -2*((c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(g/(c*d*f - a*g*e))*arctan(-sqrt(c*d^2*x
+ a*x*e^2 + (c*d*x^2 + a*d)*e)*(c*d*f - a*g*e)*sqrt(x*e + d)*sqrt(g/(c*d*f - a*g*e))/(c*d^2*g*x + a*g*x*e^2 +
(c*d*g*x^2 + a*d*g)*e)) + sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^2*d^3*f*x - a^2*g*x*e^
3 - (a*c*d*g*x^2 - a*c*d*f*x + a^2*d*g)*e^2 + (c^2*d^2*f*x^2 - a*c*d^2*g*x + a*c*d^2*f)*e)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (124) = 248\).
time = 2.42, size = 276, normalized size = 2.08 \begin {gather*} -2 \, {\left (\frac {g \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} g e^{\left (-1\right )}}{\sqrt {c d f g - a g^{2} e}}\right ) e^{\left (-1\right )}}{\sqrt {c d f g - a g^{2} e} {\left (c d f - a g e\right )}} + \frac {1}{\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} {\left (c d f - a g e\right )}}\right )} e + \frac {2 \, {\left (\sqrt {-c d^{2} e + a e^{3}} g \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g e^{\left (-1\right )}}{\sqrt {c d f g - a g^{2} e}}\right ) + \sqrt {c d f g - a g^{2} e} e\right )}}{\sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} c d f - \sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} a g e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(g*arctan(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*g*e^(-1)/sqrt(c*d*f*g - a*g^2*e))*e^(-1)/(sqrt(c*d*f*g -
a*g^2*e)*(c*d*f - a*g*e)) + 1/(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*(c*d*f - a*g*e)))*e + 2*(sqrt(-c*d^2*e
+ a*e^3)*g*arctan(sqrt(-c*d^2*e + a*e^3)*g*e^(-1)/sqrt(c*d*f*g - a*g^2*e)) + sqrt(c*d*f*g - a*g^2*e)*e)/(sqrt(
c*d*f*g - a*g^2*e)*sqrt(-c*d^2*e + a*e^3)*c*d*f - sqrt(c*d*f*g - a*g^2*e)*sqrt(-c*d^2*e + a*e^3)*a*g*e)

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

[Out]

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

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