3.9.39 \(\int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx\) [839]

Optimal. Leaf size=160 \[ -\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {2 c \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} \sqrt {g}} \]

[Out]

2*c*arctanh(g^(1/2)*(e*x+d)^(1/2)/e^(1/2)/(g*x+f)^(1/2))/e^(5/2)/g^(1/2)-2/3*(a+d*(-b*e+c*d)/e^2)*(g*x+f)^(1/2
)/(-d*g+e*f)/(e*x+d)^(3/2)+2/3*(c*(-4*d^2*g+6*d*e*f)-e*(-2*a*e*g-b*d*g+3*b*e*f))*(g*x+f)^(1/2)/e^2/(-d*g+e*f)^
2/(e*x+d)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {963, 79, 65, 223, 212} \begin {gather*} \frac {2 \sqrt {f+g x} \left (c \left (6 d e f-4 d^2 g\right )-e (-2 a e g-b d g+3 b e f)\right )}{3 e^2 \sqrt {d+e x} (e f-d g)^2}-\frac {2 \sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{3 (d+e x)^{3/2} (e f-d g)}+\frac {2 c \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} \sqrt {g}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)/((d + e*x)^(5/2)*Sqrt[f + g*x]),x]

[Out]

(-2*(a + (d*(c*d - b*e))/e^2)*Sqrt[f + g*x])/(3*(e*f - d*g)*(d + e*x)^(3/2)) + (2*(c*(6*d*e*f - 4*d^2*g) - e*(
3*b*e*f - b*d*g - 2*a*e*g))*Sqrt[f + g*x])/(3*e^2*(e*f - d*g)^2*Sqrt[d + e*x]) + (2*c*ArcTanh[(Sqrt[g]*Sqrt[d
+ e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(e^(5/2)*Sqrt[g])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 963

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

Rubi steps

\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^{5/2} \sqrt {f+g x}} \, dx &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}-\frac {2 \int \frac {\frac {c d (3 e f-d g)-e (3 b e f-b d g-2 a e g)}{2 e^2}-\frac {3}{2} c \left (f-\frac {d g}{e}\right ) x}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx}{3 (e f-d g)}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {c \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x}} \, dx}{e^2}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {(2 c) \text {Subst}\left (\int \frac {1}{\sqrt {f-\frac {d g}{e}+\frac {g x^2}{e}}} \, dx,x,\sqrt {d+e x}\right )}{e^3}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {(2 c) \text {Subst}\left (\int \frac {1}{1-\frac {g x^2}{e}} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e^3}\\ &=-\frac {2 \left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{3 (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (c \left (6 d e f-4 d^2 g\right )-e (3 b e f-b d g-2 a e g)\right ) \sqrt {f+g x}}{3 e^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {2 c \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{e^{5/2} \sqrt {g}}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 145, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {f+g x} \left (c d \left (-3 d^2 g+6 e^2 f x+d e (5 f-4 g x)\right )+e^2 (b (-2 d f-3 e f x+d g x)+a (-e f+3 d g+2 e g x))\right )}{3 e^2 (e f-d g)^2 (d+e x)^{3/2}}+\frac {2 c \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {g} \sqrt {d+e x}}\right )}{e^{5/2} \sqrt {g}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)/((d + e*x)^(5/2)*Sqrt[f + g*x]),x]

[Out]

(2*Sqrt[f + g*x]*(c*d*(-3*d^2*g + 6*e^2*f*x + d*e*(5*f - 4*g*x)) + e^2*(b*(-2*d*f - 3*e*f*x + d*g*x) + a*(-(e*
f) + 3*d*g + 2*e*g*x))))/(3*e^2*(e*f - d*g)^2*(d + e*x)^(3/2)) + (2*c*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/(Sqrt[g]
*Sqrt[d + e*x])])/(e^(5/2)*Sqrt[g])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(772\) vs. \(2(136)=272\).
time = 0.09, size = 773, normalized size = 4.83

method	result	size
default	\(\frac {\sqrt {g x +f}\, \left (3 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{2} e^{2} g^{2} x^{2}-6 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c d \,e^{3} f g \,x^{2}+3 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,e^{4} f^{2} x^{2}+6 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{3} e \,g^{2} x -12 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{2} e^{2} f g x +6 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c d \,e^{3} f^{2} x +3 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{4} g^{2}-6 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{3} e f g +3 \ln \left (\frac {2 e g x +2 \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+d g +e f}{2 \sqrt {e g}}\right ) c \,d^{2} e^{2} f^{2}+4 a \,e^{3} g x \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+2 b d \,e^{2} g x \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}-6 b \,e^{3} f x \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}-8 c \,d^{2} e g x \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+12 c d \,e^{2} f x \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+6 a d \,e^{2} g \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}-2 a \,e^{3} f \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}-4 b d \,e^{2} f \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}-6 c \,d^{3} g \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}+10 c \,d^{2} e f \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, \sqrt {e g}\right )}{3 \sqrt {e g}\, \left (d g -e f \right )^{2} \sqrt {\left (e x +d \right ) \left (g x +f \right )}\, e^{2} \left (e x +d \right )^{\frac {3}{2}}}\)	\(773\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)/(e*x+d)^(5/2)/(g*x+f)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(g*x+f)^(1/2)*(3*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^2*e^2*g^2
*x^2-6*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d*e^3*f*g*x^2+3*ln(1/2*(2
*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*e^4*f^2*x^2+6*ln(1/2*(2*e*g*x+2*((e*x+d)*
(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^3*e*g^2*x-12*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e
*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^2*e^2*f*g*x+6*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*
f)/(e*g)^(1/2))*c*d*e^3*f^2*x+3*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*
d^4*g^2-6*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^3*e*f*g+3*ln(1/2*(2*
e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^2*e^2*f^2+4*a*e^3*g*x*((e*x+d)*(g*x+f))^
(1/2)*(e*g)^(1/2)+2*b*d*e^2*g*x*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)-6*b*e^3*f*x*((e*x+d)*(g*x+f))^(1/2)*(e*g)^
(1/2)-8*c*d^2*e*g*x*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+12*c*d*e^2*f*x*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+6*a
*d*e^2*g*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)-2*a*e^3*f*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)-4*b*d*e^2*f*((e*x+d
)*(g*x+f))^(1/2)*(e*g)^(1/2)-6*c*d^3*g*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+10*c*d^2*e*f*((e*x+d)*(g*x+f))^(1/2
)*(e*g)^(1/2))/(e*g)^(1/2)/(d*g-e*f)^2/((e*x+d)*(g*x+f))^(1/2)/e^2/(e*x+d)^(3/2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(5/2)/(g*x+f)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(d*g-%e*f>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (142) = 284\).
time = 7.88, size = 770, normalized size = 4.81 \begin {gather*} \left [\frac {3 \, {\left (c d^{4} g^{2} + c f^{2} x^{2} e^{4} - 2 \, {\left (c d f g x^{2} - c d f^{2} x\right )} e^{3} + {\left (c d^{2} g^{2} x^{2} - 4 \, c d^{2} f g x + c d^{2} f^{2}\right )} e^{2} + 2 \, {\left (c d^{3} g^{2} x - c d^{3} f g\right )} e\right )} \sqrt {g} e^{\frac {1}{2}} \log \left (d^{2} g^{2} + 4 \, {\left (d g + {\left (2 \, g x + f\right )} e\right )} \sqrt {g x + f} \sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} + {\left (8 \, g^{2} x^{2} + 8 \, f g x + f^{2}\right )} e^{2} + 2 \, {\left (4 \, d g^{2} x + 3 \, d f g\right )} e\right ) - 4 \, {\left (3 \, c d^{3} g^{2} e + {\left (a f g + {\left (3 \, b f g - 2 \, a g^{2}\right )} x\right )} e^{4} + {\left (2 \, b d f g - 3 \, a d g^{2} - {\left (6 \, c d f g + b d g^{2}\right )} x\right )} e^{3} + {\left (4 \, c d^{2} g^{2} x - 5 \, c d^{2} f g\right )} e^{2}\right )} \sqrt {g x + f} \sqrt {x e + d}}{6 \, {\left (d^{4} g^{3} e^{3} + f^{2} g x^{2} e^{7} - 2 \, {\left (d f g^{2} x^{2} - d f^{2} g x\right )} e^{6} + {\left (d^{2} g^{3} x^{2} - 4 \, d^{2} f g^{2} x + d^{2} f^{2} g\right )} e^{5} + 2 \, {\left (d^{3} g^{3} x - d^{3} f g^{2}\right )} e^{4}\right )}}, -\frac {3 \, {\left (c d^{4} g^{2} + c f^{2} x^{2} e^{4} - 2 \, {\left (c d f g x^{2} - c d f^{2} x\right )} e^{3} + {\left (c d^{2} g^{2} x^{2} - 4 \, c d^{2} f g x + c d^{2} f^{2}\right )} e^{2} + 2 \, {\left (c d^{3} g^{2} x - c d^{3} f g\right )} e\right )} \sqrt {-g e} \arctan \left (\frac {{\left (d g + {\left (2 \, g x + f\right )} e\right )} \sqrt {g x + f} \sqrt {-g e} \sqrt {x e + d}}{2 \, {\left ({\left (g^{2} x^{2} + f g x\right )} e^{2} + {\left (d g^{2} x + d f g\right )} e\right )}}\right ) + 2 \, {\left (3 \, c d^{3} g^{2} e + {\left (a f g + {\left (3 \, b f g - 2 \, a g^{2}\right )} x\right )} e^{4} + {\left (2 \, b d f g - 3 \, a d g^{2} - {\left (6 \, c d f g + b d g^{2}\right )} x\right )} e^{3} + {\left (4 \, c d^{2} g^{2} x - 5 \, c d^{2} f g\right )} e^{2}\right )} \sqrt {g x + f} \sqrt {x e + d}}{3 \, {\left (d^{4} g^{3} e^{3} + f^{2} g x^{2} e^{7} - 2 \, {\left (d f g^{2} x^{2} - d f^{2} g x\right )} e^{6} + {\left (d^{2} g^{3} x^{2} - 4 \, d^{2} f g^{2} x + d^{2} f^{2} g\right )} e^{5} + 2 \, {\left (d^{3} g^{3} x - d^{3} f g^{2}\right )} e^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(5/2)/(g*x+f)^(1/2),x, algorithm="fricas")

[Out]

[1/6*(3*(c*d^4*g^2 + c*f^2*x^2*e^4 - 2*(c*d*f*g*x^2 - c*d*f^2*x)*e^3 + (c*d^2*g^2*x^2 - 4*c*d^2*f*g*x + c*d^2*
f^2)*e^2 + 2*(c*d^3*g^2*x - c*d^3*f*g)*e)*sqrt(g)*e^(1/2)*log(d^2*g^2 + 4*(d*g + (2*g*x + f)*e)*sqrt(g*x + f)*
sqrt(x*e + d)*sqrt(g)*e^(1/2) + (8*g^2*x^2 + 8*f*g*x + f^2)*e^2 + 2*(4*d*g^2*x + 3*d*f*g)*e) - 4*(3*c*d^3*g^2*
e + (a*f*g + (3*b*f*g - 2*a*g^2)*x)*e^4 + (2*b*d*f*g - 3*a*d*g^2 - (6*c*d*f*g + b*d*g^2)*x)*e^3 + (4*c*d^2*g^2
*x - 5*c*d^2*f*g)*e^2)*sqrt(g*x + f)*sqrt(x*e + d))/(d^4*g^3*e^3 + f^2*g*x^2*e^7 - 2*(d*f*g^2*x^2 - d*f^2*g*x)
*e^6 + (d^2*g^3*x^2 - 4*d^2*f*g^2*x + d^2*f^2*g)*e^5 + 2*(d^3*g^3*x - d^3*f*g^2)*e^4), -1/3*(3*(c*d^4*g^2 + c*
f^2*x^2*e^4 - 2*(c*d*f*g*x^2 - c*d*f^2*x)*e^3 + (c*d^2*g^2*x^2 - 4*c*d^2*f*g*x + c*d^2*f^2)*e^2 + 2*(c*d^3*g^2
*x - c*d^3*f*g)*e)*sqrt(-g*e)*arctan(1/2*(d*g + (2*g*x + f)*e)*sqrt(g*x + f)*sqrt(-g*e)*sqrt(x*e + d)/((g^2*x^
2 + f*g*x)*e^2 + (d*g^2*x + d*f*g)*e)) + 2*(3*c*d^3*g^2*e + (a*f*g + (3*b*f*g - 2*a*g^2)*x)*e^4 + (2*b*d*f*g -
3*a*d*g^2 - (6*c*d*f*g + b*d*g^2)*x)*e^3 + (4*c*d^2*g^2*x - 5*c*d^2*f*g)*e^2)*sqrt(g*x + f)*sqrt(x*e + d))/(d
^4*g^3*e^3 + f^2*g*x^2*e^7 - 2*(d*f*g^2*x^2 - d*f^2*g*x)*e^6 + (d^2*g^3*x^2 - 4*d^2*f*g^2*x + d^2*f^2*g)*e^5 +
2*(d^3*g^3*x - d^3*f*g^2)*e^4)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b x + c x^{2}}{\left (d + e x\right )^{\frac {5}{2}} \sqrt {f + g x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)/(e*x+d)**(5/2)/(g*x+f)**(1/2),x)

[Out]

Integral((a + b*x + c*x**2)/((d + e*x)**(5/2)*sqrt(f + g*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (142) = 284\).
time = 4.27, size = 286, normalized size = 1.79 \begin {gather*} -\frac {2 \, c \sqrt {g} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -\sqrt {g x + f} \sqrt {g} e^{\frac {1}{2}} + \sqrt {d g^{2} + {\left (g x + f\right )} g e - f g e} \right |}\right )}{{\left | g \right |}} - \frac {2 \, \sqrt {g x + f} {\left (\frac {{\left (4 \, c d^{2} g^{6} e^{2} - 6 \, c d f g^{5} e^{3} - b d g^{6} e^{3} + 3 \, b f g^{5} e^{4} - 2 \, a g^{6} e^{4}\right )} {\left (g x + f\right )}}{d^{2} g^{4} {\left | g \right |} e^{3} - 2 \, d f g^{3} {\left | g \right |} e^{4} + f^{2} g^{2} {\left | g \right |} e^{5}} + \frac {3 \, {\left (c d^{3} g^{7} e - 3 \, c d^{2} f g^{6} e^{2} + 2 \, c d f^{2} g^{5} e^{3} + b d f g^{6} e^{3} - a d g^{7} e^{3} - b f^{2} g^{5} e^{4} + a f g^{6} e^{4}\right )}}{d^{2} g^{4} {\left | g \right |} e^{3} - 2 \, d f g^{3} {\left | g \right |} e^{4} + f^{2} g^{2} {\left | g \right |} e^{5}}\right )}}{3 \, {\left (d g^{2} + {\left (g x + f\right )} g e - f g e\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^(5/2)/(g*x+f)^(1/2),x, algorithm="giac")

[Out]

-2*c*sqrt(g)*e^(-5/2)*log(abs(-sqrt(g*x + f)*sqrt(g)*e^(1/2) + sqrt(d*g^2 + (g*x + f)*g*e - f*g*e)))/abs(g) -
2/3*sqrt(g*x + f)*((4*c*d^2*g^6*e^2 - 6*c*d*f*g^5*e^3 - b*d*g^6*e^3 + 3*b*f*g^5*e^4 - 2*a*g^6*e^4)*(g*x + f)/(
d^2*g^4*abs(g)*e^3 - 2*d*f*g^3*abs(g)*e^4 + f^2*g^2*abs(g)*e^5) + 3*(c*d^3*g^7*e - 3*c*d^2*f*g^6*e^2 + 2*c*d*f
^2*g^5*e^3 + b*d*f*g^6*e^3 - a*d*g^7*e^3 - b*f^2*g^5*e^4 + a*f*g^6*e^4)/(d^2*g^4*abs(g)*e^3 - 2*d*f*g^3*abs(g)
*e^4 + f^2*g^2*abs(g)*e^5))/(d*g^2 + (g*x + f)*g*e - f*g*e)^(3/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {c\,x^2+b\,x+a}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)/((f + g*x)^(1/2)*(d + e*x)^(5/2)),x)

[Out]

int((a + b*x + c*x^2)/((f + g*x)^(1/2)*(d + e*x)^(5/2)), x)

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