3.7.96 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{3/2} (f+g x)^3} \, dx\) [696]

Optimal. Leaf size=195 \[ -\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^2 \sqrt {d+e x} (f+g x)}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2} (f+g x)^2}+\frac {3 c^2 d^2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 g^{5/2} \sqrt {c d f-a e g}} \]

[Out]

-1/2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g/(e*x+d)^(3/2)/(g*x+f)^2+3/4*c^2*d^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^(5/2)/(-a*e*g+c*d*f)^(1/2)-3/4*c*d*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^2/(g*x+f)/(e*x+d)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {876, 888, 211} \begin {gather*} \frac {3 c^2 d^2 \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 )}{4 g^{5/2} \sqrt {c d f-a e g}}-\frac {3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g^2 \sqrt {d+e x} (f+g x)}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2} (f+g x)^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)^(3/2)*(f + g*x)^3),x]

[Out]

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

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 876

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

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

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Mathematica [A]
time = 0.51, size = 135, normalized size = 0.69 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {g} (2 a e g+c d (3 f+5 g x))}{(f+g x)^2}+\frac {3 c^2 d^2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{\sqrt {c d f-a e g} \sqrt {a e+c d x}}\right )}{4 g^{5/2} \sqrt {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)^(3/2)*(f + g*x)^3),x]

[Out]

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

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Maple [A]
time = 0.14, size = 266, normalized size = 1.36

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} g^{2} x^{2}+6 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} f g x +3 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} f^{2}+5 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c d g x +2 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a e g +3 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c d f \right )}{4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, g^{2} \left (g x +f \right )^{2} \sqrt {\left (a e g -c d f \right ) g}}\)	\(266\)

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)^(3/2)/(g*x+f)^3,x,method=_RETURNVERBOSE)

[Out]

-1/4*((c*d*x+a*e)*(e*x+d))^(1/2)*(3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^2*d^2*g^2*x^2+6*arc
tanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^2*d^2*f*g*x+3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g
)^(1/2))*c^2*d^2*f^2+5*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c*d*g*x+2*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)
^(1/2)*a*e*g+3*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c*d*f)/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/g^2/(g*x+f)^2/
((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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (177) = 354\).
time = 2.12, size = 864, normalized size = 4.43 \begin {gather*} \left [-\frac {3 \, {\left (c^{2} d^{3} g^{2} x^{2} + 2 \, c^{2} d^{3} f g x + c^{2} d^{3} f^{2} + {\left (c^{2} d^{2} g^{2} x^{3} + 2 \, c^{2} d^{2} f g x^{2} + c^{2} d^{2} f^{2} x\right )} e\right )} \sqrt {-c d f g + a g^{2} e} \log \left (-\frac {c d^{2} g x - c d^{2} f + 2 \, a g x e^{2} + {\left (c d g x^{2} - c d f x + 2 \, a d g\right )} e - 2 \, \sqrt {-c d f g + a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{d g x + d f + {\left (g x^{2} + f x\right )} e}\right ) + 2 \, {\left (5 \, c^{2} d^{2} f g^{2} x + 3 \, c^{2} d^{2} f^{2} g - 2 \, a^{2} g^{3} e^{2} - {\left (5 \, a c d g^{3} x + a c d f g^{2}\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}}{8 \, {\left (c d^{2} f g^{5} x^{2} + 2 \, c d^{2} f^{2} g^{4} x + c d^{2} f^{3} g^{3} - {\left (a g^{6} x^{3} + 2 \, a f g^{5} x^{2} + a f^{2} g^{4} x\right )} e^{2} + {\left (c d f g^{5} x^{3} - a d f^{2} g^{4} + {\left (2 \, c d f^{2} g^{4} - a d g^{6}\right )} x^{2} + {\left (c d f^{3} g^{3} - 2 \, a d f g^{5}\right )} x\right )} e\right )}}, -\frac {3 \, {\left (c^{2} d^{3} g^{2} x^{2} + 2 \, c^{2} d^{3} f g x + c^{2} d^{3} f^{2} + {\left (c^{2} d^{2} g^{2} x^{3} + 2 \, c^{2} d^{2} f g x^{2} + c^{2} d^{2} f^{2} x\right )} e\right )} \sqrt {c d f g - a g^{2} e} \arctan \left (\frac {\sqrt {c d f g - a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{c d^{2} g x + a g x e^{2} + {\left (c d g x^{2} + a d g\right )} e}\right ) + {\left (5 \, c^{2} d^{2} f g^{2} x + 3 \, c^{2} d^{2} f^{2} g - 2 \, a^{2} g^{3} e^{2} - {\left (5 \, a c d g^{3} x + a c d f g^{2}\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}}{4 \, {\left (c d^{2} f g^{5} x^{2} + 2 \, c d^{2} f^{2} g^{4} x + c d^{2} f^{3} g^{3} - {\left (a g^{6} x^{3} + 2 \, a f g^{5} x^{2} + a f^{2} g^{4} x\right )} e^{2} + {\left (c d f g^{5} x^{3} - a d f^{2} g^{4} + {\left (2 \, c d f^{2} g^{4} - a d g^{6}\right )} x^{2} + {\left (c d f^{3} g^{3} - 2 \, a d f g^{5}\right )} x\right )} e\right )}}\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)^(3/2)/(g*x+f)^3,x, algorithm="fricas")

[Out]

[-1/8*(3*(c^2*d^3*g^2*x^2 + 2*c^2*d^3*f*g*x + c^2*d^3*f^2 + (c^2*d^2*g^2*x^3 + 2*c^2*d^2*f*g*x^2 + c^2*d^2*f^2
*x)*e)*sqrt(-c*d*f*g + a*g^2*e)*log(-(c*d^2*g*x - c*d^2*f + 2*a*g*x*e^2 + (c*d*g*x^2 - c*d*f*x + 2*a*d*g)*e -
2*sqrt(-c*d*f*g + a*g^2*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(d*g*x + d*f + (g*x^2 +
f*x)*e)) + 2*(5*c^2*d^2*f*g^2*x + 3*c^2*d^2*f^2*g - 2*a^2*g^3*e^2 - (5*a*c*d*g^3*x + a*c*d*f*g^2)*e)*sqrt(c*d^
2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c*d^2*f*g^5*x^2 + 2*c*d^2*f^2*g^4*x + c*d^2*f^3*g^3 - (a*g^
6*x^3 + 2*a*f*g^5*x^2 + a*f^2*g^4*x)*e^2 + (c*d*f*g^5*x^3 - a*d*f^2*g^4 + (2*c*d*f^2*g^4 - a*d*g^6)*x^2 + (c*d
*f^3*g^3 - 2*a*d*f*g^5)*x)*e), -1/4*(3*(c^2*d^3*g^2*x^2 + 2*c^2*d^3*f*g*x + c^2*d^3*f^2 + (c^2*d^2*g^2*x^3 + 2
*c^2*d^2*f*g*x^2 + c^2*d^2*f^2*x)*e)*sqrt(c*d*f*g - a*g^2*e)*arctan(sqrt(c*d*f*g - a*g^2*e)*sqrt(c*d^2*x + a*x
*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d)/(c*d^2*g*x + a*g*x*e^2 + (c*d*g*x^2 + a*d*g)*e)) + (5*c^2*d^2*f*g^2*x
+ 3*c^2*d^2*f^2*g - 2*a^2*g^3*e^2 - (5*a*c*d*g^3*x + a*c*d*f*g^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*
e)*sqrt(x*e + d))/(c*d^2*f*g^5*x^2 + 2*c*d^2*f^2*g^4*x + c*d^2*f^3*g^3 - (a*g^6*x^3 + 2*a*f*g^5*x^2 + a*f^2*g^
4*x)*e^2 + (c*d*f*g^5*x^3 - a*d*f^2*g^4 + (2*c*d*f^2*g^4 - a*d*g^6)*x^2 + (c*d*f^3*g^3 - 2*a*d*f*g^5)*x)*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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2)/(g*x+f)**3,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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^3,x, algorithm="giac")

[Out]

Timed out

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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 (f+g\,x\right )}^3\,{\left (d+e\,x\right )}^{3/2}} \,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)/((f + g*x)^3*(d + e*x)^(3/2)),x)

[Out]

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

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