3.24.61 \(\int \frac {(a+b x+c x^2)^{5/2}}{(d+e x)^3} \, dx\) [2361]
Optimal. Leaf size=331 \[ \frac {5 \left (16 c^2 d^2+5 b^2 e^2-4 c e (5 b d-a e)-4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 e^5}+\frac {5 (8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{12 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}-\frac {5 (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 \sqrt {c} e^6}+\frac {5 \sqrt {c d^2-b d e+a e^2} \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 e^6} \]
[Out]
5/12*(2*c*e*x-3*b*e+8*c*d)*(c*x^2+b*x+a)^(3/2)/e^3/(e*x+d)-1/2*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^2-5/16*(-b*e+2*c*
d)*(16*c^2*d^2+b^2*e^2-4*c*e*(-3*a*e+4*b*d))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/e^6/c^(1/2)+5/
8*(16*c^2*d^2+3*b^2*e^2-4*c*e*(-a*e+4*b*d))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(
c*x^2+b*x+a)^(1/2))*(a*e^2-b*d*e+c*d^2)^(1/2)/e^6+5/8*(16*c^2*d^2+5*b^2*e^2-4*c*e*(-a*e+5*b*d)-4*c*e*(-b*e+2*c
*d)*x)*(c*x^2+b*x+a)^(1/2)/e^5
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Rubi [A]
time = 0.36, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {746, 826, 828,
857, 635, 212, 738} \begin {gather*} -\frac {5 (2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 \sqrt {c} e^6}+\frac {5 \sqrt {a e^2-b d e+c d^2} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 e^6}+\frac {5 \sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{8 e^5}+\frac {5 \left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{12 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x + c*x^2)^(5/2)/(d + e*x)^3,x]
[Out]
(5*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(5*b*d - a*e) - 4*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(8*e^5) + (5*
(8*c*d - 3*b*e + 2*c*e*x)*(a + b*x + c*x^2)^(3/2))/(12*e^3*(d + e*x)) - (a + b*x + c*x^2)^(5/2)/(2*e*(d + e*x)
^2) - (5*(2*c*d - b*e)*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +
b*x + c*x^2])])/(16*Sqrt[c]*e^6) + (5*Sqrt[c*d^2 - b*d*e + a*e^2]*(16*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(4*b*d - a*e
))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(8*e^6)
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 635
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 738
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]
Rule 746
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 1))), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^
(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ
[2*c*d - b*e, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]
Rule 826
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m +
2*p + 2))), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Rule 828
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) && !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Rule 857
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& !IGtQ[m, 0]
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx &=-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}+\frac {5 \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx}{4 e}\\ &=\frac {5 (8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{12 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}-\frac {5 \int \frac {\left (8 b c d-3 b^2 e-4 a c e+8 c (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{d+e x} \, dx}{8 e^3}\\ &=\frac {5 \left (16 c^2 d^2+5 b^2 e^2-4 c e (5 b d-a e)-4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 e^5}+\frac {5 (8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{12 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}+\frac {5 \int \frac {2 c \left (e (b d-2 a e) \left (8 b c d-3 b^2 e-4 a c e\right )-2 d (2 c d-b e) \left (4 b c d-b^2 e-4 a c e\right )\right )-2 c (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{32 c e^5}\\ &=\frac {5 \left (16 c^2 d^2+5 b^2 e^2-4 c e (5 b d-a e)-4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 e^5}+\frac {5 (8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{12 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}-\frac {\left (5 (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 e^6}+\frac {\left (5 \left (2 c d (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )+2 c e \left (e (b d-2 a e) \left (8 b c d-3 b^2 e-4 a c e\right )-2 d (2 c d-b e) \left (4 b c d-b^2 e-4 a c e\right )\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{32 c e^6}\\ &=\frac {5 \left (16 c^2 d^2+5 b^2 e^2-4 c e (5 b d-a e)-4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 e^5}+\frac {5 (8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{12 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}-\frac {\left (5 (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{8 e^6}-\frac {\left (5 \left (2 c d (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )+2 c e \left (e (b d-2 a e) \left (8 b c d-3 b^2 e-4 a c e\right )-2 d (2 c d-b e) \left (4 b c d-b^2 e-4 a c e\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{16 c e^6}\\ &=\frac {5 \left (16 c^2 d^2+5 b^2 e^2-4 c e (5 b d-a e)-4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 e^5}+\frac {5 (8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{12 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}-\frac {5 (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 \sqrt {c} e^6}+\frac {5 \sqrt {c d^2-b d e+a e^2} \left (16 c^2 d^2-16 b c d e+3 b^2 e^2+4 a c e^2\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 e^6}\\ \end {align*}
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Mathematica [A]
time = 11.17, size = 445, normalized size = 1.34 \begin {gather*} \frac {-\frac {2 (a+x (b+c x))^{5/2}}{(d+e x)^2}+\frac {5 (2 c d-b e) (a+x (b+c x))^{5/2}}{\left (c d^2+e (-b d+a e)\right ) (d+e x)}+\frac {5 \left (-\frac {(a+x (b+c x))^{3/2} \left (3 b^2 e^2+2 c^2 d (4 d-3 e x)+c e (-11 b d+2 a e+3 b e x)\right )}{3 e^2}+\frac {-2 c^2 e \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)} \left (5 b^2 e^2+8 c^2 d (2 d-e x)+4 c e (-5 b d+a e+b e x)\right )+c^{3/2} (2 c d-b e) \left (c d^2+e (-b d+a e)\right ) \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+2 c^2 \left (16 c^2 d^2+3 b^2 e^2+4 c e (-4 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^{3/2} \tanh ^{-1}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{4 c^2 e^5}\right )}{-c d^2+e (b d-a e)}}{4 e} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^3,x]
[Out]
((-2*(a + x*(b + c*x))^(5/2))/(d + e*x)^2 + (5*(2*c*d - b*e)*(a + x*(b + c*x))^(5/2))/((c*d^2 + e*(-(b*d) + a*
e))*(d + e*x)) + (5*(-1/3*((a + x*(b + c*x))^(3/2)*(3*b^2*e^2 + 2*c^2*d*(4*d - 3*e*x) + c*e*(-11*b*d + 2*a*e +
3*b*e*x)))/e^2 + (-2*c^2*e*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)]*(5*b^2*e^2 + 8*c^2*d*(2*d - e*x)
+ 4*c*e*(-5*b*d + a*e + b*e*x)) + c^(3/2)*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(16*c^2*d^2 + b^2*e^2 + 4*c
*e*(-4*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*c^2*(16*c^2*d^2 + 3*b^2*e^2 +
4*c*e*(-4*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^
2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(4*c^2*e^5)))/(-(c*d^2) + e*(b*d - a*e)))/(4*e)
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2821\) vs.
\(2(299)=598\).
time = 0.90, size = 2822, normalized size = 8.53
| | |
| method |
result |
size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(2822\) |
| risch |
\(\text {Expression too large to display}\) |
\(10207\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*x^2+b*x+a)^(5/2)/(e*x+d)^3,x,method=_RETURNVERBOSE)
[Out]
1/e^3*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(7
/2)+3/4*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+
d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(7/2)+5/2*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(1/5*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x
+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(5/2)+1/2/e*(b*e-2*c*d)*(1/8*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(
b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+3/16*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/c*(1/
4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c
*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/
e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))))+(a*e^2-b*d*e+c*d^2)/e^2*(1/3*(c*(x+d/e)^2+1/e*(b*e-2*c
*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2/e*(b*e-2*c*d)*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2
+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/
c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)))+(a*e^2-b*d*e+c*d^2)/e^2*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2/e*(b
*e-2*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^
2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+1/e*(b
*e-2*c*d)*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e
^2)^(1/2))/(x+d/e)))))+6*c/(a*e^2-b*d*e+c*d^2)*e^2*(1/12*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e
-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(5/2)+5/24*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/c*(1/8*(
2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+3/16*(4*c*(
a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/c*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d
)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/c^(3/2)*ln((1/2
/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))))))+5/2
*c/(a*e^2-b*d*e+c*d^2)*e^2*(1/5*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(5/2)+1/2/e*(b*e
-2*c*d)*(1/8*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/
2)+3/16*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/c*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+
1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/c
^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^
(1/2))))+(a*e^2-b*d*e+c*d^2)/e^2*(1/3*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2/
e*(b*e-2*c*d)*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^
2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2
)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)))+(a*e^2-b*d*e+c*d^2)/e^2*((c*(x+d/e)^2+
1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2/e*(b*e-2*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1
/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a*e
^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(
1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))))))
________________________________________________________________________________________
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)^(5/2)/(e*x+d)^3,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(c*d^2-%e*b*d+%e^2*a>0)', see `
assume?` for
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 708 vs.
\(2 (306) = 612\).
time = 153.78, size = 2921, normalized size = 8.82 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^3,x, algorithm="fricas")
[Out]
[-1/96*(15*(32*c^3*d^5 - (b^3 + 12*a*b*c)*x^2*e^5 + 2*(3*(3*b^2*c + 4*a*c^2)*d*x^2 - (b^3 + 12*a*b*c)*d*x)*e^4
- (48*b*c^2*d^2*x^2 - 12*(3*b^2*c + 4*a*c^2)*d^2*x + (b^3 + 12*a*b*c)*d^2)*e^3 + 2*(16*c^3*d^3*x^2 - 48*b*c^2
*d^3*x + 3*(3*b^2*c + 4*a*c^2)*d^3)*e^2 + 16*(4*c^3*d^4*x - 3*b*c^2*d^4)*e)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x -
b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 30*(16*c^3*d^4 + (3*b^2*c + 4*a*c^2)*x^2*e^4 - 2
*(8*b*c^2*d*x^2 - (3*b^2*c + 4*a*c^2)*d*x)*e^3 + (16*c^3*d^2*x^2 - 32*b*c^2*d^2*x + (3*b^2*c + 4*a*c^2)*d^2)*e
^2 + 16*(2*c^3*d^3*x - b*c^2*d^3)*e)*sqrt(c*d^2 - b*d*e + a*e^2)*log(-(8*c^2*d^2*x^2 + 8*b*c*d^2*x + (b^2 + 4*
a*c)*d^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a) + (8*a*b*x + (b
^2 + 4*a*c)*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^2 + 4*a*b*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e + d^2)) -
4*(240*c^3*d^4*e + (8*c^3*x^4 + 26*b*c^2*x^3 - 54*a*b*c*x - 12*a^2*c + (33*b^2*c + 56*a*c^2)*x^2)*e^5 - 10*(2
*c^3*d*x^3 + 11*b*c^2*d*x^2 + 3*a*b*c*d - 2*(6*b^2*c + 11*a*c^2)*d*x)*e^4 + 5*(16*c^3*d^2*x^2 - 92*b*c^2*d^2*x
+ (15*b^2*c + 28*a*c^2)*d^2)*e^3 + 60*(6*c^3*d^3*x - 5*b*c^2*d^3)*e^2)*sqrt(c*x^2 + b*x + a))/(c*x^2*e^8 + 2*
c*d*x*e^7 + c*d^2*e^6), 1/48*(15*(32*c^3*d^5 - (b^3 + 12*a*b*c)*x^2*e^5 + 2*(3*(3*b^2*c + 4*a*c^2)*d*x^2 - (b^
3 + 12*a*b*c)*d*x)*e^4 - (48*b*c^2*d^2*x^2 - 12*(3*b^2*c + 4*a*c^2)*d^2*x + (b^3 + 12*a*b*c)*d^2)*e^3 + 2*(16*
c^3*d^3*x^2 - 48*b*c^2*d^3*x + 3*(3*b^2*c + 4*a*c^2)*d^3)*e^2 + 16*(4*c^3*d^4*x - 3*b*c^2*d^4)*e)*sqrt(-c)*arc
tan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 15*(16*c^3*d^4 + (3*b^2*c + 4*a*
c^2)*x^2*e^4 - 2*(8*b*c^2*d*x^2 - (3*b^2*c + 4*a*c^2)*d*x)*e^3 + (16*c^3*d^2*x^2 - 32*b*c^2*d^2*x + (3*b^2*c +
4*a*c^2)*d^2)*e^2 + 16*(2*c^3*d^3*x - b*c^2*d^3)*e)*sqrt(c*d^2 - b*d*e + a*e^2)*log(-(8*c^2*d^2*x^2 + 8*b*c*d
^2*x + (b^2 + 4*a*c)*d^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a)
+ (8*a*b*x + (b^2 + 4*a*c)*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^2 + 4*a*b*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2
*d*x*e + d^2)) + 2*(240*c^3*d^4*e + (8*c^3*x^4 + 26*b*c^2*x^3 - 54*a*b*c*x - 12*a^2*c + (33*b^2*c + 56*a*c^2)*
x^2)*e^5 - 10*(2*c^3*d*x^3 + 11*b*c^2*d*x^2 + 3*a*b*c*d - 2*(6*b^2*c + 11*a*c^2)*d*x)*e^4 + 5*(16*c^3*d^2*x^2
- 92*b*c^2*d^2*x + (15*b^2*c + 28*a*c^2)*d^2)*e^3 + 60*(6*c^3*d^3*x - 5*b*c^2*d^3)*e^2)*sqrt(c*x^2 + b*x + a))
/(c*x^2*e^8 + 2*c*d*x*e^7 + c*d^2*e^6), 1/96*(60*(16*c^3*d^4 + (3*b^2*c + 4*a*c^2)*x^2*e^4 - 2*(8*b*c^2*d*x^2
- (3*b^2*c + 4*a*c^2)*d*x)*e^3 + (16*c^3*d^2*x^2 - 32*b*c^2*d^2*x + (3*b^2*c + 4*a*c^2)*d^2)*e^2 + 16*(2*c^3*d
^3*x - b*c^2*d^3)*e)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(2*c*d*x + b*d - (b
*x + 2*a)*e)*sqrt(c*x^2 + b*x + a)/(c^2*d^2*x^2 + b*c*d^2*x + a*c*d^2 + (a*c*x^2 + a*b*x + a^2)*e^2 - (b*c*d*x
^2 + b^2*d*x + a*b*d)*e)) - 15*(32*c^3*d^5 - (b^3 + 12*a*b*c)*x^2*e^5 + 2*(3*(3*b^2*c + 4*a*c^2)*d*x^2 - (b^3
+ 12*a*b*c)*d*x)*e^4 - (48*b*c^2*d^2*x^2 - 12*(3*b^2*c + 4*a*c^2)*d^2*x + (b^3 + 12*a*b*c)*d^2)*e^3 + 2*(16*c^
3*d^3*x^2 - 48*b*c^2*d^3*x + 3*(3*b^2*c + 4*a*c^2)*d^3)*e^2 + 16*(4*c^3*d^4*x - 3*b*c^2*d^4)*e)*sqrt(c)*log(-8
*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(240*c^3*d^4*e + (8*c^3*x^
4 + 26*b*c^2*x^3 - 54*a*b*c*x - 12*a^2*c + (33*b^2*c + 56*a*c^2)*x^2)*e^5 - 10*(2*c^3*d*x^3 + 11*b*c^2*d*x^2 +
3*a*b*c*d - 2*(6*b^2*c + 11*a*c^2)*d*x)*e^4 + 5*(16*c^3*d^2*x^2 - 92*b*c^2*d^2*x + (15*b^2*c + 28*a*c^2)*d^2)
*e^3 + 60*(6*c^3*d^3*x - 5*b*c^2*d^3)*e^2)*sqrt(c*x^2 + b*x + a))/(c*x^2*e^8 + 2*c*d*x*e^7 + c*d^2*e^6), 1/48*
(30*(16*c^3*d^4 + (3*b^2*c + 4*a*c^2)*x^2*e^4 - 2*(8*b*c^2*d*x^2 - (3*b^2*c + 4*a*c^2)*d*x)*e^3 + (16*c^3*d^2*
x^2 - 32*b*c^2*d^2*x + (3*b^2*c + 4*a*c^2)*d^2)*e^2 + 16*(2*c^3*d^3*x - b*c^2*d^3)*e)*sqrt(-c*d^2 + b*d*e - a*
e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a)/(c^2*d^2*x
^2 + b*c*d^2*x + a*c*d^2 + (a*c*x^2 + a*b*x + a^2)*e^2 - (b*c*d*x^2 + b^2*d*x + a*b*d)*e)) + 15*(32*c^3*d^5 -
(b^3 + 12*a*b*c)*x^2*e^5 + 2*(3*(3*b^2*c + 4*a*c^2)*d*x^2 - (b^3 + 12*a*b*c)*d*x)*e^4 - (48*b*c^2*d^2*x^2 - 12
*(3*b^2*c + 4*a*c^2)*d^2*x + (b^3 + 12*a*b*c)*d^2)*e^3 + 2*(16*c^3*d^3*x^2 - 48*b*c^2*d^3*x + 3*(3*b^2*c + 4*a
*c^2)*d^3)*e^2 + 16*(4*c^3*d^4*x - 3*b*c^2*d^4)*e)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(
-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(240*c^3*d^4*e + (8*c^3*x^4 + 26*b*c^2*x^3 - 54*a*b*c*x - 12*a^2*c + (33*b^2*
c + 56*a*c^2)*x^2)*e^5 - 10*(2*c^3*d*x^3 + 11*b*c^2*d*x^2 + 3*a*b*c*d - 2*(6*b^2*c + 11*a*c^2)*d*x)*e^4 + 5*(1
6*c^3*d^2*x^2 - 92*b*c^2*d^2*x + (15*b^2*c + 28*a*c^2)*d^2)*e^3 + 60*(6*c^3*d^3*x - 5*b*c^2*d^3)*e^2)*sqrt(c*x
^2 + b*x + a))/(c*x^2*e^8 + 2*c*d*x*e^7 + c*d^2*e^6)]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x**2+b*x+a)**(5/2)/(e*x+d)**3,x)
[Out]
Integral((a + b*x + c*x**2)**(5/2)/(d + e*x)**3, x)
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1458 vs.
\(2 (306) = 612\).
time = 4.12, size = 1458, normalized size = 4.40 \begin {gather*} \frac {5 \, {\left (16 \, c^{3} d^{4} - 32 \, b c^{2} d^{3} e + 19 \, b^{2} c d^{2} e^{2} + 20 \, a c^{2} d^{2} e^{2} - 3 \, b^{3} d e^{3} - 20 \, a b c d e^{3} + 3 \, a b^{2} e^{4} + 4 \, a^{2} c e^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right ) e^{\left (-6\right )}}{4 \, \sqrt {-c d^{2} + b d e - a e^{2}}} + \frac {5 \, {\left (32 \, c^{3} d^{3} - 48 \, b c^{2} d^{2} e + 18 \, b^{2} c d e^{2} + 24 \, a c^{2} d e^{2} - b^{3} e^{3} - 12 \, a b c e^{3}\right )} e^{\left (-6\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{16 \, \sqrt {c}} + \frac {1}{24} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, c^{2} x e^{\left (-3\right )} - \frac {{\left (18 \, c^{4} d e^{14} - 13 \, b c^{3} e^{15}\right )} e^{\left (-18\right )}}{c^{2}}\right )} x + \frac {{\left (144 \, c^{4} d^{2} e^{13} - 162 \, b c^{3} d e^{14} + 33 \, b^{2} c^{2} e^{15} + 56 \, a c^{3} e^{15}\right )} e^{\left (-18\right )}}{c^{2}}\right )} + \frac {{\left (40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} c^{3} d^{4} e + 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {7}{2}} d^{5} - 120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b c^{\frac {5}{2}} d^{4} e + 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c^{3} d^{5} - 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b c^{2} d^{3} e^{2} - 124 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{2} c^{2} d^{4} e - 104 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a c^{3} d^{4} e + 18 \, b^{2} c^{\frac {5}{2}} d^{5} + 51 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} c^{\frac {3}{2}} d^{3} e^{2} + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{\frac {5}{2}} d^{3} e^{2} - 27 \, b^{3} c^{\frac {3}{2}} d^{4} e - 52 \, a b c^{\frac {5}{2}} d^{4} e + 49 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{2} c d^{2} e^{3} + 44 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a c^{2} d^{2} e^{3} + 59 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} c d^{3} e^{2} + 244 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c^{2} d^{3} e^{2} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{3} \sqrt {c} d^{2} e^{3} + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b c^{\frac {3}{2}} d^{2} e^{3} + 9 \, b^{4} \sqrt {c} d^{3} e^{2} + 95 \, a b^{2} c^{\frac {3}{2}} d^{3} e^{2} + 36 \, a^{2} c^{\frac {5}{2}} d^{3} e^{2} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{3} d e^{4} - 44 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b c d e^{4} - 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{4} d^{2} e^{3} - 127 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{2} c d^{2} e^{3} - 100 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} c^{2} d^{2} e^{3} - 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b^{2} \sqrt {c} d e^{4} - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} c^{\frac {3}{2}} d e^{4} - 34 \, a b^{3} \sqrt {c} d^{2} e^{3} - 104 \, a^{2} b c^{\frac {3}{2}} d^{2} e^{3} + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b^{2} e^{5} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{2} c e^{5} + 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{3} d e^{4} + 64 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b c d e^{4} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} b \sqrt {c} e^{5} + 41 \, a^{2} b^{2} \sqrt {c} d e^{4} + 36 \, a^{3} c^{\frac {3}{2}} d e^{4} - 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b^{2} e^{5} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{3} c e^{5} - 16 \, a^{3} b \sqrt {c} e^{5}\right )} e^{\left (-6\right )}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} d + b d - a e\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^3,x, algorithm="giac")
[Out]
5/4*(16*c^3*d^4 - 32*b*c^2*d^3*e + 19*b^2*c*d^2*e^2 + 20*a*c^2*d^2*e^2 - 3*b^3*d*e^3 - 20*a*b*c*d*e^3 + 3*a*b^
2*e^4 + 4*a^2*c*e^4)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))
*e^(-6)/sqrt(-c*d^2 + b*d*e - a*e^2) + 5/16*(32*c^3*d^3 - 48*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 24*a*c^2*d*e^2 - b
^3*e^3 - 12*a*b*c*e^3)*e^(-6)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/sqrt(c) + 1/24*sqrt(
c*x^2 + b*x + a)*(2*(4*c^2*x*e^(-3) - (18*c^4*d*e^14 - 13*b*c^3*e^15)*e^(-18)/c^2)*x + (144*c^4*d^2*e^13 - 162
*b*c^3*d*e^14 + 33*b^2*c^2*e^15 + 56*a*c^3*e^15)*e^(-18)/c^2) + 1/4*(40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*
c^3*d^4*e + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(7/2)*d^5 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b
*c^(5/2)*d^4*e + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^3*d^5 - 80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b
*c^2*d^3*e^2 - 124*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c^2*d^4*e - 104*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*a*c^3*d^4*e + 18*b^2*c^(5/2)*d^5 + 51*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^(3/2)*d^3*e^2 + 36*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^2*a*c^(5/2)*d^3*e^2 - 27*b^3*c^(3/2)*d^4*e - 52*a*b*c^(5/2)*d^4*e + 49*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*b^2*c*d^2*e^3 + 44*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^2*d^2*e^3 + 59*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))*b^3*c*d^3*e^2 + 244*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^2*d^3*e^2 - 3*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*sqrt(c)*d^2*e^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*c^(3/2)*d^
2*e^3 + 9*b^4*sqrt(c)*d^3*e^2 + 95*a*b^2*c^(3/2)*d^3*e^2 + 36*a^2*c^(5/2)*d^3*e^2 - 9*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*b^3*d*e^4 - 44*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c*d*e^4 - 7*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))*b^4*d^2*e^3 - 127*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*c*d^2*e^3 - 100*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*a^2*c^2*d^2*e^3 - 21*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*sqrt(c)*d*e^4 - 36*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*a^2*c^(3/2)*d*e^4 - 34*a*b^3*sqrt(c)*d^2*e^3 - 104*a^2*b*c^(3/2)*d^2*e^3 + 9*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*a*b^2*e^5 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c*e^5 + 14*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*a*b^3*d*e^4 + 64*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c*d*e^4 + 24*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^2*a^2*b*sqrt(c)*e^5 + 41*a^2*b^2*sqrt(c)*d*e^4 + 36*a^3*c^(3/2)*d*e^4 - 7*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))*a^2*b^2*e^5 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*c*e^5 - 16*a^3*b*sqrt(c)*e^5)*e^(-6
)/((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)*d + b*d - a*e)^2
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x + c*x^2)^(5/2)/(d + e*x)^3,x)
[Out]
int((a + b*x + c*x^2)^(5/2)/(d + e*x)^3, x)
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