3.7.71 \(\int \frac {(d+e x)^{3/2}}{(f+g x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [671]

Optimal. Leaf size=274 \[ -\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {5 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {15 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {15 c^2 d^2 \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 (c d f-a e g)^{7/2}} \]

[Out]

-15/4*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^(1/
2)/(-a*e*g+c*d*f)^(7/2)-2*(e*x+d)^(1/2)/(-a*e*g+c*d*f)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-5/2*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)^2/(e*x+d)^(1/2)-15/4*c*d*g*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^3/(g*x+f)/(e*x+d)^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {15 c^2 d^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 )}{4 (c d f-a e g)^{7/2}}-\frac {15 c d g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 \sqrt {d+e x} (f+g x) (c d f-a e g)^3}-\frac {5 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{(f+g x)^2 \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)^3*(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)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (5*g*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^2) - (15*c*d*g*Sqrt[a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2])/(4*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) - (15*c^2*d^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])])/(4*(c*d*f - a*e*g)^(7/2
))

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)^3 \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)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(5 g) \int \frac {\sqrt {d+e x}}{(f+g x)^3 \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)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {5 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {(15 c d 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}{4 (c d f-a e g)^2}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {5 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {15 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {\left (15 c^2 d^2 g\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 (c d f-a e g)^3}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {5 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {15 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {\left (15 c^2 d^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 )}{4 (c d f-a e g)^3}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {5 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {15 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {15 c^2 d^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 )}{4 (c d f-a e g)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.57, size = 185, normalized size = 0.68 \begin {gather*} -\frac {\sqrt {d+e x} \left (\sqrt {c d f-a e g} \left (-2 a^2 e^2 g^2+a c d e g (9 f+5 g x)+c^2 d^2 \left (8 f^2+25 f g x+15 g^2 x^2\right )\right )+15 c^2 d^2 \sqrt {g} \sqrt {a e+c d x} (f+g x)^2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{4 (c d f-a e g)^{7/2} \sqrt {(a e+c d x) (d+e x)} (f+g x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(Sqrt[d + e*x]*(Sqrt[c*d*f - a*e*g]*(-2*a^2*e^2*g^2 + a*c*d*e*g*(9*f + 5*g*x) + c^2*d^2*(8*f^2 + 25*f*g*x
+ 15*g^2*x^2)) + 15*c^2*d^2*Sqrt[g]*Sqrt[a*e + c*d*x]*(f + g*x)^2*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d
*f - a*e*g]]))/((c*d*f - a*e*g)^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)^2)

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Maple [A]
time = 0.14, size = 369, normalized size = 1.35

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \sqrt {c d x +a e}\, \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} g^{3} x^{2}+30 \sqrt {c d x +a e}\, \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} f \,g^{2} x +15 \sqrt {c d x +a e}\, \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} f^{2} g -15 \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} g^{2} x^{2}-5 \sqrt {\left (a e g -c d f \right ) g}\, a c d e \,g^{2} x -25 \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f g x +2 \sqrt {\left (a e g -c d f \right ) g}\, a^{2} e^{2} g^{2}-9 \sqrt {\left (a e g -c d f \right ) g}\, a c d e f g -8 \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f^{2}\right )}{4 \sqrt {e x +d}\, \left (c d x +a e \right ) \left (a e g -c d f \right )^{3} \left (g x +f \right )^{2} \sqrt {\left (a e g -c d f \right ) g}}\)	\(369\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(e*x+d)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(15*(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^2*d^2*g^3*x^2+30*(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^2*d^2*
f*g^2*x+15*(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^2*d^2*f^2*g-15*((a*e*g-c*d
*f)*g)^(1/2)*c^2*d^2*g^2*x^2-5*((a*e*g-c*d*f)*g)^(1/2)*a*c*d*e*g^2*x-25*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*f*g*x+
2*((a*e*g-c*d*f)*g)^(1/2)*a^2*e^2*g^2-9*((a*e*g-c*d*f)*g)^(1/2)*a*c*d*e*f*g-8*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*
f^2)/(c*d*x+a*e)/(a*e*g-c*d*f)^3/(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((e*x+d)^(3/2)/(g*x+f)^3/(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)^3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 950 vs. \(2 (257) = 514\).
time = 3.00, size = 1941, normalized size = 7.08 \begin {gather*} \left [-\frac {15 \, {\left (c^{3} d^{4} g^{2} x^{3} + 2 \, c^{3} d^{4} f g x^{2} + c^{3} d^{4} f^{2} x + {\left (a c^{2} d^{2} g^{2} x^{3} + 2 \, a c^{2} d^{2} f g x^{2} + a c^{2} d^{2} f^{2} x\right )} e^{2} + {\left (c^{3} d^{3} g^{2} x^{4} + 2 \, c^{3} d^{3} f g x^{3} + 2 \, a c^{2} d^{3} f g x + a c^{2} d^{3} f^{2} + {\left (c^{3} d^{3} f^{2} + a c^{2} d^{3} g^{2}\right )} x^{2}\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 \, {\left (15 \, c^{2} d^{2} g^{2} x^{2} + 25 \, c^{2} d^{2} f g x + 8 \, c^{2} d^{2} f^{2} - 2 \, a^{2} g^{2} e^{2} + {\left (5 \, a c d g^{2} x + 9 \, a c d f 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}}{8 \, {\left (c^{4} d^{5} f^{3} g^{2} x^{3} + 2 \, c^{4} d^{5} f^{4} g x^{2} + c^{4} d^{5} f^{5} x - {\left (a^{4} g^{5} x^{3} + 2 \, a^{4} f g^{4} x^{2} + a^{4} f^{2} g^{3} x\right )} e^{5} - {\left (a^{3} c d g^{5} x^{4} - a^{3} c d f g^{4} x^{3} + a^{4} d f^{2} g^{3} - {\left (5 \, a^{3} c d f^{2} g^{3} - a^{4} d g^{5}\right )} x^{2} - {\left (3 \, a^{3} c d f^{3} g^{2} - 2 \, a^{4} d f g^{4}\right )} x\right )} e^{4} + {\left (3 \, a^{2} c^{2} d^{2} f g^{4} x^{4} + 3 \, a^{3} c d^{2} f^{3} g^{2} + {\left (3 \, a^{2} c^{2} d^{2} f^{2} g^{3} - a^{3} c d^{2} g^{5}\right )} x^{3} - {\left (3 \, a^{2} c^{2} d^{2} f^{3} g^{2} - a^{3} c d^{2} f g^{4}\right )} x^{2} - {\left (3 \, a^{2} c^{2} d^{2} f^{4} g - 5 \, a^{3} c d^{2} f^{2} g^{3}\right )} x\right )} e^{3} - {\left (3 \, a c^{3} d^{3} f^{2} g^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} f^{4} g + {\left (5 \, a c^{3} d^{3} f^{3} g^{2} - 3 \, a^{2} c^{2} d^{3} f g^{4}\right )} x^{3} + {\left (a c^{3} d^{3} f^{4} g - 3 \, a^{2} c^{2} d^{3} f^{2} g^{3}\right )} x^{2} - {\left (a c^{3} d^{3} f^{5} - 3 \, a^{2} c^{2} d^{3} f^{3} g^{2}\right )} x\right )} e^{2} + {\left (c^{4} d^{4} f^{3} g^{2} x^{4} - a c^{3} d^{4} f^{4} g x + a c^{3} d^{4} f^{5} + {\left (2 \, c^{4} d^{4} f^{4} g - 3 \, a c^{3} d^{4} f^{2} g^{3}\right )} x^{3} + {\left (c^{4} d^{4} f^{5} - 5 \, a c^{3} d^{4} f^{3} g^{2}\right )} x^{2}\right )} e\right )}}, -\frac {15 \, {\left (c^{3} d^{4} g^{2} x^{3} + 2 \, c^{3} d^{4} f g x^{2} + c^{3} d^{4} f^{2} x + {\left (a c^{2} d^{2} g^{2} x^{3} + 2 \, a c^{2} d^{2} f g x^{2} + a c^{2} d^{2} f^{2} x\right )} e^{2} + {\left (c^{3} d^{3} g^{2} x^{4} + 2 \, c^{3} d^{3} f g x^{3} + 2 \, a c^{2} d^{3} f g x + a c^{2} d^{3} f^{2} + {\left (c^{3} d^{3} f^{2} + a c^{2} d^{3} g^{2}\right )} x^{2}\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 ) + {\left (15 \, c^{2} d^{2} g^{2} x^{2} + 25 \, c^{2} d^{2} f g x + 8 \, c^{2} d^{2} f^{2} - 2 \, a^{2} g^{2} e^{2} + {\left (5 \, a c d g^{2} x + 9 \, a c d f 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}}{4 \, {\left (c^{4} d^{5} f^{3} g^{2} x^{3} + 2 \, c^{4} d^{5} f^{4} g x^{2} + c^{4} d^{5} f^{5} x - {\left (a^{4} g^{5} x^{3} + 2 \, a^{4} f g^{4} x^{2} + a^{4} f^{2} g^{3} x\right )} e^{5} - {\left (a^{3} c d g^{5} x^{4} - a^{3} c d f g^{4} x^{3} + a^{4} d f^{2} g^{3} - {\left (5 \, a^{3} c d f^{2} g^{3} - a^{4} d g^{5}\right )} x^{2} - {\left (3 \, a^{3} c d f^{3} g^{2} - 2 \, a^{4} d f g^{4}\right )} x\right )} e^{4} + {\left (3 \, a^{2} c^{2} d^{2} f g^{4} x^{4} + 3 \, a^{3} c d^{2} f^{3} g^{2} + {\left (3 \, a^{2} c^{2} d^{2} f^{2} g^{3} - a^{3} c d^{2} g^{5}\right )} x^{3} - {\left (3 \, a^{2} c^{2} d^{2} f^{3} g^{2} - a^{3} c d^{2} f g^{4}\right )} x^{2} - {\left (3 \, a^{2} c^{2} d^{2} f^{4} g - 5 \, a^{3} c d^{2} f^{2} g^{3}\right )} x\right )} e^{3} - {\left (3 \, a c^{3} d^{3} f^{2} g^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} f^{4} g + {\left (5 \, a c^{3} d^{3} f^{3} g^{2} - 3 \, a^{2} c^{2} d^{3} f g^{4}\right )} x^{3} + {\left (a c^{3} d^{3} f^{4} g - 3 \, a^{2} c^{2} d^{3} f^{2} g^{3}\right )} x^{2} - {\left (a c^{3} d^{3} f^{5} - 3 \, a^{2} c^{2} d^{3} f^{3} g^{2}\right )} x\right )} e^{2} + {\left (c^{4} d^{4} f^{3} g^{2} x^{4} - a c^{3} d^{4} f^{4} g x + a c^{3} d^{4} f^{5} + {\left (2 \, c^{4} d^{4} f^{4} g - 3 \, a c^{3} d^{4} f^{2} g^{3}\right )} x^{3} + {\left (c^{4} d^{4} f^{5} - 5 \, a c^{3} d^{4} f^{3} g^{2}\right )} x^{2}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*(15*(c^3*d^4*g^2*x^3 + 2*c^3*d^4*f*g*x^2 + c^3*d^4*f^2*x + (a*c^2*d^2*g^2*x^3 + 2*a*c^2*d^2*f*g*x^2 + a*
c^2*d^2*f^2*x)*e^2 + (c^3*d^3*g^2*x^4 + 2*c^3*d^3*f*g*x^3 + 2*a*c^2*d^3*f*g*x + a*c^2*d^3*f^2 + (c^3*d^3*f^2 +
a*c^2*d^3*g^2)*x^2)*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*(15*c^2*d^2*g^2*x^2 + 25*c^2*d^2*f*g*x + 8*c^2*d^2*f^2 - 2*a^2*g
^2*e^2 + (5*a*c*d*g^2*x + 9*a*c*d*f*g)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^4*d^5*
f^3*g^2*x^3 + 2*c^4*d^5*f^4*g*x^2 + c^4*d^5*f^5*x - (a^4*g^5*x^3 + 2*a^4*f*g^4*x^2 + a^4*f^2*g^3*x)*e^5 - (a^3
*c*d*g^5*x^4 - a^3*c*d*f*g^4*x^3 + a^4*d*f^2*g^3 - (5*a^3*c*d*f^2*g^3 - a^4*d*g^5)*x^2 - (3*a^3*c*d*f^3*g^2 -
2*a^4*d*f*g^4)*x)*e^4 + (3*a^2*c^2*d^2*f*g^4*x^4 + 3*a^3*c*d^2*f^3*g^2 + (3*a^2*c^2*d^2*f^2*g^3 - a^3*c*d^2*g^
5)*x^3 - (3*a^2*c^2*d^2*f^3*g^2 - a^3*c*d^2*f*g^4)*x^2 - (3*a^2*c^2*d^2*f^4*g - 5*a^3*c*d^2*f^2*g^3)*x)*e^3 -
(3*a*c^3*d^3*f^2*g^3*x^4 + 3*a^2*c^2*d^3*f^4*g + (5*a*c^3*d^3*f^3*g^2 - 3*a^2*c^2*d^3*f*g^4)*x^3 + (a*c^3*d^3*
f^4*g - 3*a^2*c^2*d^3*f^2*g^3)*x^2 - (a*c^3*d^3*f^5 - 3*a^2*c^2*d^3*f^3*g^2)*x)*e^2 + (c^4*d^4*f^3*g^2*x^4 - a
*c^3*d^4*f^4*g*x + a*c^3*d^4*f^5 + (2*c^4*d^4*f^4*g - 3*a*c^3*d^4*f^2*g^3)*x^3 + (c^4*d^4*f^5 - 5*a*c^3*d^4*f^
3*g^2)*x^2)*e), -1/4*(15*(c^3*d^4*g^2*x^3 + 2*c^3*d^4*f*g*x^2 + c^3*d^4*f^2*x + (a*c^2*d^2*g^2*x^3 + 2*a*c^2*d
^2*f*g*x^2 + a*c^2*d^2*f^2*x)*e^2 + (c^3*d^3*g^2*x^4 + 2*c^3*d^3*f*g*x^3 + 2*a*c^2*d^3*f*g*x + a*c^2*d^3*f^2 +
(c^3*d^3*f^2 + a*c^2*d^3*g^2)*x^2)*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)) +
(15*c^2*d^2*g^2*x^2 + 25*c^2*d^2*f*g*x + 8*c^2*d^2*f^2 - 2*a^2*g^2*e^2 + (5*a*c*d*g^2*x + 9*a*c*d*f*g)*e)*sqrt
(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^4*d^5*f^3*g^2*x^3 + 2*c^4*d^5*f^4*g*x^2 + c^4*d^5*f^
5*x - (a^4*g^5*x^3 + 2*a^4*f*g^4*x^2 + a^4*f^2*g^3*x)*e^5 - (a^3*c*d*g^5*x^4 - a^3*c*d*f*g^4*x^3 + a^4*d*f^2*g
^3 - (5*a^3*c*d*f^2*g^3 - a^4*d*g^5)*x^2 - (3*a^3*c*d*f^3*g^2 - 2*a^4*d*f*g^4)*x)*e^4 + (3*a^2*c^2*d^2*f*g^4*x
^4 + 3*a^3*c*d^2*f^3*g^2 + (3*a^2*c^2*d^2*f^2*g^3 - a^3*c*d^2*g^5)*x^3 - (3*a^2*c^2*d^2*f^3*g^2 - a^3*c*d^2*f*
g^4)*x^2 - (3*a^2*c^2*d^2*f^4*g - 5*a^3*c*d^2*f^2*g^3)*x)*e^3 - (3*a*c^3*d^3*f^2*g^3*x^4 + 3*a^2*c^2*d^3*f^4*g
+ (5*a*c^3*d^3*f^3*g^2 - 3*a^2*c^2*d^3*f*g^4)*x^3 + (a*c^3*d^3*f^4*g - 3*a^2*c^2*d^3*f^2*g^3)*x^2 - (a*c^3*d^
3*f^5 - 3*a^2*c^2*d^3*f^3*g^2)*x)*e^2 + (c^4*d^4*f^3*g^2*x^4 - a*c^3*d^4*f^4*g*x + a*c^3*d^4*f^5 + (2*c^4*d^4*
f^4*g - 3*a*c^3*d^4*f^2*g^3)*x^3 + (c^4*d^4*f^5 - 5*a*c^3*d^4*f^3*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)**3/(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)^3/(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 )}^3\,{\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)^3*(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)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)), x)

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