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

Optimal. Leaf size=162 \[ -\frac {9 e (b d-a e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}}+\frac {9 e \sqrt {d+e x} (b d-a e)^3}{b^5}+\frac {3 e (d+e x)^{3/2} (b d-a e)^2}{b^4}+\frac {9 e (d+e x)^{5/2} (b d-a e)}{5 b^3}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {9 e (d+e x)^{7/2}}{7 b^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.18, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 47, 50, 63, 208} \begin {gather*} \frac {9 e (d+e x)^{5/2} (b d-a e)}{5 b^3}+\frac {3 e (d+e x)^{3/2} (b d-a e)^2}{b^4}+\frac {9 e \sqrt {d+e x} (b d-a e)^3}{b^5}-\frac {9 e (b d-a e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}}-\frac {(d+e x)^{9/2}}{b (a+b x)}+\frac {9 e (d+e x)^{7/2}}{7 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9*e*(b*d - a*e)^3*Sqrt[d + e*x])/b^5 + (3*e*(b*d - a*e)^2*(d + e*x)^(3/2))/b^4 + (9*e*(b*d - a*e)*(d + e*x)^(
5/2))/(5*b^3) + (9*e*(d + e*x)^(7/2))/(7*b^2) - (d + e*x)^(9/2)/(b*(a + b*x)) - (9*e*(b*d - a*e)^(7/2)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.03, size = 50, normalized size = 0.31 \begin {gather*} \frac {2 e (d+e x)^{11/2} \, _2F_1\left (2,\frac {11}{2};\frac {13}{2};-\frac {b (d+e x)}{a e-b d}\right )}{11 (a e-b d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*(d + e*x)^(11/2)*Hypergeometric2F1[2, 11/2, 13/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(11*(-(b*d) + a*e)^2)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.49, size = 300, normalized size = 1.85 \begin {gather*} \frac {e \sqrt {d+e x} \left (-315 a^4 e^4-210 a^3 b e^3 (d+e x)+1260 a^3 b d e^3-1890 a^2 b^2 d^2 e^2+42 a^2 b^2 e^2 (d+e x)^2+630 a^2 b^2 d e^2 (d+e x)+1260 a b^3 d^3 e-630 a b^3 d^2 e (d+e x)-18 a b^3 e (d+e x)^3-84 a b^3 d e (d+e x)^2-315 b^4 d^4+210 b^4 d^3 (d+e x)+42 b^4 d^2 (d+e x)^2+10 b^4 (d+e x)^4+18 b^4 d (d+e x)^3\right )}{35 b^5 (a e+b (d+e x)-b d)}-\frac {9 e (b d-a e)^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{b^{11/2} \sqrt {a e-b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Sqrt[d + e*x]*(-315*b^4*d^4 + 1260*a*b^3*d^3*e - 1890*a^2*b^2*d^2*e^2 + 1260*a^3*b*d*e^3 - 315*a^4*e^4 + 21
0*b^4*d^3*(d + e*x) - 630*a*b^3*d^2*e*(d + e*x) + 630*a^2*b^2*d*e^2*(d + e*x) - 210*a^3*b*e^3*(d + e*x) + 42*b
^4*d^2*(d + e*x)^2 - 84*a*b^3*d*e*(d + e*x)^2 + 42*a^2*b^2*e^2*(d + e*x)^2 + 18*b^4*d*(d + e*x)^3 - 18*a*b^3*e
*(d + e*x)^3 + 10*b^4*(d + e*x)^4))/(35*b^5*(-(b*d) + a*e + b*(d + e*x))) - (9*e*(b*d - a*e)^4*ArcTan[(Sqrt[b]
*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(b^(11/2)*Sqrt[-(b*d) + a*e])

________________________________________________________________________________________

fricas [B]  time = 0.43, size = 678, normalized size = 4.19 \begin {gather*} \left [-\frac {315 \, {\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} + {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (10 \, b^{4} e^{4} x^{4} - 35 \, b^{4} d^{4} + 528 \, a b^{3} d^{3} e - 1218 \, a^{2} b^{2} d^{2} e^{2} + 1050 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 2 \, {\left (29 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (26 \, b^{4} d^{2} e^{2} - 23 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (194 \, b^{4} d^{3} e - 426 \, a b^{3} d^{2} e^{2} + 357 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{70 \, {\left (b^{6} x + a b^{5}\right )}}, -\frac {315 \, {\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} + {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (10 \, b^{4} e^{4} x^{4} - 35 \, b^{4} d^{4} + 528 \, a b^{3} d^{3} e - 1218 \, a^{2} b^{2} d^{2} e^{2} + 1050 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 2 \, {\left (29 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (26 \, b^{4} d^{2} e^{2} - 23 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (194 \, b^{4} d^{3} e - 426 \, a b^{3} d^{2} e^{2} + 357 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (b^{6} x + a b^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/70*(315*(a*b^3*d^3*e - 3*a^2*b^2*d^2*e^2 + 3*a^3*b*d*e^3 - a^4*e^4 + (b^4*d^3*e - 3*a*b^3*d^2*e^2 + 3*a^2*
b^2*d*e^3 - a^3*b*e^4)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b)
)/(b*x + a)) - 2*(10*b^4*e^4*x^4 - 35*b^4*d^4 + 528*a*b^3*d^3*e - 1218*a^2*b^2*d^2*e^2 + 1050*a^3*b*d*e^3 - 31
5*a^4*e^4 + 2*(29*b^4*d*e^3 - 9*a*b^3*e^4)*x^3 + 6*(26*b^4*d^2*e^2 - 23*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 + 2*(
194*b^4*d^3*e - 426*a*b^3*d^2*e^2 + 357*a^2*b^2*d*e^3 - 105*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^6*x + a*b^5), -1/3
5*(315*(a*b^3*d^3*e - 3*a^2*b^2*d^2*e^2 + 3*a^3*b*d*e^3 - a^4*e^4 + (b^4*d^3*e - 3*a*b^3*d^2*e^2 + 3*a^2*b^2*d
*e^3 - a^3*b*e^4)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (10*b^4*
e^4*x^4 - 35*b^4*d^4 + 528*a*b^3*d^3*e - 1218*a^2*b^2*d^2*e^2 + 1050*a^3*b*d*e^3 - 315*a^4*e^4 + 2*(29*b^4*d*e
^3 - 9*a*b^3*e^4)*x^3 + 6*(26*b^4*d^2*e^2 - 23*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 + 2*(194*b^4*d^3*e - 426*a*b^3
*d^2*e^2 + 357*a^2*b^2*d*e^3 - 105*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^6*x + a*b^5)]

________________________________________________________________________________________

giac [B]  time = 0.20, size = 387, normalized size = 2.39 \begin {gather*} \frac {9 \, {\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{5}} - \frac {\sqrt {x e + d} b^{4} d^{4} e - 4 \, \sqrt {x e + d} a b^{3} d^{3} e^{2} + 6 \, \sqrt {x e + d} a^{2} b^{2} d^{2} e^{3} - 4 \, \sqrt {x e + d} a^{3} b d e^{4} + \sqrt {x e + d} a^{4} e^{5}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{5}} + \frac {2 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{12} e + 14 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{12} d e + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{12} d^{2} e + 140 \, \sqrt {x e + d} b^{12} d^{3} e - 14 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{11} e^{2} - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{11} d e^{2} - 420 \, \sqrt {x e + d} a b^{11} d^{2} e^{2} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{10} e^{3} + 420 \, \sqrt {x e + d} a^{2} b^{10} d e^{3} - 140 \, \sqrt {x e + d} a^{3} b^{9} e^{4}\right )}}{35 \, b^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9*(b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2
*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5) - (sqrt(x*e + d)*b^4*d^4*e - 4*sqrt(x*e + d)*a*b^3*d^3*e^2 + 6*sqrt(x*
e + d)*a^2*b^2*d^2*e^3 - 4*sqrt(x*e + d)*a^3*b*d*e^4 + sqrt(x*e + d)*a^4*e^5)/(((x*e + d)*b - b*d + a*e)*b^5)
+ 2/35*(5*(x*e + d)^(7/2)*b^12*e + 14*(x*e + d)^(5/2)*b^12*d*e + 35*(x*e + d)^(3/2)*b^12*d^2*e + 140*sqrt(x*e
+ d)*b^12*d^3*e - 14*(x*e + d)^(5/2)*a*b^11*e^2 - 70*(x*e + d)^(3/2)*a*b^11*d*e^2 - 420*sqrt(x*e + d)*a*b^11*d
^2*e^2 + 35*(x*e + d)^(3/2)*a^2*b^10*e^3 + 420*sqrt(x*e + d)*a^2*b^10*d*e^3 - 140*sqrt(x*e + d)*a^3*b^9*e^4)/b
^14

________________________________________________________________________________________

maple [B]  time = 0.09, size = 539, normalized size = 3.33 \begin {gather*} \frac {9 a^{4} e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{5}}-\frac {36 a^{3} d \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {54 a^{2} d^{2} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{3}}-\frac {36 a \,d^{3} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{2}}+\frac {9 d^{4} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b}-\frac {\sqrt {e x +d}\, a^{4} e^{5}}{\left (b e x +a e \right ) b^{5}}+\frac {4 \sqrt {e x +d}\, a^{3} d \,e^{4}}{\left (b e x +a e \right ) b^{4}}-\frac {6 \sqrt {e x +d}\, a^{2} d^{2} e^{3}}{\left (b e x +a e \right ) b^{3}}+\frac {4 \sqrt {e x +d}\, a \,d^{3} e^{2}}{\left (b e x +a e \right ) b^{2}}-\frac {\sqrt {e x +d}\, d^{4} e}{\left (b e x +a e \right ) b}-\frac {8 \sqrt {e x +d}\, a^{3} e^{4}}{b^{5}}+\frac {24 \sqrt {e x +d}\, a^{2} d \,e^{3}}{b^{4}}-\frac {24 \sqrt {e x +d}\, a \,d^{2} e^{2}}{b^{3}}+\frac {8 \sqrt {e x +d}\, d^{3} e}{b^{2}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} a^{2} e^{3}}{b^{4}}-\frac {4 \left (e x +d \right )^{\frac {3}{2}} a d \,e^{2}}{b^{3}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} d^{2} e}{b^{2}}-\frac {4 \left (e x +d \right )^{\frac {5}{2}} a \,e^{2}}{5 b^{3}}+\frac {4 \left (e x +d \right )^{\frac {5}{2}} d e}{5 b^{2}}+\frac {2 \left (e x +d \right )^{\frac {7}{2}} e}{7 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*e*(e*x+d)^(7/2)/b^2-4/5/b^3*(e*x+d)^(5/2)*a*e^2+4/5*e/b^2*(e*x+d)^(5/2)*d+2/b^4*(e*x+d)^(3/2)*a^2*e^3-4/b^
3*(e*x+d)^(3/2)*a*d*e^2+2*e/b^2*(e*x+d)^(3/2)*d^2-8/b^5*e^4*a^3*(e*x+d)^(1/2)+24/b^4*a^2*d*e^3*(e*x+d)^(1/2)-2
4/b^3*a*d^2*e^2*(e*x+d)^(1/2)+8*e/b^2*d^3*(e*x+d)^(1/2)-1/b^5*(e*x+d)^(1/2)/(b*e*x+a*e)*a^4*e^5+4/b^4*(e*x+d)^
(1/2)/(b*e*x+a*e)*a^3*d*e^4-6/b^3*(e*x+d)^(1/2)/(b*e*x+a*e)*a^2*d^2*e^3+4/b^2*(e*x+d)^(1/2)/(b*e*x+a*e)*a*d^3*
e^2-e/b*(e*x+d)^(1/2)/(b*e*x+a*e)*d^4+9/b^5/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^
4*e^5-36/b^4/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*d*e^4+54/b^3/(b*(a*e-b*d))^(1
/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*d^2*e^3-36/b^2/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/
(b*(a*e-b*d))^(1/2))*a*d^3*e^2+9*e/b/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*d^4

________________________________________________________________________________________

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

________________________________________________________________________________________

mupad [B]  time = 0.61, size = 352, normalized size = 2.17 \begin {gather*} \left (\frac {\left (\frac {2\,e\,{\left (2\,b^2\,d-2\,a\,b\,e\right )}^2}{b^6}-\frac {2\,e\,{\left (a\,e-b\,d\right )}^2}{b^4}\right )\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^2}-\frac {2\,e\,\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (a\,e-b\,d\right )}^2}{b^6}\right )\,\sqrt {d+e\,x}+\left (\frac {2\,e\,{\left (2\,b^2\,d-2\,a\,b\,e\right )}^2}{3\,b^6}-\frac {2\,e\,{\left (a\,e-b\,d\right )}^2}{3\,b^4}\right )\,{\left (d+e\,x\right )}^{3/2}-\frac {\sqrt {d+e\,x}\,\left (a^4\,e^5-4\,a^3\,b\,d\,e^4+6\,a^2\,b^2\,d^2\,e^3-4\,a\,b^3\,d^3\,e^2+b^4\,d^4\,e\right )}{b^6\,\left (d+e\,x\right )-b^6\,d+a\,b^5\,e}+\frac {2\,e\,{\left (d+e\,x\right )}^{7/2}}{7\,b^2}+\frac {2\,e\,\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,b^4}+\frac {9\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,{\left (a\,e-b\,d\right )}^{7/2}\,\sqrt {d+e\,x}}{a^4\,e^5-4\,a^3\,b\,d\,e^4+6\,a^2\,b^2\,d^2\,e^3-4\,a\,b^3\,d^3\,e^2+b^4\,d^4\,e}\right )\,{\left (a\,e-b\,d\right )}^{7/2}}{b^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((((2*e*(2*b^2*d - 2*a*b*e)^2)/b^6 - (2*e*(a*e - b*d)^2)/b^4)*(2*b^2*d - 2*a*b*e))/b^2 - (2*e*(2*b^2*d - 2*a*b
*e)*(a*e - b*d)^2)/b^6)*(d + e*x)^(1/2) + ((2*e*(2*b^2*d - 2*a*b*e)^2)/(3*b^6) - (2*e*(a*e - b*d)^2)/(3*b^4))*
(d + e*x)^(3/2) - ((d + e*x)^(1/2)*(a^4*e^5 + b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4)
)/(b^6*(d + e*x) - b^6*d + a*b^5*e) + (2*e*(d + e*x)^(7/2))/(7*b^2) + (2*e*(2*b^2*d - 2*a*b*e)*(d + e*x)^(5/2)
)/(5*b^4) + (9*e*atan((b^(1/2)*e*(a*e - b*d)^(7/2)*(d + e*x)^(1/2))/(a^4*e^5 + b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6
*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4))*(a*e - b*d)^(7/2))/b^(11/2)

________________________________________________________________________________________

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)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Timed out

________________________________________________________________________________________