3.21.26 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=316 \[ \frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {2 (2 c d-b e)^{5/2} (e f-d g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}} \]

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Rubi [A]  time = 0.59, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {794, 664, 660, 208} \begin {gather*} \frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {2 (2 c d-b e)^{5/2} (e f-d g) \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 )}{e^2}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}} \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)^(7/2),x]

[Out]

(2*(2*c*d - b*e)^2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*Sqrt[d + e*x]) + (2*(2*c*d - b*
e)*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(d + e*x)^(3/2)) + (2*(e*f - d*g)*(d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*e^2*(d + e*x)^(5/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2)
)/(7*c*e^2*(d + e*x)^(7/2)) - (2*(2*c*d - b*e)^(5/2)*(e*f - d*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])])/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 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 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), 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] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 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)^{7/2}} \, dx &=-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}-\frac {\left (2 \left (\frac {7}{2} e \left (-2 c e^2 f+b e^2 g\right )-\frac {7}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx}{7 c e^3}\\ &=\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\frac {((2 c d-b e) (e f-d 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}{e}\\ &=\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\frac {\left ((2 c d-b e)^2 (e f-d g)\right ) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{e}\\ &=\frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\frac {\left ((2 c d-b e)^3 (e f-d g)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{e}\\ &=\frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\left (2 (2 c d-b e)^3 (e f-d g)\right ) \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 {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}-\frac {2 (2 c d-b e)^{5/2} (e f-d 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 )}{e^2}\\ \end {align*}

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Mathematica [A]  time = 0.57, size = 197, normalized size = 0.62 \begin {gather*} \frac {2 ((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac {7 c (e f-d g) \left (\sqrt {c (d-e x)-b e} \left (23 b^2 e^2+b c e (11 e x-81 d)+c^2 \left (73 d^2-16 d e x+3 e^2 x^2\right )\right )-15 (2 c d-b e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {-b e+c d-c e x}}{\sqrt {2 c d-b e}}\right )\right )}{15 (c (d-e x)-b e)^{5/2}}+g (b e-c d+c e x)\right )}{7 c e^2 (d+e x)^{5/2}} \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)^(7/2),x]

[Out]

(2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*(g*(-(c*d) + b*e + c*e*x) + (7*c*(e*f - d*g)*(Sqrt[-(b*e) + c*(d -
e*x)]*(23*b^2*e^2 + b*c*e*(-81*d + 11*e*x) + c^2*(73*d^2 - 16*d*e*x + 3*e^2*x^2)) - 15*(2*c*d - b*e)^(5/2)*Ar
cTanh[Sqrt[c*d - b*e - c*e*x]/Sqrt[2*c*d - b*e]]))/(15*(-(b*e) + c*(d - e*x))^(5/2))))/(7*c*e^2*(d + e*x)^(5/2
))

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IntegrateAlgebraic [A]  time = 4.62, size = 347, normalized size = 1.10 \begin {gather*} \frac {2 \sqrt {-b e (d+e x)-c (d+e x)^2+2 c d (d+e x)} \left (15 b^3 e^3 g+45 b^2 c e^2 g (d+e x)-251 b^2 c d e^2 g+161 b^2 c e^3 f+824 b c^2 d^2 e g+77 b c^2 e^2 f (d+e x)-644 b c^2 d e^2 f+45 b c^2 e g (d+e x)^2-257 b c^2 d e g (d+e x)-764 c^3 d^3 g+644 c^3 d^2 e f+334 c^3 d^2 g (d+e x)+21 c^3 e f (d+e x)^2-154 c^3 d e f (d+e x)+15 c^3 g (d+e x)^3-111 c^3 d g (d+e x)^2\right )}{105 c e^2 \sqrt {d+e x}}+\frac {2 (b e-2 c d)^{5/2} (d g-e f) \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 )}{e^2} \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)^(7/2),x]

[Out]

(2*Sqrt[2*c*d*(d + e*x) - b*e*(d + e*x) - c*(d + e*x)^2]*(644*c^3*d^2*e*f - 644*b*c^2*d*e^2*f + 161*b^2*c*e^3*
f - 764*c^3*d^3*g + 824*b*c^2*d^2*e*g - 251*b^2*c*d*e^2*g + 15*b^3*e^3*g - 154*c^3*d*e*f*(d + e*x) + 77*b*c^2*
e^2*f*(d + e*x) + 334*c^3*d^2*g*(d + e*x) - 257*b*c^2*d*e*g*(d + e*x) + 45*b^2*c*e^2*g*(d + e*x) + 21*c^3*e*f*
(d + e*x)^2 - 111*c^3*d*g*(d + e*x)^2 + 45*b*c^2*e*g*(d + e*x)^2 + 15*c^3*g*(d + e*x)^3))/(105*c*e^2*Sqrt[d +
e*x]) + (2*(-2*c*d + b*e)^(5/2)*(-(e*f) + d*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)))])/e^2

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fricas [A]  time = 0.43, size = 949, normalized size = 3.00 \begin {gather*} \left [-\frac {105 \, \sqrt {2 \, c d - b e} {\left ({\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} f - {\left (4 \, c^{3} d^{4} - 4 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )} g + {\left ({\left (4 \, c^{3} d^{2} e^{2} - 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} f - {\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} g\right )} x\right )} \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 (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f - {\left (22 \, c^{3} d e^{2} - 15 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (73 \, c^{3} d^{2} e - 81 \, b c^{2} d e^{2} + 23 \, b^{2} c e^{3}\right )} f - {\left (526 \, c^{3} d^{3} - 612 \, b c^{2} d^{2} e + 206 \, b^{2} c d e^{2} - 15 \, b^{3} e^{3}\right )} g - {\left (7 \, {\left (16 \, c^{3} d e^{2} - 11 \, b c^{2} e^{3}\right )} f - {\left (157 \, c^{3} d^{2} e - 167 \, b c^{2} d e^{2} + 45 \, b^{2} c 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}}{105 \, {\left (c e^{3} x + c d e^{2}\right )}}, -\frac {2 \, {\left (105 \, \sqrt {-2 \, c d + b e} {\left ({\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} f - {\left (4 \, c^{3} d^{4} - 4 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )} g + {\left ({\left (4 \, c^{3} d^{2} e^{2} - 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} f - {\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} g\right )} x\right )} \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 (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f - {\left (22 \, c^{3} d e^{2} - 15 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (73 \, c^{3} d^{2} e - 81 \, b c^{2} d e^{2} + 23 \, b^{2} c e^{3}\right )} f - {\left (526 \, c^{3} d^{3} - 612 \, b c^{2} d^{2} e + 206 \, b^{2} c d e^{2} - 15 \, b^{3} e^{3}\right )} g - {\left (7 \, {\left (16 \, c^{3} d e^{2} - 11 \, b c^{2} e^{3}\right )} f - {\left (157 \, c^{3} d^{2} e - 167 \, b c^{2} d e^{2} + 45 \, b^{2} c 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}\right )}}{105 \, {\left (c e^{3} x + c d 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)^(7/2),x, algorithm="fricas")

[Out]

[-1/105*(105*sqrt(2*c*d - b*e)*((4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f - (4*c^3*d^4 - 4*b*c^2*d^3*e +
b^2*c*d^2*e^2)*g + ((4*c^3*d^2*e^2 - 4*b*c^2*d*e^3 + b^2*c*e^4)*f - (4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*
e^3)*g)*x)*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*(15*c^3*e^3*g*x^3 + 3*(7*c^3*e^3*f - (22
*c^3*d*e^2 - 15*b*c^2*e^3)*g)*x^2 + 7*(73*c^3*d^2*e - 81*b*c^2*d*e^2 + 23*b^2*c*e^3)*f - (526*c^3*d^3 - 612*b*
c^2*d^2*e + 206*b^2*c*d*e^2 - 15*b^3*e^3)*g - (7*(16*c^3*d*e^2 - 11*b*c^2*e^3)*f - (157*c^3*d^2*e - 167*b*c^2*
d*e^2 + 45*b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d))/(c*e^3*x + c*d*e^2), -2/
105*(105*sqrt(-2*c*d + b*e)*((4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f - (4*c^3*d^4 - 4*b*c^2*d^3*e + b^
2*c*d^2*e^2)*g + ((4*c^3*d^2*e^2 - 4*b*c^2*d*e^3 + b^2*c*e^4)*f - (4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3
)*g)*x)*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)) - (15*c^3*e^3*g*x^3 + 3*(7*c^3*e^3*f - (22*c^3*d*e^2 - 15*b*c^2*e^3)*g)*x^2 + 7*(73*c^3*d^
2*e - 81*b*c^2*d*e^2 + 23*b^2*c*e^3)*f - (526*c^3*d^3 - 612*b*c^2*d^2*e + 206*b^2*c*d*e^2 - 15*b^3*e^3)*g - (7
*(16*c^3*d*e^2 - 11*b*c^2*e^3)*f - (157*c^3*d^2*e - 167*b*c^2*d*e^2 + 45*b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*
e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d))/(c*e^3*x + c*d*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)^(7/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.07, size = 956, normalized size = 3.03 \begin {gather*} \frac {2 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (105 b^{3} c d \,e^{3} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-105 b^{3} c \,e^{4} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-630 b^{2} c^{2} d^{2} e^{2} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+630 b^{2} c^{2} d \,e^{3} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+1260 b \,c^{3} d^{3} e g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-1260 b \,c^{3} d^{2} e^{2} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-840 c^{4} d^{4} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+840 c^{4} d^{3} e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+15 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} e^{3} g \,x^{3}+45 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b \,c^{2} e^{3} g \,x^{2}-66 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} d \,e^{2} g \,x^{2}+21 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} e^{3} f \,x^{2}+45 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} c \,e^{3} g x -167 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b \,c^{2} d \,e^{2} g x +77 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b \,c^{2} e^{3} f x +157 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} d^{2} e g x -112 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} d \,e^{2} f x +15 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, b^{3} e^{3} g -206 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, b^{2} c d \,e^{2} g +161 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, b^{2} c \,e^{3} f +612 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, b \,c^{2} d^{2} e g -567 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, b \,c^{2} d \,e^{2} f -526 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, c^{3} d^{3} g +511 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, c^{3} d^{2} e f \right )}{105 \sqrt {e x +d}\, \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c \,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)^(7/2),x)

[Out]

2/105*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(15*x^3*c^3*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+45*x^2
*b*c^2*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-66*x^2*c^3*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1
/2)+21*x^2*c^3*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1
/2))*b^3*c*d*e^3*g-105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^3*c*e^4*f-630*arctan((-c*e*x-b*e+c*d
)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c^2*d^2*e^2*g+630*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c^2*d*e^
3*f+1260*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3*d^3*e*g-1260*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*
e-2*c*d)^(1/2))*b*c^3*d^2*e^2*f-840*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^4*g+840*arctan((-c*
e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^3*e*f+45*x*b^2*c*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-16
7*x*b*c^2*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+77*x*b*c^2*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)
^(1/2)+157*x*c^3*d^2*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-112*x*c^3*d*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*
e-2*c*d)^(1/2)+15*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*b^3*e^3*g-206*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1
/2)*b^2*c*d*e^2*g+161*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*b^2*c*e^3*f+612*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c
*d)^(1/2)*b*c^2*d^2*e*g-567*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*b*c^2*d*e^2*f-526*(b*e-2*c*d)^(1/2)*(-c*e
*x-b*e+c*d)^(1/2)*c^3*d^3*g+511*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*c^3*d^2*e*f)/(e*x+d)^(1/2)/(-c*e*x-b*
e+c*d)^(1/2)/c/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 {7}{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)^(7/2),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)^(7/2), 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 )}^{7/2}} \,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)^(7/2),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)^(7/2), 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)**(7/2),x)

[Out]

Timed out

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