3.21.28 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=372 \[ -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (4 b e g-11 c d g+3 c e f)}{4 e^2 (d+e x)^{7/2} (2 c d-b e)}+\frac {5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-11 c d g+3 c e f)}{12 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac {5 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-11 c d g+3 c e f)}{4 e^2 \sqrt {d+e x}}-\frac {5 c \sqrt {2 c d-b e} (4 b e g-11 c d g+3 c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{4 e^2} \]

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Rubi [A]  time = 0.60, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {792, 662, 664, 660, 208} \begin {gather*} -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (4 b e g-11 c d g+3 c e f)}{4 e^2 (d+e x)^{7/2} (2 c d-b e)}+\frac {5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-11 c d g+3 c e f)}{12 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac {5 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-11 c d g+3 c e f)}{4 e^2 \sqrt {d+e x}}-\frac {5 c \sqrt {2 c d-b e} (4 b e g-11 c d g+3 c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{4 e^2} \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)^(11/2),x]

[Out]

(5*c*(3*c*e*f - 11*c*d*g + 4*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^2*Sqrt[d + e*x]) + (5*c*(3
*c*e*f - 11*c*d*g + 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(12*e^2*(2*c*d - b*e)*(d + e*x)^(3/2
)) + ((3*c*e*f - 11*c*d*g + 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(4*e^2*(2*c*d - b*e)*(d + e*
x)^(7/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(2*e^2*(2*c*d - b*e)*(d + e*x)^(11/2)) -
(5*c*Sqrt[2*c*d - b*e]*(3*c*e*f - 11*c*d*g + 4*b*e*g)*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt
[2*c*d - b*e]*Sqrt[d + e*x])])/(4*e^2)

Rule 208

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

Rule 660

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

Rule 662

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 + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(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] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && IntegerQ[2*p]

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)^{11/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac {(3 c e f-11 c d g+4 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)^{9/2}} \, dx}{4 e (2 c d-b e)}\\ &=\frac {(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac {(5 c (3 c e f-11 c d g+4 b e g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx}{8 e (2 c d-b e)}\\ &=\frac {5 c (3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac {(5 c (3 c e f-11 c d g+4 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{8 e}\\ &=\frac {5 c (3 c e f-11 c d g+4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 \sqrt {d+e x}}+\frac {5 c (3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac {(5 c (2 c d-b e) (3 c e f-11 c d g+4 b e g)) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 e}\\ &=\frac {5 c (3 c e f-11 c d g+4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 \sqrt {d+e x}}+\frac {5 c (3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac {1}{4} (5 c (2 c d-b e) (3 c e f-11 c d g+4 b e g)) \operatorname {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right )\\ &=\frac {5 c (3 c e f-11 c d g+4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 \sqrt {d+e x}}+\frac {5 c (3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac {5 c \sqrt {2 c d-b e} (3 c e f-11 c d g+4 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2}\\ \end {align*}

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Mathematica [C]  time = 0.25, size = 130, normalized size = 0.35 \begin {gather*} \frac {((d+e x) (c (d-e x)-b e))^{7/2} \left (\frac {c (d+e x)^2 (4 b e g-11 c d g+3 c e f) \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {-c d+b e+c e x}{b e-2 c d}\right )}{e (b e-2 c d)^2}+\frac {7 d g}{e}-7 f\right )}{14 e (d+e x)^{11/2} (2 c d-b e)} \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)^(11/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(7/2)*(-7*f + (7*d*g)/e + (c*(3*c*e*f - 11*c*d*g + 4*b*e*g)*(d + e*x)^2*Hy
pergeometric2F1[2, 7/2, 9/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])/(e*(-2*c*d + b*e)^2)))/(14*e*(2*c*d - b*e
)*(d + e*x)^(11/2))

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IntegrateAlgebraic [A]  time = 5.37, size = 360, normalized size = 0.97 \begin {gather*} \frac {\sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2} \left (-12 b^2 e^2 g (d+e x)+6 b^2 d e^2 g-6 b^2 e^3 f-24 b c d^2 e g-27 b c e^2 f (d+e x)+24 b c d e^2 f+75 b c d e g (d+e x)+56 b c e g (d+e x)^2+24 c^2 d^3 g-24 c^2 d^2 e f-102 c^2 d^2 g (d+e x)+54 c^2 d e f (d+e x)+24 c^2 e f (d+e x)^2-136 c^2 d g (d+e x)^2+8 c^2 g (d+e x)^3\right )}{12 e^2 (d+e x)^{5/2}}-\frac {5 \left (4 b^2 c e^2 g-19 b c^2 d e g+3 b c^2 e^2 f+22 c^3 d^2 g-6 c^3 d e f\right ) \tan ^{-1}\left (\frac {\sqrt {b e-2 c d} \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2}}{\sqrt {d+e x} (b e+c (d+e x)-2 c d)}\right )}{4 e^2 \sqrt {b e-2 c d}} \end {gather*}

Antiderivative was successfully verified.

[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)^(11/2),x]

[Out]

(Sqrt[(2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2]*(-24*c^2*d^2*e*f + 24*b*c*d*e^2*f - 6*b^2*e^3*f + 24*c^2*d^3*g
- 24*b*c*d^2*e*g + 6*b^2*d*e^2*g + 54*c^2*d*e*f*(d + e*x) - 27*b*c*e^2*f*(d + e*x) - 102*c^2*d^2*g*(d + e*x) +
75*b*c*d*e*g*(d + e*x) - 12*b^2*e^2*g*(d + e*x) + 24*c^2*e*f*(d + e*x)^2 - 136*c^2*d*g*(d + e*x)^2 + 56*b*c*e
*g*(d + e*x)^2 + 8*c^2*g*(d + e*x)^3))/(12*e^2*(d + e*x)^(5/2)) - (5*(-6*c^3*d*e*f + 3*b*c^2*e^2*f + 22*c^3*d^
2*g - 19*b*c^2*d*e*g + 4*b^2*c*e^2*g)*ArcTan[(Sqrt[-2*c*d + b*e]*Sqrt[(2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2]
)/(Sqrt[d + e*x]*(-2*c*d + b*e + c*(d + e*x)))])/(4*e^2*Sqrt[-2*c*d + b*e])

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fricas [A]  time = 0.45, size = 959, normalized size = 2.58 \begin {gather*} \left [\frac {15 \, {\left (3 \, c^{2} d^{3} e f + {\left (3 \, c^{2} e^{4} f - {\left (11 \, c^{2} d e^{3} - 4 \, b c e^{4}\right )} g\right )} x^{3} + 3 \, {\left (3 \, c^{2} d e^{3} f - {\left (11 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3}\right )} g\right )} x^{2} - {\left (11 \, c^{2} d^{4} - 4 \, b c d^{3} e\right )} g + 3 \, {\left (3 \, c^{2} d^{2} e^{2} f - {\left (11 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2}\right )} g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x + 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (8 \, c^{2} e^{3} g x^{3} + 8 \, {\left (3 \, c^{2} e^{3} f - 7 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} + 3 \, {\left (18 \, c^{2} d^{2} e - b c d e^{2} - 2 \, b^{2} e^{3}\right )} f - {\left (206 \, c^{2} d^{3} - 107 \, b c d^{2} e + 6 \, b^{2} d e^{2}\right )} g + {\left (3 \, {\left (34 \, c^{2} d e^{2} - 9 \, b c e^{3}\right )} f - {\left (350 \, c^{2} d^{2} e - 187 \, b c d e^{2} + 12 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{24 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}}, -\frac {15 \, {\left (3 \, c^{2} d^{3} e f + {\left (3 \, c^{2} e^{4} f - {\left (11 \, c^{2} d e^{3} - 4 \, b c e^{4}\right )} g\right )} x^{3} + 3 \, {\left (3 \, c^{2} d e^{3} f - {\left (11 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3}\right )} g\right )} x^{2} - {\left (11 \, c^{2} d^{4} - 4 \, b c d^{3} e\right )} g + 3 \, {\left (3 \, c^{2} d^{2} e^{2} f - {\left (11 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) - {\left (8 \, c^{2} e^{3} g x^{3} + 8 \, {\left (3 \, c^{2} e^{3} f - 7 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} + 3 \, {\left (18 \, c^{2} d^{2} e - b c d e^{2} - 2 \, b^{2} e^{3}\right )} f - {\left (206 \, c^{2} d^{3} - 107 \, b c d^{2} e + 6 \, b^{2} d e^{2}\right )} g + {\left (3 \, {\left (34 \, c^{2} d e^{2} - 9 \, b c e^{3}\right )} f - {\left (350 \, c^{2} d^{2} e - 187 \, b c d e^{2} + 12 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{12 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}}\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)^(11/2),x, algorithm="fricas")

[Out]

[1/24*(15*(3*c^2*d^3*e*f + (3*c^2*e^4*f - (11*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(3*c^2*d*e^3*f - (11*c^2*d^2*e
^2 - 4*b*c*d*e^3)*g)*x^2 - (11*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(3*c^2*d^2*e^2*f - (11*c^2*d^3*e - 4*b*c*d^2*e^2)*
g)*x)*sqrt(2*c*d - b*e)*log(-(c*e^2*x^2 - 3*c*d^2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x + 2*sqrt(-c*e^2*x^2 - b*e^2*
x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(8*c^2*e^3*g*x^3 + 8*(3*c^2
*e^3*f - 7*(2*c^2*d*e^2 - b*c*e^3)*g)*x^2 + 3*(18*c^2*d^2*e - b*c*d*e^2 - 2*b^2*e^3)*f - (206*c^2*d^3 - 107*b*
c*d^2*e + 6*b^2*d*e^2)*g + (3*(34*c^2*d*e^2 - 9*b*c*e^3)*f - (350*c^2*d^2*e - 187*b*c*d*e^2 + 12*b^2*e^3)*g)*x
)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d))/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2), -
1/12*(15*(3*c^2*d^3*e*f + (3*c^2*e^4*f - (11*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(3*c^2*d*e^3*f - (11*c^2*d^2*e^
2 - 4*b*c*d*e^3)*g)*x^2 - (11*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(3*c^2*d^2*e^2*f - (11*c^2*d^3*e - 4*b*c*d^2*e^2)*g
)*x)*sqrt(-2*c*d + b*e)*arctan(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b*e)*sqrt(e*x + d)/(c*
e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)) - (8*c^2*e^3*g*x^3 + 8*(3*c^2*e^3*f - 7*(2*c^2*d*e^2 - b*c*e^3)*g)*x^2 + 3
*(18*c^2*d^2*e - b*c*d*e^2 - 2*b^2*e^3)*f - (206*c^2*d^3 - 107*b*c*d^2*e + 6*b^2*d*e^2)*g + (3*(34*c^2*d*e^2 -
9*b*c*e^3)*f - (350*c^2*d^2*e - 187*b*c*d*e^2 + 12*b^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*
sqrt(e*x + d))/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2)]

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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)^(11/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.08, size = 1189, normalized size = 3.20 \begin {gather*} -\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (60 b^{2} c \,e^{4} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-285 b \,c^{2} d \,e^{3} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+45 b \,c^{2} e^{4} f \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+330 c^{3} d^{2} e^{2} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-90 c^{3} d \,e^{3} f \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+120 b^{2} c d \,e^{3} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-570 b \,c^{2} d^{2} e^{2} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+90 b \,c^{2} d \,e^{3} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+660 c^{3} d^{3} e g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-180 c^{3} d^{2} e^{2} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+60 b^{2} c \,d^{2} e^{2} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-285 b \,c^{2} d^{3} e g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+45 b \,c^{2} d^{2} e^{2} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+330 c^{3} d^{4} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-90 c^{3} d^{3} e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-8 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} e^{3} g \,x^{3}-56 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{3} g \,x^{2}+112 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d \,e^{2} g \,x^{2}-24 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} e^{3} f \,x^{2}+12 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} e^{3} g x -187 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c d \,e^{2} g x +27 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{3} f x +350 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{2} e g x -102 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d \,e^{2} f x +6 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} d \,e^{2} g +6 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} e^{3} f -107 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,d^{2} e g +3 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c d \,e^{2} f +206 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{3} g -54 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{2} e f \right )}{12 \left (e x +d \right )^{\frac {5}{2}} \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, e^{2}} \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)^(11/2),x)

[Out]

-1/12*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(330*c^3*d^4*g*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-1
87*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c*d*e^2*g*x-90*c^3*d^3*e*f*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*
c*d)^(1/2))+6*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^2*e^3*f+206*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*
c^2*d^3*g-56*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c*e^3*g*x^2+112*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/
2)*c^2*d*e^2*g*x^2+27*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c*e^3*f*x+350*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c
*d)^(1/2)*c^2*d^2*e*g*x-102*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^2*d*e^2*f*x+3*(-c*e*x-b*e+c*d)^(1/2)*(b
*e-2*c*d)^(1/2)*b*c*d*e^2*f-285*b*c^2*d*e^3*g*x^2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-570*b*c^2*d
^2*e^2*g*x*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+45*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2)
)*x^2*b*c^2*e^4*f+60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b^2*c*e^4*g+45*b*c^2*d^2*e^2*f*arcta
n((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-8*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^2*e^3*g*x^3+60*b^2*c*
d^2*e^2*g*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+330*c^3*d^2*e^2*g*x^2*arctan((-c*e*x-b*e+c*d)^(1/2)
/(b*e-2*c*d)^(1/2))-90*c^3*d*e^3*f*x^2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+660*c^3*d^3*e*g*x*arct
an((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-180*c^3*d^2*e^2*f*x*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/
2))-285*b*c^2*d^3*e*g*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-24*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(
1/2)*c^2*e^3*f*x^2+12*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^2*e^3*g*x+6*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d
)^(1/2)*b^2*d*e^2*g-54*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^2*d^2*e*f+90*b*c^2*d*e^3*f*x*arctan((-c*e*x-
b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+120*b^2*c*d*e^3*g*x*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-107*(-c
*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c*d^2*e*g)/(e*x+d)^(5/2)/(-c*e*x-b*e+c*d)^(1/2)/e^2/(b*e-2*c*d)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {11}{2}}}\,{d x} \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)^(11/2),x, algorithm="maxima")

[Out]

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(11/2), x)

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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 )}^{11/2}} \,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)^(11/2),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)^(11/2), x)

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sympy [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)**(11/2),x)

[Out]

Timed out

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