3.20.69 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^2} \, dx\)
Optimal. Leaf size=354 \[ \frac {(2 c d-b e)^4 (-3 b e g-4 c d g+10 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{256 c^{5/2} e^2}+\frac {(b+2 c x) (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g-4 c d g+10 c e f)}{128 c^2 e}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g-4 c d g+10 c e f)}{48 c e}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2 (2 c d-b e)}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-3 b e g-4 c d g+10 c e f)}{15 e^2 (2 c d-b e)} \]
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Rubi [A] time = 0.61, antiderivative size = 354, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 5, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used =
{792, 664, 612, 621, 204} \begin {gather*} \frac {(b+2 c x) (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g-4 c d g+10 c e f)}{128 c^2 e}+\frac {(2 c d-b e)^4 (-3 b e g-4 c d g+10 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{256 c^{5/2} e^2}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g-4 c d g+10 c e f)}{48 c e}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2 (2 c d-b e)}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-3 b e g-4 c d g+10 c e f)}{15 e^2 (2 c d-b e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^2,x]
[Out]
((2*c*d - b*e)^2*(10*c*e*f - 4*c*d*g - 3*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(128*c^
2*e) + ((10*c*e*f - 4*c*d*g - 3*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(48*c*e) + ((1
0*c*e*f - 4*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(15*e^2*(2*c*d - b*e)) + (2*(e*f - d
*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^2) + ((2*c*d - b*e)^4*(10*c*e*
f - 4*c*d*g - 3*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(256*c^(
5/2)*e^2)
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 612
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]
Rule 621
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 664
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 + 2*p + 1)), x] - Dist[(p*(2*c*d - b*e))/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*
(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a
*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]
Rule 792
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(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] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] && !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^2}+\frac {(10 c e f-4 c d g-3 b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{d+e x} \, dx}{3 e (2 c d-b e)}\\ &=\frac {(10 c e f-4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 e^2 (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^2}-\frac {((-2 c d+b e) (10 c e f-4 c d g-3 b e g)) \int \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{6 e (2 c d-b e)}\\ &=\frac {(10 c e f-4 c d g-3 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 c e}+\frac {(10 c e f-4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 e^2 (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^2}+\frac {\left ((2 c d-b e)^2 (10 c e f-4 c d g-3 b e g)\right ) \int \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{32 c e}\\ &=\frac {(2 c d-b e)^2 (10 c e f-4 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{128 c^2 e}+\frac {(10 c e f-4 c d g-3 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 c e}+\frac {(10 c e f-4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 e^2 (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^2}+\frac {\left ((2 c d-b e)^4 (10 c e f-4 c d g-3 b e g)\right ) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{256 c^2 e}\\ &=\frac {(2 c d-b e)^2 (10 c e f-4 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{128 c^2 e}+\frac {(10 c e f-4 c d g-3 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 c e}+\frac {(10 c e f-4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 e^2 (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^2}+\frac {\left ((2 c d-b e)^4 (10 c e f-4 c d g-3 b e g)\right ) \operatorname {Subst}\left (\int \frac {1}{-4 c e^2-x^2} \, dx,x,\frac {-b e^2-2 c e^2 x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{128 c^2 e}\\ &=\frac {(2 c d-b e)^2 (10 c e f-4 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{128 c^2 e}+\frac {(10 c e f-4 c d g-3 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 c e}+\frac {(10 c e f-4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 e^2 (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^2}+\frac {(2 c d-b e)^4 (10 c e f-4 c d g-3 b e g) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{256 c^{5/2} e^2}\\ \end {align*}
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Mathematica [A] time = 4.41, size = 379, normalized size = 1.07 \begin {gather*} \frac {(b e-c d+c e x)^2 ((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac {\sqrt {\frac {b e-c d+c e x}{b e-2 c d}} (-3 b e g-4 c d g+10 c e f) \left (2 c^2 e^9 (d+e x)^2 (e (2 c d-b e))^{3/2} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}} \left (59 b^2 e^2+4 b c e (17 e x-42 d)+4 c^2 \left (31 d^2-22 d e x+6 e^2 x^2\right )\right )-15 e^8 (b e-2 c d)^4 \left (\sqrt {c} e^{5/2} \sqrt {d+e x} (b e-2 c d) \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )+c e^2 (d+e x) \sqrt {e (2 c d-b e)} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}\right )\right )}{128 c^2 e^7 (d+e x)^2 \sqrt {e (2 c d-b e)} (b e-c d+c e x)^4}-3 e^3 g\right )}{15 c e^5} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^2,x]
[Out]
((-(c*d) + b*e + c*e*x)^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-3*e^3*g + ((10*c*e*f - 4*c*d*g - 3*b*e*g)
*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)]*(2*c^2*e^9*(e*(2*c*d - b*e))^(3/2)*(d + e*x)^2*Sqrt[(-(c*d) + b*e
+ c*e*x)/(-2*c*d + b*e)]*(59*b^2*e^2 + 4*b*c*e*(-42*d + 17*e*x) + 4*c^2*(31*d^2 - 22*d*e*x + 6*e^2*x^2)) - 15
*e^8*(-2*c*d + b*e)^4*(c*e^2*Sqrt[e*(2*c*d - b*e)]*(d + e*x)*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] + Sqr
t[c]*e^(5/2)*(-2*c*d + b*e)*Sqrt[d + e*x]*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]])))/(12
8*c^2*e^7*Sqrt[e*(2*c*d - b*e)]*(d + e*x)^2*(-(c*d) + b*e + c*e*x)^4)))/(15*c*e^5)
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IntegrateAlgebraic [F] time = 180.21, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^2,x]
[Out]
$Aborted
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fricas [A] time = 0.92, size = 1097, normalized size = 3.10 \begin {gather*} \left [\frac {15 \, {\left (10 \, {\left (16 \, c^{5} d^{4} e - 32 \, b c^{4} d^{3} e^{2} + 24 \, b^{2} c^{3} d^{2} e^{3} - 8 \, b^{3} c^{2} d e^{4} + b^{4} c e^{5}\right )} f - {\left (64 \, c^{5} d^{5} - 80 \, b c^{4} d^{4} e + 40 \, b^{3} c^{2} d^{2} e^{3} - 20 \, b^{4} c d e^{4} + 3 \, b^{5} e^{5}\right )} g\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} + 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) + 4 \, {\left (384 \, c^{5} e^{4} g x^{4} + 48 \, {\left (10 \, c^{5} e^{4} f - {\left (20 \, c^{5} d e^{3} - 21 \, b c^{4} e^{4}\right )} g\right )} x^{3} - 8 \, {\left (10 \, {\left (16 \, c^{5} d e^{3} - 17 \, b c^{4} e^{4}\right )} f - {\left (64 \, c^{5} d^{2} e^{2} - 164 \, b c^{4} d e^{3} + 93 \, b^{2} c^{3} e^{4}\right )} g\right )} x^{2} + 10 \, {\left (128 \, c^{5} d^{3} e - 156 \, b c^{4} d^{2} e^{2} + 28 \, b^{2} c^{3} d e^{3} + 15 \, b^{3} c^{2} e^{4}\right )} f - {\left (896 \, c^{5} d^{4} - 1392 \, b c^{4} d^{3} e + 796 \, b^{2} c^{3} d^{2} e^{2} - 240 \, b^{3} c^{2} d e^{3} + 45 \, b^{4} c e^{4}\right )} g + 2 \, {\left (10 \, {\left (36 \, c^{5} d^{2} e^{2} - 100 \, b c^{4} d e^{3} + 59 \, b^{2} c^{3} e^{4}\right )} f + {\left (240 \, c^{5} d^{3} e - 284 \, b c^{4} d^{2} e^{2} + 64 \, b^{2} c^{3} d e^{3} + 15 \, b^{3} c^{2} e^{4}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{7680 \, c^{3} e^{2}}, -\frac {15 \, {\left (10 \, {\left (16 \, c^{5} d^{4} e - 32 \, b c^{4} d^{3} e^{2} + 24 \, b^{2} c^{3} d^{2} e^{3} - 8 \, b^{3} c^{2} d e^{4} + b^{4} c e^{5}\right )} f - {\left (64 \, c^{5} d^{5} - 80 \, b c^{4} d^{4} e + 40 \, b^{3} c^{2} d^{2} e^{3} - 20 \, b^{4} c d e^{4} + 3 \, b^{5} e^{5}\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) - 2 \, {\left (384 \, c^{5} e^{4} g x^{4} + 48 \, {\left (10 \, c^{5} e^{4} f - {\left (20 \, c^{5} d e^{3} - 21 \, b c^{4} e^{4}\right )} g\right )} x^{3} - 8 \, {\left (10 \, {\left (16 \, c^{5} d e^{3} - 17 \, b c^{4} e^{4}\right )} f - {\left (64 \, c^{5} d^{2} e^{2} - 164 \, b c^{4} d e^{3} + 93 \, b^{2} c^{3} e^{4}\right )} g\right )} x^{2} + 10 \, {\left (128 \, c^{5} d^{3} e - 156 \, b c^{4} d^{2} e^{2} + 28 \, b^{2} c^{3} d e^{3} + 15 \, b^{3} c^{2} e^{4}\right )} f - {\left (896 \, c^{5} d^{4} - 1392 \, b c^{4} d^{3} e + 796 \, b^{2} c^{3} d^{2} e^{2} - 240 \, b^{3} c^{2} d e^{3} + 45 \, b^{4} c e^{4}\right )} g + 2 \, {\left (10 \, {\left (36 \, c^{5} d^{2} e^{2} - 100 \, b c^{4} d e^{3} + 59 \, b^{2} c^{3} e^{4}\right )} f + {\left (240 \, c^{5} d^{3} e - 284 \, b c^{4} d^{2} e^{2} + 64 \, b^{2} c^{3} d e^{3} + 15 \, b^{3} c^{2} e^{4}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{3840 \, c^{3} e^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^2,x, algorithm="fricas")
[Out]
[1/7680*(15*(10*(16*c^5*d^4*e - 32*b*c^4*d^3*e^2 + 24*b^2*c^3*d^2*e^3 - 8*b^3*c^2*d*e^4 + b^4*c*e^5)*f - (64*c
^5*d^5 - 80*b*c^4*d^4*e + 40*b^3*c^2*d^2*e^3 - 20*b^4*c*d*e^4 + 3*b^5*e^5)*g)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b
*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2 + 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt
(-c)) + 4*(384*c^5*e^4*g*x^4 + 48*(10*c^5*e^4*f - (20*c^5*d*e^3 - 21*b*c^4*e^4)*g)*x^3 - 8*(10*(16*c^5*d*e^3 -
17*b*c^4*e^4)*f - (64*c^5*d^2*e^2 - 164*b*c^4*d*e^3 + 93*b^2*c^3*e^4)*g)*x^2 + 10*(128*c^5*d^3*e - 156*b*c^4*
d^2*e^2 + 28*b^2*c^3*d*e^3 + 15*b^3*c^2*e^4)*f - (896*c^5*d^4 - 1392*b*c^4*d^3*e + 796*b^2*c^3*d^2*e^2 - 240*b
^3*c^2*d*e^3 + 45*b^4*c*e^4)*g + 2*(10*(36*c^5*d^2*e^2 - 100*b*c^4*d*e^3 + 59*b^2*c^3*e^4)*f + (240*c^5*d^3*e
- 284*b*c^4*d^2*e^2 + 64*b^2*c^3*d*e^3 + 15*b^3*c^2*e^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^
3*e^2), -1/3840*(15*(10*(16*c^5*d^4*e - 32*b*c^4*d^3*e^2 + 24*b^2*c^3*d^2*e^3 - 8*b^3*c^2*d*e^4 + b^4*c*e^5)*f
- (64*c^5*d^5 - 80*b*c^4*d^4*e + 40*b^3*c^2*d^2*e^3 - 20*b^4*c*d*e^4 + 3*b^5*e^5)*g)*sqrt(c)*arctan(1/2*sqrt(
-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) -
2*(384*c^5*e^4*g*x^4 + 48*(10*c^5*e^4*f - (20*c^5*d*e^3 - 21*b*c^4*e^4)*g)*x^3 - 8*(10*(16*c^5*d*e^3 - 17*b*c
^4*e^4)*f - (64*c^5*d^2*e^2 - 164*b*c^4*d*e^3 + 93*b^2*c^3*e^4)*g)*x^2 + 10*(128*c^5*d^3*e - 156*b*c^4*d^2*e^2
+ 28*b^2*c^3*d*e^3 + 15*b^3*c^2*e^4)*f - (896*c^5*d^4 - 1392*b*c^4*d^3*e + 796*b^2*c^3*d^2*e^2 - 240*b^3*c^2*
d*e^3 + 45*b^4*c*e^4)*g + 2*(10*(36*c^5*d^2*e^2 - 100*b*c^4*d*e^3 + 59*b^2*c^3*e^4)*f + (240*c^5*d^3*e - 284*b
*c^4*d^2*e^2 + 64*b^2*c^3*d*e^3 + 15*b^3*c^2*e^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^3*e^2)]
________________________________________________________________________________________
giac [F(-1)] 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((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^2,x, algorithm="giac")
[Out]
Timed out
________________________________________________________________________________________
maple [B] time = 0.08, size = 4215, normalized size = 11.91 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^2,x)
[Out]
5/128*e^5/c/(-b*e^2+2*c*d*e)*b^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/
e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d*g-1/16*g*b^2/c*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)
-9/32*g*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d^2-1/8*g*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(
x+d/e))^(3/2)*x+2/3*c/(-b*e^2+2*c*d*e)*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)*f-5/24*e^2/(-b*e^2+2*
c*d*e)*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*f+25/8*e^3*c^2/(-b*e^2+2*c*d*e)*b^2/(c*e^2)^(1/2)
*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^
3*f-15/8*e^2*c^2/(-b*e^2+2*c*d*e)*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d^2*f+15/8*e*c^2/(-b*e
^2+2*c*d*e)*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d^3*g+15/16*e^3*c/(-b*e^2+2*c*d*e)*b^2*(-(x+
d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d*f+5/12*e*c/(-b*e^2+2*c*d*e)*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d
*e)*(x+d/e))^(3/2)*x*d*g+25/8*e*c^3/(-b*e^2+2*c*d*e)*b/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2
*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^5*g+25/16*e^3*c/(-b*e^2+2*c*d*e)*b^3/(c*e^
2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(
1/2))*d^3*g-25/8*e^2*c^3/(-b*e^2+2*c*d*e)*b/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e
^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^4*f-25/16*e^4*c/(-b*e^2+2*c*d*e)*b^3/(c*e^2)^(1/2)*ar
ctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^2*f
-25/8*e^2*c^2/(-b*e^2+2*c*d*e)*b^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+
d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^4*g-15/16*e^2*c/(-b*e^2+2*c*d*e)*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2
+2*c*d*e)*(x+d/e))^(1/2)*x*d^2*g-15/32*e^2/(-b*e^2+2*c*d*e)*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1
/2)*d^2*g+15/32*e^3/(-b*e^2+2*c*d*e)*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d*f-15/16*g*b*c^2/(
c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e
))^(1/2))*d^4-9/16*g*b*c*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d^2+9/64*g*e*b^3/c*(-(x+d/e)^2*c*
e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d-3/256*g*e^4*b^5/c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^
2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))-15/32*g*e^2*b^3/(c*e^2)^(1/2)*arctan((c*e
^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^2-3/64*g*e^2
*b^3/c*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x+9/32*g*e*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+
d/e))^(1/2)*x*d+1/4*g/e*c*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x+3/8*g/e*c^2*d^3*(-(x+d/e)^2*c*
e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x+5/24*e/(-b*e^2+2*c*d*e)*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^
(3/2)*d*g-5/64*e^4/c/(-b*e^2+2*c*d*e)*b^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*f-5/32*e^4/(-b*e^2
+2*c*d*e)*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*f+2/3/e^2/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-(x+d/
e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)*f+1/5*g/e^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)+3/16*
g/e*c*d^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b-3/128*g*e^2*b^4/c^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*
c*d*e)*(x+d/e))^(1/2)+1/8*g/e*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*b+3/8*g/e*c^3*d^5/(c*e^2)^(1
/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))
-2/3/e^3/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)*d*g-5/4*c^4/(-b*e^2+2*c*
d*e)*d^6/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d
*e)*(x+d/e))^(1/2))*g-2/3/e*c/(-b*e^2+2*c*d*e)*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)*d*g-5/8*c^2/(
-b*e^2+2*c*d*e)*d^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b*g-5/6*c^2/(-b*e^2+2*c*d*e)*d^2*(-(x+d/
e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x*g-5/4*c^3/(-b*e^2+2*c*d*e)*d^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)
*(x+d/e))^(1/2)*x*g-5/12*c/(-b*e^2+2*c*d*e)*d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*b*g+15/16*e*
c/(-b*e^2+2*c*d*e)*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d^3*g+5/4*e*c^3/(-b*e^2+2*c*d*e)*d^3*
(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*f+5/12*e*c/(-b*e^2+2*c*d*e)*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*
c*d*e)*(x+d/e))^(3/2)*b*f+5/4*e*c^4/(-b*e^2+2*c*d*e)*d^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2
+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*f+25/64*e^5/(-b*e^2+2*c*d*e)*b^4/(c*e^2)^(
1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)
)*d*f-5/128*e^6/c/(-b*e^2+2*c*d*e)*b^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(
-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*f-25/64*e^4/(-b*e^2+2*c*d*e)*b^4/(c*e^2)^(1/2)*arctan((c*e^2
)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^2*g+15/128*g*e
^3*b^4/c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d
*e)*(x+d/e))^(1/2))*d-5/12*e^2*c/(-b*e^2+2*c*d*e)*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x*f+5/8*
e*c^2/(-b*e^2+2*c*d*e)*d^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b*f+5/6*e*c^2/(-b*e^2+2*c*d*e)*d*
(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x*f-15/16*e^2*c/(-b*e^2+2*c*d*e)*b^2*(-(x+d/e)^2*c*e^2+(-b*e
^2+2*c*d*e)*(x+d/e))^(1/2)*d^2*f+5/64*e^3/c/(-b*e^2+2*c*d*e)*b^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(
1/2)*d*g+5/32*e^3/(-b*e^2+2*c*d*e)*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d*g+15/16*g*e*b^2*c
/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d
/e))^(1/2))*d^3
________________________________________________________________________________________
maxima [B] time = 1.76, size = 1751, normalized size = 4.95
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^2,x, algorithm="maxima")
[Out]
5/4*b*c^3*d^3*f*arcsin(2*c*e*x/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))/(-c)^(3/2) - 5/8*c^4*d
^4*f*arcsin(2*c*e*x/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))/((-c)^(3/2)*e) - 15/16*b^2*c^2*d^
2*e*f*arcsin(2*c*e*x/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))/(-c)^(3/2) + 5/16*b^3*c*d*e^2*f*
arcsin(2*c*e*x/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))/(-c)^(3/2) - 5/128*b^4*e^3*f*arcsin(2*
c*e*x/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))/(-c)^(3/2) + 1/4*c^4*d^5*g*arcsin(2*c*e*x/(2*c*
d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))/((-c)^(3/2)*e^2) - 5/16*b*c^3*d^4*g*arcsin(2*c*e*x/(2*c*d
- b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))/((-c)^(3/2)*e) + 5/32*b^3*c*d^2*e*g*arcsin(2*c*e*x/(2*c*d -
b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))/(-c)^(3/2) - 5/64*b^4*d*e^2*g*arcsin(2*c*e*x/(2*c*d - b*e) + 4
*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))/(-c)^(3/2) + 3/256*b^5*e^3*g*arcsin(2*c*e*x/(2*c*d - b*e) + 4*c*d/(2*c
*d - b*e) - b*e/(2*c*d - b*e))/((-c)^(3/2)*c) + 5/8*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*c^
2*d^2*f*x - 5/8*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*b*c*d*e*f*x + 5/32*sqrt(c*e^2*x^2 + 4*
c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*b^2*e^2*f*x + 1/16*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e
)*b*c*d^2*g*x - 1/4*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*c^2*d^3*g*x/e + 1/8*sqrt(c*e^2*x^2
+ 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*b^2*d*e*g*x - 3/64*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 -
b*d*e)*b^3*e^2*g*x/c - 25/16*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*b*c*d^2*f + 5/4*sqrt(c*e^
2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*c^2*d^3*f/e + 5/8*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^
2 - b*d*e)*b^2*d*e*f - 5/64*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*b^3*e^2*f/c + 7/32*sqrt(c*
e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*b^2*d^2*g - 1/2*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^
2 - b*d*e)*c^2*d^4*g/e^2 + 1/4*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*b*c*d^3*g/e - 5/32*sqrt
(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*b^3*d*e*g/c + 3/128*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x +
3*c*d^2 - b*d*e)*b^4*e^2*g/c^2 - 1/8*(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*b*g*x + 1/4*(-c*e^2*x^2 - b
*e^2*x + c*d^2 - b*d*e)^(3/2)*c*d*g*x/e - 5/24*(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*b*f + 5/12*(-c*e^2
*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*c*d*f/e - 1/16*(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*b^2*g/c - 1/
4*(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*d*g/(e^3*x + d*e^2) - 5/12*(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*
e)^(3/2)*c*d^2*g/e^2 + 1/3*(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*b*d*g/e + 1/4*(-c*e^2*x^2 - b*e^2*x +
c*d^2 - b*d*e)^(5/2)*f/(e^2*x + d*e) + 1/5*(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*g/e^2
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^2,x)
[Out]
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^2, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**2,x)
[Out]
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(f + g*x)/(d + e*x)**2, x)
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