3.7.70 \(\int \frac {(d+e x)^{3/2}}{(f+g x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [670]

Optimal. Leaf size=202 \[ -\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {3 c d \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{(c d f-a e g)^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {882, 886, 888, 211} \begin {gather*} -\frac {3 c d \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)^{5/2}}-\frac {3 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{(f+g 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)^2*(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)*(f + g*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (3*g*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])/((c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)) - (3*c*d*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)^(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 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 886

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)/((n + 1)*(c*e*f + c*d*g - b*e*g))),
x] - Dist[c*e*((m - n - 2)/((n + 1)*(c*e*f + c*d*g - b*e*g))), Int[(d + e*x)^m*(f + g*x)^(n + 1)*(a + b*x + c
*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, 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[n, -1] && IntegerQ[2*p]

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

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Mathematica [A]
time = 0.39, size = 141, normalized size = 0.70 \begin {gather*} -\frac {\sqrt {d+e x} \left (\sqrt {c d f-a e g} (a e g+c d (2 f+3 g x))+3 c d \sqrt {g} \sqrt {a e+c d x} (f+g 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)^{5/2} \sqrt {(a e+c d x) (d+e x)} (f+g x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.16, size = 215, normalized size = 1.06

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \sqrt {c d x +a e}\, \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c d \,g^{2} x +3 \sqrt {c d x +a e}\, \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c d f g -3 \sqrt {\left (a e g -c d f \right ) g}\, c d g x -\sqrt {\left (a e g -c d f \right ) g}\, a e g -2 \sqrt {\left (a e g -c d f \right ) g}\, c d f \right )}{\sqrt {e x +d}\, \left (c d x +a e \right ) \left (a e g -c d f \right )^{2} \left (g x +f \right ) \sqrt {\left (a e g -c d f \right ) g}}\)	\(215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (192) = 384\).
time = 1.82, size = 1097, normalized size = 5.43 \begin {gather*} \left [\frac {3 \, {\left (c^{2} d^{3} g x^{2} + c^{2} d^{3} f x + {\left (a c d g x^{2} + a c d f x\right )} e^{2} + {\left (c^{2} d^{2} g x^{3} + c^{2} d^{2} f x^{2} + a c d^{2} g x + a c d^{2} f\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} {\left (3 \, c d g x + 2 \, c d f + a g e\right )} \sqrt {x e + d}}{2 \, {\left (c^{3} d^{4} f^{2} g x^{2} + c^{3} d^{4} f^{3} x + {\left (a^{3} g^{3} x^{2} + a^{3} f g^{2} x\right )} e^{4} + {\left (a^{2} c d g^{3} x^{3} - a^{2} c d f g^{2} x^{2} + a^{3} d f g^{2} - {\left (2 \, a^{2} c d f^{2} g - a^{3} d g^{3}\right )} x\right )} e^{3} - {\left (2 \, a c^{2} d^{2} f g^{2} x^{3} + 2 \, a^{2} c d^{2} f^{2} g + {\left (a c^{2} d^{2} f^{2} g - a^{2} c d^{2} g^{3}\right )} x^{2} - {\left (a c^{2} d^{2} f^{3} - a^{2} c d^{2} f g^{2}\right )} x\right )} e^{2} + {\left (c^{3} d^{3} f^{2} g x^{3} - a c^{2} d^{3} f^{2} g x + a c^{2} d^{3} f^{3} + {\left (c^{3} d^{3} f^{3} - 2 \, a c^{2} d^{3} f g^{2}\right )} x^{2}\right )} e\right )}}, -\frac {3 \, {\left (c^{2} d^{3} g x^{2} + c^{2} d^{3} f x + {\left (a c d g x^{2} + a c d f x\right )} e^{2} + {\left (c^{2} d^{2} g x^{3} + c^{2} d^{2} f x^{2} + a c d^{2} g x + a c d^{2} f\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} {\left (3 \, c d g x + 2 \, c d f + a g e\right )} \sqrt {x e + d}}{c^{3} d^{4} f^{2} g x^{2} + c^{3} d^{4} f^{3} x + {\left (a^{3} g^{3} x^{2} + a^{3} f g^{2} x\right )} e^{4} + {\left (a^{2} c d g^{3} x^{3} - a^{2} c d f g^{2} x^{2} + a^{3} d f g^{2} - {\left (2 \, a^{2} c d f^{2} g - a^{3} d g^{3}\right )} x\right )} e^{3} - {\left (2 \, a c^{2} d^{2} f g^{2} x^{3} + 2 \, a^{2} c d^{2} f^{2} g + {\left (a c^{2} d^{2} f^{2} g - a^{2} c d^{2} g^{3}\right )} x^{2} - {\left (a c^{2} d^{2} f^{3} - a^{2} c d^{2} f g^{2}\right )} x\right )} e^{2} + {\left (c^{3} d^{3} f^{2} g x^{3} - a c^{2} d^{3} f^{2} g x + a c^{2} d^{3} f^{3} + {\left (c^{3} d^{3} f^{3} - 2 \, a c^{2} d^{3} f g^{2}\right )} x^{2}\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)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/2*(3*(c^2*d^3*g*x^2 + c^2*d^3*f*x + (a*c*d*g*x^2 + a*c*d*f*x)*e^2 + (c^2*d^2*g*x^3 + c^2*d^2*f*x^2 + a*c*d^
2*g*x + a*c*d^2*f)*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)*(3*c*d*g*x + 2*c*d*f +
a*g*e)*sqrt(x*e + d))/(c^3*d^4*f^2*g*x^2 + c^3*d^4*f^3*x + (a^3*g^3*x^2 + a^3*f*g^2*x)*e^4 + (a^2*c*d*g^3*x^3
- a^2*c*d*f*g^2*x^2 + a^3*d*f*g^2 - (2*a^2*c*d*f^2*g - a^3*d*g^3)*x)*e^3 - (2*a*c^2*d^2*f*g^2*x^3 + 2*a^2*c*d
^2*f^2*g + (a*c^2*d^2*f^2*g - a^2*c*d^2*g^3)*x^2 - (a*c^2*d^2*f^3 - a^2*c*d^2*f*g^2)*x)*e^2 + (c^3*d^3*f^2*g*x
^3 - a*c^2*d^3*f^2*g*x + a*c^2*d^3*f^3 + (c^3*d^3*f^3 - 2*a*c^2*d^3*f*g^2)*x^2)*e), -(3*(c^2*d^3*g*x^2 + c^2*d
^3*f*x + (a*c*d*g*x^2 + a*c*d*f*x)*e^2 + (c^2*d^2*g*x^3 + c^2*d^2*f*x^2 + a*c*d^2*g*x + a*c*d^2*f)*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)*(3*
c*d*g*x + 2*c*d*f + a*g*e)*sqrt(x*e + d))/(c^3*d^4*f^2*g*x^2 + c^3*d^4*f^3*x + (a^3*g^3*x^2 + a^3*f*g^2*x)*e^4
+ (a^2*c*d*g^3*x^3 - a^2*c*d*f*g^2*x^2 + a^3*d*f*g^2 - (2*a^2*c*d*f^2*g - a^3*d*g^3)*x)*e^3 - (2*a*c^2*d^2*f*
g^2*x^3 + 2*a^2*c*d^2*f^2*g + (a*c^2*d^2*f^2*g - a^2*c*d^2*g^3)*x^2 - (a*c^2*d^2*f^3 - a^2*c*d^2*f*g^2)*x)*e^2
+ (c^3*d^3*f^2*g*x^3 - a*c^2*d^3*f^2*g*x + a*c^2*d^3*f^3 + (c^3*d^3*f^3 - 2*a*c^2*d^3*f*g^2)*x^2)*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)**2/(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 [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)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

Timed out

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

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