3.1.86 \(\int \frac {\sqrt {a+c x^2} (d+e x+f x^2)}{(g+h x)^5} \, dx\) [86]

Optimal. Leaf size=313 \[ -\frac {\left (4 c^2 d g^2+4 a^2 f h^2-a c \left (f g^2-h (5 e g-d h)\right )\right ) (a h-c g x) \sqrt {a+c x^2}}{8 \left (c g^2+a h^2\right )^3 (g+h x)^2}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}+\frac {\left (4 a h^2 (2 f g-e h)+c g \left (3 f g^2+h (e g-5 d h)\right )\right ) \left (a+c x^2\right )^{3/2}}{12 h \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac {a c \left (4 c^2 d g^2+4 a^2 f h^2-a c \left (f g^2-h (5 e g-d h)\right )\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{8 \left (c g^2+a h^2\right )^{7/2}} \]

[Out]

-1/4*(d*h^2-e*g*h+f*g^2)*(c*x^2+a)^(3/2)/h/(a*h^2+c*g^2)/(h*x+g)^4+1/12*(4*a*h^2*(-e*h+2*f*g)+c*g*(3*f*g^2+h*(
-5*d*h+e*g)))*(c*x^2+a)^(3/2)/h/(a*h^2+c*g^2)^2/(h*x+g)^3-1/8*a*c*(4*c^2*d*g^2+4*a^2*f*h^2-a*c*(f*g^2-h*(-d*h+
5*e*g)))*arctanh((-c*g*x+a*h)/(a*h^2+c*g^2)^(1/2)/(c*x^2+a)^(1/2))/(a*h^2+c*g^2)^(7/2)-1/8*(4*c^2*d*g^2+4*a^2*
f*h^2-a*c*(f*g^2-h*(-d*h+5*e*g)))*(-c*g*x+a*h)*(c*x^2+a)^(1/2)/(a*h^2+c*g^2)^3/(h*x+g)^2

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Rubi [A]
time = 0.25, antiderivative size = 312, 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 = {1665, 821, 735, 739, 212} \begin {gather*} -\frac {\sqrt {a+c x^2} (a h-c g x) \left (4 a^2 f h^2-a c \left (f g^2-h (5 e g-d h)\right )+4 c^2 d g^2\right )}{8 (g+h x)^2 \left (a h^2+c g^2\right )^3}-\frac {a c \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (4 a^2 f h^2-a c \left (f g^2-h (5 e g-d h)\right )+4 c^2 d g^2\right )}{8 \left (a h^2+c g^2\right )^{7/2}}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{4 h (g+h x)^4 \left (a h^2+c g^2\right )}+\frac {\left (a+c x^2\right )^{3/2} \left (4 a h^2 (2 f g-e h)+c g h (e g-5 d h)+3 c f g^3\right )}{12 h (g+h x)^3 \left (a h^2+c g^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*((4*c^2*d*g^2 + 4*a^2*f*h^2 - a*c*(f*g^2 - h*(5*e*g - d*h)))*(a*h - c*g*x)*Sqrt[a + c*x^2])/((c*g^2 + a*h
^2)^3*(g + h*x)^2) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(3/2))/(4*h*(c*g^2 + a*h^2)*(g + h*x)^4) + ((3*c*f*g
^3 + c*g*h*(e*g - 5*d*h) + 4*a*h^2*(2*f*g - e*h))*(a + c*x^2)^(3/2))/(12*h*(c*g^2 + a*h^2)^2*(g + h*x)^3) - (a
*c*(4*c^2*d*g^2 + 4*a^2*f*h^2 - a*c*(f*g^2 - h*(5*e*g - d*h)))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt
[a + c*x^2])])/(8*(c*g^2 + a*h^2)^(7/2))

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 735

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(-2*a*e + (2*c
*d)*x)*((a + c*x^2)^p/(2*(m + 1)*(c*d^2 + a*e^2))), x] - Dist[4*a*c*(p/(2*(m + 1)*(c*d^2 + a*e^2))), Int[(d +
e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2
, 0] && GtQ[p, 0]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 1665

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx &=-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac {\int \frac {\left (-4 (c d g-a f g+a e h)-\left (4 a f h+c \left (e g+\frac {3 f g^2}{h}-d h\right )\right ) x\right ) \sqrt {a+c x^2}}{(g+h x)^4} \, dx}{4 \left (c g^2+a h^2\right )}\\ &=-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}+\frac {\left (3 c f g^3+c g h (e g-5 d h)+4 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{3/2}}{12 h \left (c g^2+a h^2\right )^2 (g+h x)^3}+\frac {\left (4 c^2 d g^2+4 a^2 f h^2-a c \left (f g^2-h (5 e g-d h)\right )\right ) \int \frac {\sqrt {a+c x^2}}{(g+h x)^3} \, dx}{4 \left (c g^2+a h^2\right )^2}\\ &=-\frac {\left (4 c^2 d g^2+4 a^2 f h^2-a c \left (f g^2-h (5 e g-d h)\right )\right ) (a h-c g x) \sqrt {a+c x^2}}{8 \left (c g^2+a h^2\right )^3 (g+h x)^2}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}+\frac {\left (3 c f g^3+c g h (e g-5 d h)+4 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{3/2}}{12 h \left (c g^2+a h^2\right )^2 (g+h x)^3}+\frac {\left (a c \left (4 c^2 d g^2+4 a^2 f h^2-a c \left (f g^2-h (5 e g-d h)\right )\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{8 \left (c g^2+a h^2\right )^3}\\ &=-\frac {\left (4 c^2 d g^2+4 a^2 f h^2-a c \left (f g^2-h (5 e g-d h)\right )\right ) (a h-c g x) \sqrt {a+c x^2}}{8 \left (c g^2+a h^2\right )^3 (g+h x)^2}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}+\frac {\left (3 c f g^3+c g h (e g-5 d h)+4 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{3/2}}{12 h \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac {\left (a c \left (4 c^2 d g^2+4 a^2 f h^2-a c \left (f g^2-h (5 e g-d h)\right )\right )\right ) \text {Subst}\left (\int \frac {1}{c g^2+a h^2-x^2} \, dx,x,\frac {a h-c g x}{\sqrt {a+c x^2}}\right )}{8 \left (c g^2+a h^2\right )^3}\\ &=-\frac {\left (4 c^2 d g^2+4 a^2 f h^2-a c \left (f g^2-h (5 e g-d h)\right )\right ) (a h-c g x) \sqrt {a+c x^2}}{8 \left (c g^2+a h^2\right )^3 (g+h x)^2}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}+\frac {\left (3 c f g^3+c g h (e g-5 d h)+4 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{3/2}}{12 h \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac {a c \left (4 c^2 d g^2+4 a^2 f h^2-a c \left (f g^2-h (5 e g-d h)\right )\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{8 \left (c g^2+a h^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 10.86, size = 439, normalized size = 1.40 \begin {gather*} \frac {1}{24} \left (-\frac {\sqrt {a+c x^2} \left (6 \left (c g^2+a h^2\right )^3 \left (f g^2+h (-e g+d h)\right )-2 \left (c g^2+a h^2\right )^2 \left (9 c f g^3+c g h (-5 e g+d h)-4 a h^2 (-2 f g+e h)\right ) (g+h x)+\left (c g^2+a h^2\right ) \left (12 a^2 f h^4+2 c^2 \left (9 f g^4-g^2 h (e g+d h)\right )+a c h^2 \left (35 f g^2+h (-7 e g+3 d h)\right )\right ) (g+h x)^2-c \left (4 a^2 h^4 (7 f g-2 e h)+a c g h^2 \left (19 f g^2+h (9 e g-13 d h)\right )+2 c^2 \left (3 f g^5+g^3 h (e g+d h)\right )\right ) (g+h x)^3\right )}{\left (c g^2 h+a h^3\right )^3 (g+h x)^4}+\frac {3 a c \left (4 c^2 d g^2+4 a^2 f h^2-a c \left (f g^2+h (-5 e g+d h)\right )\right ) \log (g+h x)}{\left (c g^2+a h^2\right )^{7/2}}-\frac {3 a c \left (4 c^2 d g^2+4 a^2 f h^2-a c \left (f g^2+h (-5 e g+d h)\right )\right ) \log \left (a h-c g x+\sqrt {c g^2+a h^2} \sqrt {a+c x^2}\right )}{\left (c g^2+a h^2\right )^{7/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((Sqrt[a + c*x^2]*(6*(c*g^2 + a*h^2)^3*(f*g^2 + h*(-(e*g) + d*h)) - 2*(c*g^2 + a*h^2)^2*(9*c*f*g^3 + c*g*h*(
-5*e*g + d*h) - 4*a*h^2*(-2*f*g + e*h))*(g + h*x) + (c*g^2 + a*h^2)*(12*a^2*f*h^4 + 2*c^2*(9*f*g^4 - g^2*h*(e*
g + d*h)) + a*c*h^2*(35*f*g^2 + h*(-7*e*g + 3*d*h)))*(g + h*x)^2 - c*(4*a^2*h^4*(7*f*g - 2*e*h) + a*c*g*h^2*(1
9*f*g^2 + h*(9*e*g - 13*d*h)) + 2*c^2*(3*f*g^5 + g^3*h*(e*g + d*h)))*(g + h*x)^3))/((c*g^2*h + a*h^3)^3*(g + h
*x)^4)) + (3*a*c*(4*c^2*d*g^2 + 4*a^2*f*h^2 - a*c*(f*g^2 + h*(-5*e*g + d*h)))*Log[g + h*x])/(c*g^2 + a*h^2)^(7
/2) - (3*a*c*(4*c^2*d*g^2 + 4*a^2*f*h^2 - a*c*(f*g^2 + h*(-5*e*g + d*h)))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2
]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(7/2))/24

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3902\) vs. \(2(293)=586\).
time = 0.08, size = 3903, normalized size = 12.47

method	result	size
default	\(\text {Expression too large to display}\)	\(3903\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*h^2-e*g*h+f*g^2)/h^7*(-1/4/(a*h^2+c*g^2)*h^2/(x+1/h*g)^4*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2
)^(3/2)+5/4*c*g*h/(a*h^2+c*g^2)*(-1/3/(a*h^2+c*g^2)*h^2/(x+1/h*g)^3*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*
g^2)/h^2)^(3/2)+c*g*h/(a*h^2+c*g^2)*(-1/2/(a*h^2+c*g^2)*h^2/(x+1/h*g)^2*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^
2+c*g^2)/h^2)^(3/2)+1/2*c*g*h/(a*h^2+c*g^2)*(-1/(a*h^2+c*g^2)*h^2/(x+1/h*g)*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(
a*h^2+c*g^2)/h^2)^(3/2)-c*g*h/(a*h^2+c*g^2)*(((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)-c^(1/2)
*g/h*ln((-c*g/h+c*(x+1/h*g))/c^(1/2)+((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2))-(a*h^2+c*g^2)/
h^2/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+1/h*g)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+1/h*g)
^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+1/h*g)))+2*c/(a*h^2+c*g^2)*h^2*(1/4*(2*c*(x+1/h*g)-2*c*g/h
)/c*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)+1/8*(4*c*(a*h^2+c*g^2)/h^2-4*c^2*g^2/h^2)/c^(3/2
)*ln((-c*g/h+c*(x+1/h*g))/c^(1/2)+((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2))))+1/2*c/(a*h^2+c*
g^2)*h^2*(((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)-c^(1/2)*g/h*ln((-c*g/h+c*(x+1/h*g))/c^(1/2
)+((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2))-(a*h^2+c*g^2)/h^2/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2
*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+1/h*g)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2
)/h^2)^(1/2))/(x+1/h*g)))))-1/4*c/(a*h^2+c*g^2)*h^2*(-1/2/(a*h^2+c*g^2)*h^2/(x+1/h*g)^2*((x+1/h*g)^2*c-2*c*g/h
*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(3/2)+1/2*c*g*h/(a*h^2+c*g^2)*(-1/(a*h^2+c*g^2)*h^2/(x+1/h*g)*((x+1/h*g)^2*c-2*c
*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(3/2)-c*g*h/(a*h^2+c*g^2)*(((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^
2)^(1/2)-c^(1/2)*g/h*ln((-c*g/h+c*(x+1/h*g))/c^(1/2)+((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)
)-(a*h^2+c*g^2)/h^2/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+1/h*g)+2*((a*h^2+c*g^2)/h^2)^
(1/2)*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+1/h*g)))+2*c/(a*h^2+c*g^2)*h^2*(1/4*(2*c*(
x+1/h*g)-2*c*g/h)/c*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)+1/8*(4*c*(a*h^2+c*g^2)/h^2-4*c^2
*g^2/h^2)/c^(3/2)*ln((-c*g/h+c*(x+1/h*g))/c^(1/2)+((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2))))
+1/2*c/(a*h^2+c*g^2)*h^2*(((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)-c^(1/2)*g/h*ln((-c*g/h+c*(
x+1/h*g))/c^(1/2)+((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2))-(a*h^2+c*g^2)/h^2/((a*h^2+c*g^2)/
h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+1/h*g)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+1/h*g)^2*c-2*c*g/h*(x+1/h
*g)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+1/h*g)))))+(e*h-2*f*g)/h^6*(-1/3/(a*h^2+c*g^2)*h^2/(x+1/h*g)^3*((x+1/h*g)^2*c
-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(3/2)+c*g*h/(a*h^2+c*g^2)*(-1/2/(a*h^2+c*g^2)*h^2/(x+1/h*g)^2*((x+1/h*g)
^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(3/2)+1/2*c*g*h/(a*h^2+c*g^2)*(-1/(a*h^2+c*g^2)*h^2/(x+1/h*g)*((x+1/
h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(3/2)-c*g*h/(a*h^2+c*g^2)*(((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h
^2+c*g^2)/h^2)^(1/2)-c^(1/2)*g/h*ln((-c*g/h+c*(x+1/h*g))/c^(1/2)+((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2
)/h^2)^(1/2))-(a*h^2+c*g^2)/h^2/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+1/h*g)+2*((a*h^2+
c*g^2)/h^2)^(1/2)*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+1/h*g)))+2*c/(a*h^2+c*g^2)*h^2
*(1/4*(2*c*(x+1/h*g)-2*c*g/h)/c*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)+1/8*(4*c*(a*h^2+c*g^
2)/h^2-4*c^2*g^2/h^2)/c^(3/2)*ln((-c*g/h+c*(x+1/h*g))/c^(1/2)+((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h
^2)^(1/2))))+1/2*c/(a*h^2+c*g^2)*h^2*(((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)-c^(1/2)*g/h*ln
((-c*g/h+c*(x+1/h*g))/c^(1/2)+((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2))-(a*h^2+c*g^2)/h^2/((a
*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+1/h*g)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+1/h*g)^2*c-2*
c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+1/h*g)))))+f/h^5*(-1/2/(a*h^2+c*g^2)*h^2/(x+1/h*g)^2*((x+1/h*g)^2
*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(3/2)+1/2*c*g*h/(a*h^2+c*g^2)*(-1/(a*h^2+c*g^2)*h^2/(x+1/h*g)*((x+1/h*
g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(3/2)-c*g*h/(a*h^2+c*g^2)*(((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2
+c*g^2)/h^2)^(1/2)-c^(1/2)*g/h*ln((-c*g/h+c*(x+1/h*g))/c^(1/2)+((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/
h^2)^(1/2))-(a*h^2+c*g^2)/h^2/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+1/h*g)+2*((a*h^2+c*
g^2)/h^2)^(1/2)*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+1/h*g)))+2*c/(a*h^2+c*g^2)*h^2*(
1/4*(2*c*(x+1/h*g)-2*c*g/h)/c*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)+1/8*(4*c*(a*h^2+c*g^2)
/h^2-4*c^2*g^2/h^2)/c^(3/2)*ln((-c*g/h+c*(x+1/h*g))/c^(1/2)+((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2
)^(1/2))))+1/2*c/(a*h^2+c*g^2)*h^2*(((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)-c^(1/2)*g/h*ln((
-c*g/h+c*(x+1/h*g))/c^(1/2)+((x+1/h*g)^2*c-2*c*...

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 3416 vs. \(2 (297) = 594\).
time = 0.43, size = 3416, normalized size = 10.91 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/8*sqrt(c*x^2 + a)*c^3*f*g^5/(c^3*g^6*h^4*x + 3*a*c^2*g^4*h^6*x + 3*a^2*c*g^2*h^8*x + a^3*h^10*x + c^3*g^7*h
^3 + 3*a*c^2*g^5*h^5 + 3*a^2*c*g^3*h^7 + a^3*g*h^9) - 5/8*(c*x^2 + a)^(3/2)*c^2*f*g^4/(c^3*g^6*h^3*x^2 + 3*a*c
^2*g^4*h^5*x^2 + 3*a^2*c*g^2*h^7*x^2 + a^3*h^9*x^2 + 2*c^3*g^7*h^2*x + 6*a*c^2*g^5*h^4*x + 6*a^2*c*g^3*h^6*x +
2*a^3*g*h^8*x + c^3*g^8*h + 3*a*c^2*g^6*h^3 + 3*a^2*c*g^4*h^5 + a^3*g^2*h^7) + 5/8*sqrt(c*x^2 + a)*c^3*f*g^4/
(c^3*g^6*h^3 + 3*a*c^2*g^4*h^5 + 3*a^2*c*g^2*h^7 + a^3*h^9) + 5/8*sqrt(c*x^2 + a)*c^3*g^4*e/(c^3*g^6*h^3*x + 3
*a*c^2*g^4*h^5*x + 3*a^2*c*g^2*h^7*x + a^3*h^9*x + c^3*g^7*h^2 + 3*a*c^2*g^5*h^4 + 3*a^2*c*g^3*h^6 + a^3*g*h^8
) - 5/8*sqrt(c*x^2 + a)*c^3*d*g^3/(c^3*g^6*h^2*x + 3*a*c^2*g^4*h^4*x + 3*a^2*c*g^2*h^6*x + a^3*h^8*x + c^3*g^7
*h + 3*a*c^2*g^5*h^3 + 3*a^2*c*g^3*h^5 + a^3*g*h^7) + 5/8*(c*x^2 + a)^(3/2)*c^2*g^3*e/(c^3*g^6*h^2*x^2 + 3*a*c
^2*g^4*h^4*x^2 + 3*a^2*c*g^2*h^6*x^2 + a^3*h^8*x^2 + 2*c^3*g^7*h*x + 6*a*c^2*g^5*h^3*x + 6*a^2*c*g^3*h^5*x + 2
*a^3*g*h^7*x + c^3*g^8 + 3*a*c^2*g^6*h^2 + 3*a^2*c*g^4*h^4 + a^3*g^2*h^6) - 5/8*sqrt(c*x^2 + a)*c^3*g^3*e/(c^3
*g^6*h^2 + 3*a*c^2*g^4*h^4 + 3*a^2*c*g^2*h^6 + a^3*h^8) - 5/8*(c*x^2 + a)^(3/2)*c^2*d*g^2/(c^3*g^6*h*x^2 + 3*a
*c^2*g^4*h^3*x^2 + 3*a^2*c*g^2*h^5*x^2 + a^3*h^7*x^2 + 2*c^3*g^7*x + 6*a*c^2*g^5*h^2*x + 6*a^2*c*g^3*h^4*x + 2
*a^3*g*h^6*x + c^3*g^8/h + 3*a*c^2*g^6*h + 3*a^2*c*g^4*h^3 + a^3*g^2*h^5) + 5/8*sqrt(c*x^2 + a)*c^3*d*g^2/(c^3
*g^6*h + 3*a*c^2*g^4*h^3 + 3*a^2*c*g^2*h^5 + a^3*h^7) - 5/12*(c*x^2 + a)^(3/2)*c*f*g^3/(c^2*g^4*h^4*x^3 + 2*a*
c*g^2*h^6*x^3 + a^2*h^8*x^3 + 3*c^2*g^5*h^3*x^2 + 6*a*c*g^3*h^5*x^2 + 3*a^2*g*h^7*x^2 + 3*c^2*g^6*h^2*x + 6*a*
c*g^4*h^4*x + 3*a^2*g^2*h^6*x + c^2*g^7*h + 2*a*c*g^5*h^3 + a^2*g^3*h^5) + 9/8*sqrt(c*x^2 + a)*c^2*f*g^3/(c^2*
g^4*h^4*x + 2*a*c*g^2*h^6*x + a^2*h^8*x + c^2*g^5*h^3 + 2*a*c*g^3*h^5 + a^2*g*h^7) + 9/8*(c*x^2 + a)^(3/2)*c*f
*g^2/(c^2*g^4*h^3*x^2 + 2*a*c*g^2*h^5*x^2 + a^2*h^7*x^2 + 2*c^2*g^5*h^2*x + 4*a*c*g^3*h^4*x + 2*a^2*g*h^6*x +
c^2*g^6*h + 2*a*c*g^4*h^3 + a^2*g^2*h^5) - 9/8*sqrt(c*x^2 + a)*c^2*f*g^2/(c^2*g^4*h^3 + 2*a*c*g^2*h^5 + a^2*h^
7) + 5/12*(c*x^2 + a)^(3/2)*c*g^2*e/(c^2*g^4*h^3*x^3 + 2*a*c*g^2*h^5*x^3 + a^2*h^7*x^3 + 3*c^2*g^5*h^2*x^2 + 6
*a*c*g^3*h^4*x^2 + 3*a^2*g*h^6*x^2 + 3*c^2*g^6*h*x + 6*a*c*g^4*h^3*x + 3*a^2*g^2*h^5*x + c^2*g^7 + 2*a*c*g^5*h
^2 + a^2*g^3*h^4) - 5/8*sqrt(c*x^2 + a)*c^2*g^2*e/(c^2*g^4*h^3*x + 2*a*c*g^2*h^5*x + a^2*h^7*x + c^2*g^5*h^2 +
2*a*c*g^3*h^4 + a^2*g*h^6) - 5/12*(c*x^2 + a)^(3/2)*c*d*g/(c^2*g^4*h^2*x^3 + 2*a*c*g^2*h^4*x^3 + a^2*h^6*x^3
+ 3*c^2*g^5*h*x^2 + 6*a*c*g^3*h^3*x^2 + 3*a^2*g*h^5*x^2 + 3*c^2*g^6*x + 6*a*c*g^4*h^2*x + 3*a^2*g^2*h^4*x + c^
2*g^7/h + 2*a*c*g^5*h + a^2*g^3*h^3) + 1/8*sqrt(c*x^2 + a)*c^2*d*g/(c^2*g^4*h^2*x + 2*a*c*g^2*h^4*x + a^2*h^6*
x + c^2*g^5*h + 2*a*c*g^3*h^3 + a^2*g*h^5) - 1/4*(c*x^2 + a)^(3/2)*f*g^2/(c*g^2*h^5*x^4 + a*h^7*x^4 + 4*c*g^3*
h^4*x^3 + 4*a*g*h^6*x^3 + 6*c*g^4*h^3*x^2 + 6*a*g^2*h^5*x^2 + 4*c*g^5*h^2*x + 4*a*g^3*h^4*x + c*g^6*h + a*g^4*
h^3) - 5/8*(c*x^2 + a)^(3/2)*c*g*e/(c^2*g^4*h^2*x^2 + 2*a*c*g^2*h^4*x^2 + a^2*h^6*x^2 + 2*c^2*g^5*h*x + 4*a*c*
g^3*h^3*x + 2*a^2*g*h^5*x + c^2*g^6 + 2*a*c*g^4*h^2 + a^2*g^2*h^4) + 5/8*sqrt(c*x^2 + a)*c^2*g*e/(c^2*g^4*h^2
+ 2*a*c*g^2*h^4 + a^2*h^6) + 1/8*(c*x^2 + a)^(3/2)*c*d/(c^2*g^4*h*x^2 + 2*a*c*g^2*h^3*x^2 + a^2*h^5*x^2 + 2*c^
2*g^5*x + 4*a*c*g^3*h^2*x + 2*a^2*g*h^4*x + c^2*g^6/h + 2*a*c*g^4*h + a^2*g^2*h^3) - 1/8*sqrt(c*x^2 + a)*c^2*d
/(c^2*g^4*h + 2*a*c*g^2*h^3 + a^2*h^5) + 2/3*(c*x^2 + a)^(3/2)*f*g/(c*g^2*h^4*x^3 + a*h^6*x^3 + 3*c*g^3*h^3*x^
2 + 3*a*g*h^5*x^2 + 3*c*g^4*h^2*x + 3*a*g^2*h^4*x + c*g^5*h + a*g^3*h^3) - 1/2*sqrt(c*x^2 + a)*c*f*g/(c*g^2*h^
4*x + a*h^6*x + c*g^3*h^3 + a*g*h^5) + 1/4*(c*x^2 + a)^(3/2)*g*e/(c*g^2*h^4*x^4 + a*h^6*x^4 + 4*c*g^3*h^3*x^3
+ 4*a*g*h^5*x^3 + 6*c*g^4*h^2*x^2 + 6*a*g^2*h^4*x^2 + 4*c*g^5*h*x + 4*a*g^3*h^3*x + c*g^6 + a*g^4*h^2) - 1/4*(
c*x^2 + a)^(3/2)*d/(c*g^2*h^3*x^4 + a*h^5*x^4 + 4*c*g^3*h^2*x^3 + 4*a*g*h^4*x^3 + 6*c*g^4*h*x^2 + 6*a*g^2*h^3*
x^2 + 4*c*g^5*x + 4*a*g^3*h^2*x + c*g^6/h + a*g^4*h) - 1/2*(c*x^2 + a)^(3/2)*f/(c*g^2*h^3*x^2 + a*h^5*x^2 + 2*
c*g^3*h^2*x + 2*a*g*h^4*x + c*g^4*h + a*g^2*h^3) + 1/2*sqrt(c*x^2 + a)*c*f/(c*g^2*h^3 + a*h^5) - 1/3*(c*x^2 +
a)^(3/2)*e/(c*g^2*h^3*x^3 + a*h^5*x^3 + 3*c*g^3*h^2*x^2 + 3*a*g*h^4*x^2 + 3*c*g^4*h*x + 3*a*g^2*h^3*x + c*g^5
+ a*g^3*h^2) - 5/8*c^4*f*g^6*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^
2/h^2)^(7/2)*h^11) - 5/8*c^4*d*g^4*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a
+ c*g^2/h^2)^(7/2)*h^9) + 7/4*c^3*f*g^4*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))
/((a + c*g^2/h^2)^(5/2)*h^9) + 3/4*c^3*d*g^2*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x +
g)))/((a + c*g^2/h^2)^(5/2)*h^7) - 13/8*c^2*f*g^2*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs
(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h^7) - 1/8*c^2*d*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*ab
s(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h^5) + 1/2*...

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1272 vs. \(2 (297) = 594\).
time = 53.46, size = 2571, normalized size = 8.21 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(3*((4*a*c^3*d - a^2*c^2*f)*g^6 - (a^2*c^2*d - 4*a^3*c*f)*g^4*h^2 + ((4*a*c^3*d - a^2*c^2*f)*g^2*h^4 - (
a^2*c^2*d - 4*a^3*c*f)*h^6)*x^4 + 4*((4*a*c^3*d - a^2*c^2*f)*g^3*h^3 - (a^2*c^2*d - 4*a^3*c*f)*g*h^5)*x^3 + 6*
((4*a*c^3*d - a^2*c^2*f)*g^4*h^2 - (a^2*c^2*d - 4*a^3*c*f)*g^2*h^4)*x^2 + 4*((4*a*c^3*d - a^2*c^2*f)*g^5*h - (
a^2*c^2*d - 4*a^3*c*f)*g^3*h^3)*x + 5*(a^2*c^2*g*h^5*x^4 + 4*a^2*c^2*g^2*h^4*x^3 + 6*a^2*c^2*g^3*h^3*x^2 + 4*a
^2*c^2*g^4*h^2*x + a^2*c^2*g^5*h)*e)*sqrt(c*g^2 + a*h^2)*log((2*a*c*g*h*x - a*c*g^2 - 2*a^2*h^2 - (2*c^2*g^2 +
a*c*h^2)*x^2 - 2*sqrt(c*g^2 + a*h^2)*(c*g*x - a*h)*sqrt(c*x^2 + a))/(h^2*x^2 + 2*g*h*x + g^2)) - 2*(6*a^4*d*h
^7 + (28*a*c^3*d - 13*a^2*c^2*f)*g^6*h + (47*a^2*c^2*d - 11*a^3*c*f)*g^4*h^3 + (25*a^3*c*d + 2*a^4*f)*g^2*h^5
- (6*c^4*f*g^7 + (2*c^4*d + 25*a*c^3*f)*g^5*h^2 - (11*a*c^3*d - 47*a^2*c^2*f)*g^3*h^4 - (13*a^2*c^2*d - 28*a^3
*c*f)*g*h^6)*x^3 - (4*(2*c^4*d + a*c^3*f)*g^6*h - (32*a*c^3*d - 41*a^2*c^2*f)*g^4*h^3 - (43*a^2*c^2*d - 25*a^3
*c*f)*g^2*h^5 - 3*(a^3*c*d + 4*a^4*f)*h^7)*x^2 - (3*(4*c^4*d + a*c^3*f)*g^7 - (25*a*c^3*d - 43*a^2*c^2*f)*g^5*
h^2 - (41*a^2*c^2*d - 32*a^3*c*f)*g^3*h^4 - 4*(a^3*c*d + 2*a^4*f)*g*h^6)*x - (8*a*c^3*g^7 - a^2*c^2*g^5*h^2 -
11*a^3*c*g^3*h^4 - 2*a^4*g*h^6 + (2*c^4*g^6*h + 11*a*c^3*g^4*h^3 + a^2*c^2*g^2*h^5 - 8*a^3*c*h^7)*x^3 + (8*c^4
*g^7 + 44*a*c^3*g^5*h^2 + 19*a^2*c^2*g^3*h^4 - 17*a^3*c*g*h^6)*x^2 + (17*a*c^3*g^6*h - 19*a^2*c^2*g^4*h^3 - 44
*a^3*c*g^2*h^5 - 8*a^4*h^7)*x)*e)*sqrt(c*x^2 + a))/(c^4*g^12 + 4*a*c^3*g^10*h^2 + 6*a^2*c^2*g^8*h^4 + 4*a^3*c*
g^6*h^6 + a^4*g^4*h^8 + (c^4*g^8*h^4 + 4*a*c^3*g^6*h^6 + 6*a^2*c^2*g^4*h^8 + 4*a^3*c*g^2*h^10 + a^4*h^12)*x^4
+ 4*(c^4*g^9*h^3 + 4*a*c^3*g^7*h^5 + 6*a^2*c^2*g^5*h^7 + 4*a^3*c*g^3*h^9 + a^4*g*h^11)*x^3 + 6*(c^4*g^10*h^2 +
4*a*c^3*g^8*h^4 + 6*a^2*c^2*g^6*h^6 + 4*a^3*c*g^4*h^8 + a^4*g^2*h^10)*x^2 + 4*(c^4*g^11*h + 4*a*c^3*g^9*h^3 +
6*a^2*c^2*g^7*h^5 + 4*a^3*c*g^5*h^7 + a^4*g^3*h^9)*x), -1/24*(3*((4*a*c^3*d - a^2*c^2*f)*g^6 - (a^2*c^2*d - 4
*a^3*c*f)*g^4*h^2 + ((4*a*c^3*d - a^2*c^2*f)*g^2*h^4 - (a^2*c^2*d - 4*a^3*c*f)*h^6)*x^4 + 4*((4*a*c^3*d - a^2*
c^2*f)*g^3*h^3 - (a^2*c^2*d - 4*a^3*c*f)*g*h^5)*x^3 + 6*((4*a*c^3*d - a^2*c^2*f)*g^4*h^2 - (a^2*c^2*d - 4*a^3*
c*f)*g^2*h^4)*x^2 + 4*((4*a*c^3*d - a^2*c^2*f)*g^5*h - (a^2*c^2*d - 4*a^3*c*f)*g^3*h^3)*x + 5*(a^2*c^2*g*h^5*x
^4 + 4*a^2*c^2*g^2*h^4*x^3 + 6*a^2*c^2*g^3*h^3*x^2 + 4*a^2*c^2*g^4*h^2*x + a^2*c^2*g^5*h)*e)*sqrt(-c*g^2 - a*h
^2)*arctan(sqrt(-c*g^2 - a*h^2)*(c*g*x - a*h)*sqrt(c*x^2 + a)/(a*c*g^2 + a^2*h^2 + (c^2*g^2 + a*c*h^2)*x^2)) +
(6*a^4*d*h^7 + (28*a*c^3*d - 13*a^2*c^2*f)*g^6*h + (47*a^2*c^2*d - 11*a^3*c*f)*g^4*h^3 + (25*a^3*c*d + 2*a^4*
f)*g^2*h^5 - (6*c^4*f*g^7 + (2*c^4*d + 25*a*c^3*f)*g^5*h^2 - (11*a*c^3*d - 47*a^2*c^2*f)*g^3*h^4 - (13*a^2*c^2
*d - 28*a^3*c*f)*g*h^6)*x^3 - (4*(2*c^4*d + a*c^3*f)*g^6*h - (32*a*c^3*d - 41*a^2*c^2*f)*g^4*h^3 - (43*a^2*c^2
*d - 25*a^3*c*f)*g^2*h^5 - 3*(a^3*c*d + 4*a^4*f)*h^7)*x^2 - (3*(4*c^4*d + a*c^3*f)*g^7 - (25*a*c^3*d - 43*a^2*
c^2*f)*g^5*h^2 - (41*a^2*c^2*d - 32*a^3*c*f)*g^3*h^4 - 4*(a^3*c*d + 2*a^4*f)*g*h^6)*x - (8*a*c^3*g^7 - a^2*c^2
*g^5*h^2 - 11*a^3*c*g^3*h^4 - 2*a^4*g*h^6 + (2*c^4*g^6*h + 11*a*c^3*g^4*h^3 + a^2*c^2*g^2*h^5 - 8*a^3*c*h^7)*x
^3 + (8*c^4*g^7 + 44*a*c^3*g^5*h^2 + 19*a^2*c^2*g^3*h^4 - 17*a^3*c*g*h^6)*x^2 + (17*a*c^3*g^6*h - 19*a^2*c^2*g
^4*h^3 - 44*a^3*c*g^2*h^5 - 8*a^4*h^7)*x)*e)*sqrt(c*x^2 + a))/(c^4*g^12 + 4*a*c^3*g^10*h^2 + 6*a^2*c^2*g^8*h^4
+ 4*a^3*c*g^6*h^6 + a^4*g^4*h^8 + (c^4*g^8*h^4 + 4*a*c^3*g^6*h^6 + 6*a^2*c^2*g^4*h^8 + 4*a^3*c*g^2*h^10 + a^4
*h^12)*x^4 + 4*(c^4*g^9*h^3 + 4*a*c^3*g^7*h^5 + 6*a^2*c^2*g^5*h^7 + 4*a^3*c*g^3*h^9 + a^4*g*h^11)*x^3 + 6*(c^4
*g^10*h^2 + 4*a*c^3*g^8*h^4 + 6*a^2*c^2*g^6*h^6 + 4*a^3*c*g^4*h^8 + a^4*g^2*h^10)*x^2 + 4*(c^4*g^11*h + 4*a*c^
3*g^9*h^3 + 6*a^2*c^2*g^7*h^5 + 4*a^3*c*g^5*h^7 + a^4*g^3*h^9)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^5,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 {\sqrt {c\,x^2+a}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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