3.4.47 \(\int \frac {(d+e x)^{9/2}}{(b x+c x^2)^2} \, dx\)

Optimal. Leaf size=251 \[ -\frac {(c d-b e)^{7/2} (5 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{7/2}}+\frac {d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}+\frac {e (d+e x)^{3/2} \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 b^2 c^2}+\frac {e \sqrt {d+e x} (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{b^2 c^3}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}+\frac {e (d+e x)^{5/2} (2 c d-b e)}{b^2 c} \]

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Rubi [A]  time = 0.53, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {738, 824, 826, 1166, 208} \begin {gather*} \frac {e (d+e x)^{3/2} \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 b^2 c^2}+\frac {e \sqrt {d+e x} (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{b^2 c^3}-\frac {(c d-b e)^{7/2} (5 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{7/2}}+\frac {d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(d+e x)^{7/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}+\frac {e (d+e x)^{5/2} (2 c d-b e)}{b^2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(9/2)/(b*x + c*x^2)^2,x]

[Out]

(e*(2*c*d - b*e)*(c^2*d^2 - b*c*d*e + 5*b^2*e^2)*Sqrt[d + e*x])/(b^2*c^3) + (e*(6*c^2*d^2 - 6*b*c*d*e + 5*b^2*
e^2)*(d + e*x)^(3/2))/(3*b^2*c^2) + (e*(2*c*d - b*e)*(d + e*x)^(5/2))/(b^2*c) - ((d + e*x)^(7/2)*(b*d + (2*c*d
- b*e)*x))/(b^2*(b*x + c*x^2)) + (d^(7/2)*(4*c*d - 9*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 - ((c*d - b*e)^
(7/2)*(4*c*d + 5*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(7/2))

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 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, 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] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 824

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

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {(d+e x)^{5/2} \left (\frac {1}{2} d (4 c d-9 b e)-\frac {5}{2} e (2 c d-b e) x\right )}{b x+c x^2} \, dx}{b^2}\\ &=\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} c d^2 (4 c d-9 b e)-\frac {1}{2} e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) x\right )}{b x+c x^2} \, dx}{b^2 c}\\ &=\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} c^2 d^3 (4 c d-9 b e)-\frac {1}{2} e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) x\right )}{b x+c x^2} \, dx}{b^2 c^2}\\ &=\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} c^3 d^4 (4 c d-9 b e)+\frac {1}{2} e \left (2 c^4 d^4-4 b c^3 d^3 e-14 b^2 c^2 d^2 e^2+16 b^3 c d e^3-5 b^4 e^4\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 c^3}\\ &=\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {\frac {1}{2} c^3 d^4 e (4 c d-9 b e)-\frac {1}{2} d e \left (2 c^4 d^4-4 b c^3 d^3 e-14 b^2 c^2 d^2 e^2+16 b^3 c d e^3-5 b^4 e^4\right )+\frac {1}{2} e \left (2 c^4 d^4-4 b c^3 d^3 e-14 b^2 c^2 d^2 e^2+16 b^3 c d e^3-5 b^4 e^4\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^2 c^3}\\ &=\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\left (c d^4 (4 c d-9 b e)\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{b^3}+\frac {\left ((c d-b e)^4 (4 c d+5 b e)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{b^3 c^3}\\ &=\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^3}+\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) (d+e x)^{3/2}}{3 b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{5/2}}{b^2 c}-\frac {(d+e x)^{7/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}+\frac {d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(c d-b e)^{7/2} (4 c d+5 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.37, size = 202, normalized size = 0.80 \begin {gather*} \frac {\frac {b \sqrt {d+e x} \left (-15 b^4 e^4 x+2 b^3 c e^3 x (19 d-5 e x)+2 b^2 c^2 e^2 x \left (-9 d^2+13 d e x+e^2 x^2\right )-3 b c^3 d^3 (d-4 e x)-6 c^4 d^4 x\right )}{c^3 x (b+c x)}-\frac {3 (c d-b e)^{7/2} (5 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{c^{7/2}}+3 d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{3 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(9/2)/(b*x + c*x^2)^2,x]

[Out]

((b*Sqrt[d + e*x]*(-6*c^4*d^4*x - 15*b^4*e^4*x + 2*b^3*c*e^3*x*(19*d - 5*e*x) - 3*b*c^3*d^3*(d - 4*e*x) + 2*b^
2*c^2*e^2*x*(-9*d^2 + 13*d*e*x + e^2*x^2)))/(c^3*x*(b + c*x)) + 3*d^(7/2)*(4*c*d - 9*b*e)*ArcTanh[Sqrt[d + e*x
]/Sqrt[d]] - (3*(c*d - b*e)^(7/2)*(4*c*d + 5*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/c^(7/2))/(
3*b^3)

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IntegrateAlgebraic [A]  time = 0.59, size = 329, normalized size = 1.31 \begin {gather*} -\frac {(5 b e+4 c d) (b e-c d)^{7/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b^3 c^{7/2}}+\frac {\left (4 c d^{9/2}-9 b d^{7/2} e\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {\sqrt {d+e x} \left (15 b^4 e^4 (d+e x)-15 b^4 d e^4+48 b^3 c d^2 e^3-58 b^3 c d e^3 (d+e x)+10 b^3 c e^3 (d+e x)^2-42 b^2 c^2 d^3 e^2+64 b^2 c^2 d^2 e^2 (d+e x)-20 b^2 c^2 d e^2 (d+e x)^2-2 b^2 c^2 e^2 (d+e x)^3+15 b c^3 d^4 e-12 b c^3 d^3 e (d+e x)-6 c^4 d^5+6 c^4 d^4 (d+e x)\right )}{3 b^2 c^3 x (b e+c (d+e x)-c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(9/2)/(b*x + c*x^2)^2,x]

[Out]

-1/3*(Sqrt[d + e*x]*(-6*c^4*d^5 + 15*b*c^3*d^4*e - 42*b^2*c^2*d^3*e^2 + 48*b^3*c*d^2*e^3 - 15*b^4*d*e^4 + 6*c^
4*d^4*(d + e*x) - 12*b*c^3*d^3*e*(d + e*x) + 64*b^2*c^2*d^2*e^2*(d + e*x) - 58*b^3*c*d*e^3*(d + e*x) + 15*b^4*
e^4*(d + e*x) - 20*b^2*c^2*d*e^2*(d + e*x)^2 + 10*b^3*c*e^3*(d + e*x)^2 - 2*b^2*c^2*e^2*(d + e*x)^3))/(b^2*c^3
*x*(-(c*d) + b*e + c*(d + e*x))) - ((-(c*d) + b*e)^(7/2)*(4*c*d + 5*b*e)*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sq
rt[d + e*x])/(c*d - b*e)])/(b^3*c^(7/2)) + ((4*c*d^(9/2) - 9*b*d^(7/2)*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3

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fricas [A]  time = 3.33, size = 1561, normalized size = 6.22

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

[-1/6*(3*((4*c^5*d^4 - 7*b*c^4*d^3*e - 3*b^2*c^3*d^2*e^2 + 11*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 + (4*b*c^4*d^4
- 7*b^2*c^3*d^3*e - 3*b^3*c^2*d^2*e^2 + 11*b^4*c*d*e^3 - 5*b^5*e^4)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d
- b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + 3*((4*c^5*d^4 - 9*b*c^4*d^3*e)*x^2 + (4*b*c^4*d^4
- 9*b^2*c^3*d^3*e)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^3*c^2*e^4*x^3 - 3*b^2*c^3*
d^4 + 2*(13*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 - (6*b*c^4*d^4 - 12*b^2*c^3*d^3*e + 18*b^3*c^2*d^2*e^2 - 38*b^4*c
*d*e^3 + 15*b^5*e^4)*x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*c^3*x), -1/6*(6*((4*c^5*d^4 - 7*b*c^4*d^3*e - 3*b^2*
c^3*d^2*e^2 + 11*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 + (4*b*c^4*d^4 - 7*b^2*c^3*d^3*e - 3*b^3*c^2*d^2*e^2 + 11*b^
4*c*d*e^3 - 5*b^5*e^4)*x)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3*(
(4*c^5*d^4 - 9*b*c^4*d^3*e)*x^2 + (4*b*c^4*d^4 - 9*b^2*c^3*d^3*e)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d
) + 2*d)/x) - 2*(2*b^3*c^2*e^4*x^3 - 3*b^2*c^3*d^4 + 2*(13*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 - (6*b*c^4*d^4 - 1
2*b^2*c^3*d^3*e + 18*b^3*c^2*d^2*e^2 - 38*b^4*c*d*e^3 + 15*b^5*e^4)*x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*c^3*x
), -1/6*(6*((4*c^5*d^4 - 9*b*c^4*d^3*e)*x^2 + (4*b*c^4*d^4 - 9*b^2*c^3*d^3*e)*x)*sqrt(-d)*arctan(sqrt(e*x + d)
*sqrt(-d)/d) + 3*((4*c^5*d^4 - 7*b*c^4*d^3*e - 3*b^2*c^3*d^2*e^2 + 11*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 + (4*b*
c^4*d^4 - 7*b^2*c^3*d^3*e - 3*b^3*c^2*d^2*e^2 + 11*b^4*c*d*e^3 - 5*b^5*e^4)*x)*sqrt((c*d - b*e)/c)*log((c*e*x
+ 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - 2*(2*b^3*c^2*e^4*x^3 - 3*b^2*c^3*d^4 + 2*(
13*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 - (6*b*c^4*d^4 - 12*b^2*c^3*d^3*e + 18*b^3*c^2*d^2*e^2 - 38*b^4*c*d*e^3 +
15*b^5*e^4)*x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*c^3*x), -1/3*(3*((4*c^5*d^4 - 7*b*c^4*d^3*e - 3*b^2*c^3*d^2*e
^2 + 11*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 + (4*b*c^4*d^4 - 7*b^2*c^3*d^3*e - 3*b^3*c^2*d^2*e^2 + 11*b^4*c*d*e^3
- 5*b^5*e^4)*x)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3*((4*c^5*d^
4 - 9*b*c^4*d^3*e)*x^2 + (4*b*c^4*d^4 - 9*b^2*c^3*d^3*e)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (2*b^3
*c^2*e^4*x^3 - 3*b^2*c^3*d^4 + 2*(13*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 - (6*b*c^4*d^4 - 12*b^2*c^3*d^3*e + 18*b
^3*c^2*d^2*e^2 - 38*b^4*c*d*e^3 + 15*b^5*e^4)*x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*c^3*x)]

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giac [A]  time = 0.28, size = 436, normalized size = 1.74 \begin {gather*} -\frac {{\left (4 \, c d^{5} - 9 \, b d^{4} e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} + \frac {{\left (4 \, c^{5} d^{5} - 11 \, b c^{4} d^{4} e + 4 \, b^{2} c^{3} d^{3} e^{2} + 14 \, b^{3} c^{2} d^{2} e^{3} - 16 \, b^{4} c d e^{4} + 5 \, b^{5} e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c^{3}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{4} e^{3} + 12 \, \sqrt {x e + d} c^{4} d e^{3} - 6 \, \sqrt {x e + d} b c^{3} e^{4}\right )}}{3 \, c^{6}} - \frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{4} d^{4} e - 2 \, \sqrt {x e + d} c^{4} d^{5} e - 4 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{3} d^{3} e^{2} + 5 \, \sqrt {x e + d} b c^{3} d^{4} e^{2} + 6 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{2} e^{3} - 6 \, \sqrt {x e + d} b^{2} c^{2} d^{3} e^{3} - 4 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c d e^{4} + 4 \, \sqrt {x e + d} b^{3} c d^{2} e^{4} + {\left (x e + d\right )}^{\frac {3}{2}} b^{4} e^{5} - \sqrt {x e + d} b^{4} d e^{5}}{{\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )} b^{2} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

-(4*c*d^5 - 9*b*d^4*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) + (4*c^5*d^5 - 11*b*c^4*d^4*e + 4*b^2*c^3
*d^3*e^2 + 14*b^3*c^2*d^2*e^3 - 16*b^4*c*d*e^4 + 5*b^5*e^5)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt
(-c^2*d + b*c*e)*b^3*c^3) + 2/3*((x*e + d)^(3/2)*c^4*e^3 + 12*sqrt(x*e + d)*c^4*d*e^3 - 6*sqrt(x*e + d)*b*c^3*
e^4)/c^6 - (2*(x*e + d)^(3/2)*c^4*d^4*e - 2*sqrt(x*e + d)*c^4*d^5*e - 4*(x*e + d)^(3/2)*b*c^3*d^3*e^2 + 5*sqrt
(x*e + d)*b*c^3*d^4*e^2 + 6*(x*e + d)^(3/2)*b^2*c^2*d^2*e^3 - 6*sqrt(x*e + d)*b^2*c^2*d^3*e^3 - 4*(x*e + d)^(3
/2)*b^3*c*d*e^4 + 4*sqrt(x*e + d)*b^3*c*d^2*e^4 + (x*e + d)^(3/2)*b^4*e^5 - sqrt(x*e + d)*b^4*d*e^5)/(((x*e +
d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)*b^2*c^3)

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maple [B]  time = 0.08, size = 515, normalized size = 2.05 \begin {gather*} \frac {5 b^{2} e^{5} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, c^{3}}-\frac {16 b d \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, c^{2}}+\frac {4 d^{3} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b}-\frac {11 c \,d^{4} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{2}}+\frac {4 c^{2} d^{5} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {14 d^{2} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, c}-\frac {\sqrt {e x +d}\, b^{2} e^{5}}{\left (c e x +b e \right ) c^{3}}+\frac {4 \sqrt {e x +d}\, b d \,e^{4}}{\left (c e x +b e \right ) c^{2}}+\frac {4 \sqrt {e x +d}\, d^{3} e^{2}}{\left (c e x +b e \right ) b}-\frac {\sqrt {e x +d}\, c \,d^{4} e}{\left (c e x +b e \right ) b^{2}}-\frac {6 \sqrt {e x +d}\, d^{2} e^{3}}{\left (c e x +b e \right ) c}-\frac {9 d^{\frac {7}{2}} e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2}}+\frac {4 c \,d^{\frac {9}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3}}-\frac {4 \sqrt {e x +d}\, b \,e^{4}}{c^{3}}+\frac {8 \sqrt {e x +d}\, d \,e^{3}}{c^{2}}-\frac {\sqrt {e x +d}\, d^{4}}{b^{2} x}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} e^{3}}{3 c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(9/2)/(c*x^2+b*x)^2,x)

[Out]

2/3*e^3/c^2*(e*x+d)^(3/2)-4*e^4/c^3*(e*x+d)^(1/2)*b+8*e^3/c^2*(e*x+d)^(1/2)*d-e^5/c^3*b^2*(e*x+d)^(1/2)/(c*e*x
+b*e)+4*e^4/c^2*b*(e*x+d)^(1/2)/(c*e*x+b*e)*d-6*e^3/c*(e*x+d)^(1/2)/(c*e*x+b*e)*d^2+4*e^2/b*(e*x+d)^(1/2)/(c*e
*x+b*e)*d^3-e*c/b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*d^4+5*e^5/c^3*b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e
-c*d)*c)^(1/2)*c)-16*e^4/c^2*b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*d+14*e^3/c/((b*
e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*d^2+4*e^2/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/
2)/((b*e-c*d)*c)^(1/2)*c)*d^3-11*e*c/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*d^4+4
*c^2/b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*d^5-d^4/b^2*(e*x+d)^(1/2)/x-9*e*d^(7/
2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))+4*d^(9/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*c

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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((e*x+d)^(9/2)/(c*x^2+b*x)^2,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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

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mupad [B]  time = 1.02, size = 3360, normalized size = 13.39

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(9/2)/(b*x + c*x^2)^2,x)

[Out]

(((d + e*x)^(3/2)*(b^4*e^5 + 2*c^4*d^4*e - 4*b*c^3*d^3*e^2 + 6*b^2*c^2*d^2*e^3 - 4*b^3*c*d*e^4))/b^2 - ((d + e
*x)^(1/2)*(b^4*d*e^5 + 2*c^4*d^5*e - 5*b*c^3*d^4*e^2 - 4*b^3*c*d^2*e^4 + 6*b^2*c^2*d^3*e^3))/b^2)/((2*c^4*d -
b*c^3*e)*(d + e*x) - c^4*(d + e*x)^2 - c^4*d^2 + b*c^3*d*e) + (2*e^3*(d + e*x)^(3/2))/(3*c^2) + (2*e^3*(4*c^2*
d - 2*b*c*e)*(d + e*x)^(1/2))/c^4 - (atan(((((((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 5
6*b^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(9*b*e - 4*c*d)*(d^7)^(
1/2)*(d + e*x)^(1/2))/(b^7*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2))/(2*b^3) + (2*(d + e*x)^(1/2)*(25*b^10*e^12 + 32*
c^10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*e^6 + 504*b^5*c
^5*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c*d*e^11))/(b^4*c^5))*(
9*b*e - 4*c*d)*(d^7)^(1/2)*1i)/(2*b^3) - (((((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 56*
b^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) - ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(9*b*e - 4*c*d)*(d^7)^(1/
2)*(d + e*x)^(1/2))/(b^7*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2))/(2*b^3) - (2*(d + e*x)^(1/2)*(25*b^10*e^12 + 32*c^
10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*e^6 + 504*b^5*c^5
*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c*d*e^11))/(b^4*c^5))*(9*
b*e - 4*c*d)*(d^7)^(1/2)*1i)/(2*b^3))/((((((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 56*b^
8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(9*b*e - 4*c*d)*(d^7)^(1/2)
*(d + e*x)^(1/2))/(b^7*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2))/(2*b^3) + (2*(d + e*x)^(1/2)*(25*b^10*e^12 + 32*c^10
*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*e^6 + 504*b^5*c^5*d
^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c*d*e^11))/(b^4*c^5))*(9*b*
e - 4*c*d)*(d^7)^(1/2))/(2*b^3) - (2*(225*b^10*d^4*e^13 + 32*c^10*d^14*e^3 - 224*b*c^9*d^13*e^4 - 1540*b^9*c*d
^5*e^12 + 326*b^2*c^8*d^12*e^5 + 956*b^3*c^7*d^11*e^6 - 3430*b^4*c^6*d^10*e^7 + 3048*b^5*c^5*d^9*e^8 + 1659*b^
6*c^4*d^8*e^9 - 5256*b^7*c^3*d^7*e^10 + 4204*b^8*c^2*d^6*e^11))/(b^6*c^5) + (((((20*b^10*c^4*d*e^7 + 8*b^6*c^8
*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 56*b^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) - ((4*b^7*c^7*e^3 - 8*b^6*c
^8*d*e^2)*(9*b*e - 4*c*d)*(d^7)^(1/2)*(d + e*x)^(1/2))/(b^7*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2))/(2*b^3) - (2*(d
+ e*x)^(1/2)*(25*b^10*e^12 + 32*c^10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5
- 420*b^4*c^6*d^6*e^6 + 504*b^5*c^5*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10
- 160*b^9*c*d*e^11))/(b^4*c^5))*(9*b*e - 4*c*d)*(d^7)^(1/2))/(2*b^3)))*(9*b*e - 4*c*d)*(d^7)^(1/2)*1i)/b^3 + (
atan((((-c^7*(b*e - c*d)^7)^(1/2)*((2*(d + e*x)^(1/2)*(25*b^10*e^12 + 32*c^10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 2
34*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*e^6 + 504*b^5*c^5*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408
*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c*d*e^11))/(b^4*c^5) + ((-c^7*(b*e - c*d)^7)^(1/2)*(5*b*e +
4*c*d)*((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 56*b^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)
/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(-c^7*(b*e - c*d)^7)^(1/2)*(5*b*e + 4*c*d)*(d + e*x)^(1/2))/(b
^7*c^12)))/(2*b^3*c^7))*(5*b*e + 4*c*d)*1i)/(2*b^3*c^7) + ((-c^7*(b*e - c*d)^7)^(1/2)*((2*(d + e*x)^(1/2)*(25*
b^10*e^12 + 32*c^10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*
e^6 + 504*b^5*c^5*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c*d*e^11
))/(b^4*c^5) - ((-c^7*(b*e - c*d)^7)^(1/2)*(5*b*e + 4*c*d)*((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^
7*d^4*e^4 + 56*b^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) - ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(-c^7*(b*e
- c*d)^7)^(1/2)*(5*b*e + 4*c*d)*(d + e*x)^(1/2))/(b^7*c^12)))/(2*b^3*c^7))*(5*b*e + 4*c*d)*1i)/(2*b^3*c^7))/(
(2*(225*b^10*d^4*e^13 + 32*c^10*d^14*e^3 - 224*b*c^9*d^13*e^4 - 1540*b^9*c*d^5*e^12 + 326*b^2*c^8*d^12*e^5 + 9
56*b^3*c^7*d^11*e^6 - 3430*b^4*c^6*d^10*e^7 + 3048*b^5*c^5*d^9*e^8 + 1659*b^6*c^4*d^8*e^9 - 5256*b^7*c^3*d^7*e
^10 + 4204*b^8*c^2*d^6*e^11))/(b^6*c^5) - ((-c^7*(b*e - c*d)^7)^(1/2)*((2*(d + e*x)^(1/2)*(25*b^10*e^12 + 32*c
^10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*e^6 + 504*b^5*c^
5*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^8*c^2*d^2*e^10 - 160*b^9*c*d*e^11))/(b^4*c^5) + (
(-c^7*(b*e - c*d)^7)^(1/2)*(5*b*e + 4*c*d)*((20*b^10*c^4*d*e^7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 56*b
^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) + ((4*b^7*c^7*e^3 - 8*b^6*c^8*d*e^2)*(-c^7*(b*e - c*d)^7)^(1/2)
*(5*b*e + 4*c*d)*(d + e*x)^(1/2))/(b^7*c^12)))/(2*b^3*c^7))*(5*b*e + 4*c*d))/(2*b^3*c^7) + ((-c^7*(b*e - c*d)^
7)^(1/2)*((2*(d + e*x)^(1/2)*(25*b^10*e^12 + 32*c^10*d^10*e^2 - 160*b*c^9*d^9*e^3 + 234*b^2*c^8*d^8*e^4 + 24*b
^3*c^7*d^7*e^5 - 420*b^4*c^6*d^6*e^6 + 504*b^5*c^5*d^5*e^7 - 42*b^6*c^4*d^4*e^8 - 408*b^7*c^3*d^3*e^9 + 396*b^
8*c^2*d^2*e^10 - 160*b^9*c*d*e^11))/(b^4*c^5) - ((-c^7*(b*e - c*d)^7)^(1/2)*(5*b*e + 4*c*d)*((20*b^10*c^4*d*e^
7 + 8*b^6*c^8*d^5*e^3 - 20*b^7*c^7*d^4*e^4 + 56*b^8*c^6*d^3*e^5 - 64*b^9*c^5*d^2*e^6)/(b^6*c^5) - ((4*b^7*c^7*
e^3 - 8*b^6*c^8*d*e^2)*(-c^7*(b*e - c*d)^7)^(1/2)*(5*b*e + 4*c*d)*(d + e*x)^(1/2))/(b^7*c^12)))/(2*b^3*c^7))*(
5*b*e + 4*c*d))/(2*b^3*c^7)))*(-c^7*(b*e - c*d)^7)^(1/2)*(5*b*e + 4*c*d)*1i)/(b^3*c^7)

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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((e*x+d)**(9/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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