3.14.67 \(\int \frac {(d+e x)^{13/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=224 \[ -\frac {3003 e^5 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{15/2}}+\frac {3003 e^5 \sqrt {d+e x} (b d-a e)}{128 b^7}-\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {1001 e^5 (d+e x)^{3/2}}{128 b^6} \]

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Rubi [A]  time = 0.14, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 10, 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 {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}+\frac {3003 e^5 \sqrt {d+e x} (b d-a e)}{128 b^7}-\frac {3003 e^5 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{15/2}}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {1001 e^5 (d+e x)^{3/2}}{128 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(3003*e^5*(b*d - a*e)*Sqrt[d + e*x])/(128*b^7) + (1001*e^5*(d + e*x)^(3/2))/(128*b^6) - (3003*e^4*(d + e*x)^(5
/2))/(640*b^5*(a + b*x)) - (429*e^3*(d + e*x)^(7/2))/(320*b^4*(a + b*x)^2) - (143*e^2*(d + e*x)^(9/2))/(240*b^
3*(a + b*x)^3) - (13*e*(d + e*x)^(11/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(13/2)/(5*b*(a + b*x)^5) - (3003*e^5
*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(15/2))

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)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^{13/2}}{(a+b x)^6} \, dx\\ &=-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {(13 e) \int \frac {(d+e x)^{11/2}}{(a+b x)^5} \, dx}{10 b}\\ &=-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (143 e^2\right ) \int \frac {(d+e x)^{9/2}}{(a+b x)^4} \, dx}{80 b^2}\\ &=-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (429 e^3\right ) \int \frac {(d+e x)^{7/2}}{(a+b x)^3} \, dx}{160 b^3}\\ &=-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (3003 e^4\right ) \int \frac {(d+e x)^{5/2}}{(a+b x)^2} \, dx}{640 b^4}\\ &=-\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (3003 e^5\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{256 b^5}\\ &=\frac {1001 e^5 (d+e x)^{3/2}}{128 b^6}-\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (3003 e^5 (b d-a e)\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{256 b^6}\\ &=\frac {3003 e^5 (b d-a e) \sqrt {d+e x}}{128 b^7}+\frac {1001 e^5 (d+e x)^{3/2}}{128 b^6}-\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (3003 e^5 (b d-a e)^2\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^7}\\ &=\frac {3003 e^5 (b d-a e) \sqrt {d+e x}}{128 b^7}+\frac {1001 e^5 (d+e x)^{3/2}}{128 b^6}-\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (3003 e^4 (b d-a e)^2\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 b^7}\\ &=\frac {3003 e^5 (b d-a e) \sqrt {d+e x}}{128 b^7}+\frac {1001 e^5 (d+e x)^{3/2}}{128 b^6}-\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}-\frac {3003 e^5 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{15/2}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 52, normalized size = 0.23 \begin {gather*} \frac {2 e^5 (d+e x)^{15/2} \, _2F_1\left (6,\frac {15}{2};\frac {17}{2};-\frac {b (d+e x)}{a e-b d}\right )}{15 (a e-b d)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*e^5*(d + e*x)^(15/2)*Hypergeometric2F1[6, 15/2, 17/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(15*(-(b*d) + a*e)^
6)

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IntegrateAlgebraic [B]  time = 2.76, size = 531, normalized size = 2.37 \begin {gather*} \frac {e^5 \sqrt {d+e x} \left (-45045 a^6 e^6-210210 a^5 b e^5 (d+e x)+270270 a^5 b d e^5-675675 a^4 b^2 d^2 e^4-384384 a^4 b^2 e^4 (d+e x)^2+1051050 a^4 b^2 d e^4 (d+e x)+900900 a^3 b^3 d^3 e^3-2102100 a^3 b^3 d^2 e^3 (d+e x)-338910 a^3 b^3 e^3 (d+e x)^3+1537536 a^3 b^3 d e^3 (d+e x)^2-675675 a^2 b^4 d^4 e^2+2102100 a^2 b^4 d^3 e^2 (d+e x)-2306304 a^2 b^4 d^2 e^2 (d+e x)^2-137995 a^2 b^4 e^2 (d+e x)^4+1016730 a^2 b^4 d e^2 (d+e x)^3+270270 a b^5 d^5 e-1051050 a b^5 d^4 e (d+e x)+1537536 a b^5 d^3 e (d+e x)^2-1016730 a b^5 d^2 e (d+e x)^3-16640 a b^5 e (d+e x)^5+275990 a b^5 d e (d+e x)^4-45045 b^6 d^6+210210 b^6 d^5 (d+e x)-384384 b^6 d^4 (d+e x)^2+338910 b^6 d^3 (d+e x)^3-137995 b^6 d^2 (d+e x)^4+1280 b^6 (d+e x)^6+16640 b^6 d (d+e x)^5\right )}{1920 b^7 (a e+b (d+e x)-b d)^5}-\frac {3003 e^5 (b d-a e)^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 b^{15/2} \sqrt {a e-b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(e^5*Sqrt[d + e*x]*(-45045*b^6*d^6 + 270270*a*b^5*d^5*e - 675675*a^2*b^4*d^4*e^2 + 900900*a^3*b^3*d^3*e^3 - 67
5675*a^4*b^2*d^2*e^4 + 270270*a^5*b*d*e^5 - 45045*a^6*e^6 + 210210*b^6*d^5*(d + e*x) - 1051050*a*b^5*d^4*e*(d
+ e*x) + 2102100*a^2*b^4*d^3*e^2*(d + e*x) - 2102100*a^3*b^3*d^2*e^3*(d + e*x) + 1051050*a^4*b^2*d*e^4*(d + e*
x) - 210210*a^5*b*e^5*(d + e*x) - 384384*b^6*d^4*(d + e*x)^2 + 1537536*a*b^5*d^3*e*(d + e*x)^2 - 2306304*a^2*b
^4*d^2*e^2*(d + e*x)^2 + 1537536*a^3*b^3*d*e^3*(d + e*x)^2 - 384384*a^4*b^2*e^4*(d + e*x)^2 + 338910*b^6*d^3*(
d + e*x)^3 - 1016730*a*b^5*d^2*e*(d + e*x)^3 + 1016730*a^2*b^4*d*e^2*(d + e*x)^3 - 338910*a^3*b^3*e^3*(d + e*x
)^3 - 137995*b^6*d^2*(d + e*x)^4 + 275990*a*b^5*d*e*(d + e*x)^4 - 137995*a^2*b^4*e^2*(d + e*x)^4 + 16640*b^6*d
*(d + e*x)^5 - 16640*a*b^5*e*(d + e*x)^5 + 1280*b^6*(d + e*x)^6))/(1920*b^7*(-(b*d) + a*e + b*(d + e*x))^5) -
(3003*e^5*(b*d - a*e)^2*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(128*b^(15/2)*Sqrt[-(b
*d) + a*e])

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fricas [B]  time = 0.45, size = 1234, normalized size = 5.51

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(45045*(a^5*b*d*e^5 - a^6*e^6 + (b^6*d*e^5 - a*b^5*e^6)*x^5 + 5*(a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 + 10*
(a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 10*(a^3*b^3*d*e^5 - a^4*b^2*e^6)*x^2 + 5*(a^4*b^2*d*e^5 - a^5*b*e^6)*x)*sq
rt((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*(1280*b^6*e
^6*x^6 - 384*b^6*d^6 - 624*a*b^5*d^5*e - 1144*a^2*b^4*d^4*e^2 - 2574*a^3*b^3*d^3*e^3 - 9009*a^4*b^2*d^2*e^4 +
60060*a^5*b*d*e^5 - 45045*a^6*e^6 + 1280*(19*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 - 5*(7119*b^6*d^2*e^4 - 38558*a*b^5
*d*e^5 + 27599*a^2*b^4*e^6)*x^4 - 10*(2107*b^6*d^3*e^3 + 7917*a*b^5*d^2*e^4 - 46475*a^2*b^4*d*e^5 + 33891*a^3*
b^3*e^6)*x^3 - 2*(5012*b^6*d^4*e^2 + 11557*a*b^5*d^3*e^3 + 42042*a^2*b^4*d^2*e^4 - 260403*a^3*b^3*d*e^5 + 1921
92*a^4*b^2*e^6)*x^2 - 2*(1464*b^6*d^5*e + 2704*a*b^5*d^4*e^2 + 6149*a^2*b^4*d^3*e^3 + 21879*a^3*b^3*d^2*e^4 -
141141*a^4*b^2*d*e^5 + 105105*a^5*b*e^6)*x)*sqrt(e*x + d))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3
*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7), -1/1920*(45045*(a^5*b*d*e^5 - a^6*e^6 + (b^6*d*e^5 - a*b^5*e^6)*x^5 + 5*(a*
b^5*d*e^5 - a^2*b^4*e^6)*x^4 + 10*(a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 10*(a^3*b^3*d*e^5 - a^4*b^2*e^6)*x^2 + 5
*(a^4*b^2*d*e^5 - a^5*b*e^6)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e))
- (1280*b^6*e^6*x^6 - 384*b^6*d^6 - 624*a*b^5*d^5*e - 1144*a^2*b^4*d^4*e^2 - 2574*a^3*b^3*d^3*e^3 - 9009*a^4*
b^2*d^2*e^4 + 60060*a^5*b*d*e^5 - 45045*a^6*e^6 + 1280*(19*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 - 5*(7119*b^6*d^2*e^4
- 38558*a*b^5*d*e^5 + 27599*a^2*b^4*e^6)*x^4 - 10*(2107*b^6*d^3*e^3 + 7917*a*b^5*d^2*e^4 - 46475*a^2*b^4*d*e^
5 + 33891*a^3*b^3*e^6)*x^3 - 2*(5012*b^6*d^4*e^2 + 11557*a*b^5*d^3*e^3 + 42042*a^2*b^4*d^2*e^4 - 260403*a^3*b^
3*d*e^5 + 192192*a^4*b^2*e^6)*x^2 - 2*(1464*b^6*d^5*e + 2704*a*b^5*d^4*e^2 + 6149*a^2*b^4*d^3*e^3 + 21879*a^3*
b^3*d^2*e^4 - 141141*a^4*b^2*d*e^5 + 105105*a^5*b*e^6)*x)*sqrt(e*x + d))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^1
0*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7)]

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giac [B]  time = 0.27, size = 613, normalized size = 2.74 \begin {gather*} \frac {3003 \, {\left (b^{2} d^{2} e^{5} - 2 \, a b d e^{6} + a^{2} e^{7}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, \sqrt {-b^{2} d + a b e} b^{7}} - \frac {35595 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{6} d^{2} e^{5} - 121310 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{6} d^{3} e^{5} + 160384 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} d^{4} e^{5} - 96290 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} d^{5} e^{5} + 22005 \, \sqrt {x e + d} b^{6} d^{6} e^{5} - 71190 \, {\left (x e + d\right )}^{\frac {9}{2}} a b^{5} d e^{6} + 363930 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{5} d^{2} e^{6} - 641536 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{5} d^{3} e^{6} + 481450 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{5} d^{4} e^{6} - 132030 \, \sqrt {x e + d} a b^{5} d^{5} e^{6} + 35595 \, {\left (x e + d\right )}^{\frac {9}{2}} a^{2} b^{4} e^{7} - 363930 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{4} d e^{7} + 962304 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{4} d^{2} e^{7} - 962900 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{4} d^{3} e^{7} + 330075 \, \sqrt {x e + d} a^{2} b^{4} d^{4} e^{7} + 121310 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{3} b^{3} e^{8} - 641536 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{3} d e^{8} + 962900 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{3} d^{2} e^{8} - 440100 \, \sqrt {x e + d} a^{3} b^{3} d^{3} e^{8} + 160384 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{4} b^{2} e^{9} - 481450 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b^{2} d e^{9} + 330075 \, \sqrt {x e + d} a^{4} b^{2} d^{2} e^{9} + 96290 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{5} b e^{10} - 132030 \, \sqrt {x e + d} a^{5} b d e^{10} + 22005 \, \sqrt {x e + d} a^{6} e^{11}}{1920 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{7}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{12} e^{5} + 18 \, \sqrt {x e + d} b^{12} d e^{5} - 18 \, \sqrt {x e + d} a b^{11} e^{6}\right )}}{3 \, b^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

3003/128*(b^2*d^2*e^5 - 2*a*b*d*e^6 + a^2*e^7)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b
*e)*b^7) - 1/1920*(35595*(x*e + d)^(9/2)*b^6*d^2*e^5 - 121310*(x*e + d)^(7/2)*b^6*d^3*e^5 + 160384*(x*e + d)^(
5/2)*b^6*d^4*e^5 - 96290*(x*e + d)^(3/2)*b^6*d^5*e^5 + 22005*sqrt(x*e + d)*b^6*d^6*e^5 - 71190*(x*e + d)^(9/2)
*a*b^5*d*e^6 + 363930*(x*e + d)^(7/2)*a*b^5*d^2*e^6 - 641536*(x*e + d)^(5/2)*a*b^5*d^3*e^6 + 481450*(x*e + d)^
(3/2)*a*b^5*d^4*e^6 - 132030*sqrt(x*e + d)*a*b^5*d^5*e^6 + 35595*(x*e + d)^(9/2)*a^2*b^4*e^7 - 363930*(x*e + d
)^(7/2)*a^2*b^4*d*e^7 + 962304*(x*e + d)^(5/2)*a^2*b^4*d^2*e^7 - 962900*(x*e + d)^(3/2)*a^2*b^4*d^3*e^7 + 3300
75*sqrt(x*e + d)*a^2*b^4*d^4*e^7 + 121310*(x*e + d)^(7/2)*a^3*b^3*e^8 - 641536*(x*e + d)^(5/2)*a^3*b^3*d*e^8 +
962900*(x*e + d)^(3/2)*a^3*b^3*d^2*e^8 - 440100*sqrt(x*e + d)*a^3*b^3*d^3*e^8 + 160384*(x*e + d)^(5/2)*a^4*b^
2*e^9 - 481450*(x*e + d)^(3/2)*a^4*b^2*d*e^9 + 330075*sqrt(x*e + d)*a^4*b^2*d^2*e^9 + 96290*(x*e + d)^(3/2)*a^
5*b*e^10 - 132030*sqrt(x*e + d)*a^5*b*d*e^10 + 22005*sqrt(x*e + d)*a^6*e^11)/(((x*e + d)*b - b*d + a*e)^5*b^7)
+ 2/3*((x*e + d)^(3/2)*b^12*e^5 + 18*sqrt(x*e + d)*b^12*d*e^5 - 18*sqrt(x*e + d)*a*b^11*e^6)/b^18

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maple [B]  time = 0.08, size = 908, normalized size = 4.05 \begin {gather*} -\frac {1467 \sqrt {e x +d}\, a^{6} e^{11}}{128 \left (b e x +a e \right )^{5} b^{7}}+\frac {4401 \sqrt {e x +d}\, a^{5} d \,e^{10}}{64 \left (b e x +a e \right )^{5} b^{6}}-\frac {22005 \sqrt {e x +d}\, a^{4} d^{2} e^{9}}{128 \left (b e x +a e \right )^{5} b^{5}}+\frac {7335 \sqrt {e x +d}\, a^{3} d^{3} e^{8}}{32 \left (b e x +a e \right )^{5} b^{4}}-\frac {22005 \sqrt {e x +d}\, a^{2} d^{4} e^{7}}{128 \left (b e x +a e \right )^{5} b^{3}}+\frac {4401 \sqrt {e x +d}\, a \,d^{5} e^{6}}{64 \left (b e x +a e \right )^{5} b^{2}}-\frac {1467 \sqrt {e x +d}\, d^{6} e^{5}}{128 \left (b e x +a e \right )^{5} b}-\frac {9629 \left (e x +d \right )^{\frac {3}{2}} a^{5} e^{10}}{192 \left (b e x +a e \right )^{5} b^{6}}+\frac {48145 \left (e x +d \right )^{\frac {3}{2}} a^{4} d \,e^{9}}{192 \left (b e x +a e \right )^{5} b^{5}}-\frac {48145 \left (e x +d \right )^{\frac {3}{2}} a^{3} d^{2} e^{8}}{96 \left (b e x +a e \right )^{5} b^{4}}+\frac {48145 \left (e x +d \right )^{\frac {3}{2}} a^{2} d^{3} e^{7}}{96 \left (b e x +a e \right )^{5} b^{3}}-\frac {48145 \left (e x +d \right )^{\frac {3}{2}} a \,d^{4} e^{6}}{192 \left (b e x +a e \right )^{5} b^{2}}+\frac {9629 \left (e x +d \right )^{\frac {3}{2}} d^{5} e^{5}}{192 \left (b e x +a e \right )^{5} b}-\frac {1253 \left (e x +d \right )^{\frac {5}{2}} a^{4} e^{9}}{15 \left (b e x +a e \right )^{5} b^{5}}+\frac {5012 \left (e x +d \right )^{\frac {5}{2}} a^{3} d \,e^{8}}{15 \left (b e x +a e \right )^{5} b^{4}}-\frac {2506 \left (e x +d \right )^{\frac {5}{2}} a^{2} d^{2} e^{7}}{5 \left (b e x +a e \right )^{5} b^{3}}+\frac {5012 \left (e x +d \right )^{\frac {5}{2}} a \,d^{3} e^{6}}{15 \left (b e x +a e \right )^{5} b^{2}}-\frac {1253 \left (e x +d \right )^{\frac {5}{2}} d^{4} e^{5}}{15 \left (b e x +a e \right )^{5} b}-\frac {12131 \left (e x +d \right )^{\frac {7}{2}} a^{3} e^{8}}{192 \left (b e x +a e \right )^{5} b^{4}}+\frac {12131 \left (e x +d \right )^{\frac {7}{2}} a^{2} d \,e^{7}}{64 \left (b e x +a e \right )^{5} b^{3}}-\frac {12131 \left (e x +d \right )^{\frac {7}{2}} a \,d^{2} e^{6}}{64 \left (b e x +a e \right )^{5} b^{2}}+\frac {12131 \left (e x +d \right )^{\frac {7}{2}} d^{3} e^{5}}{192 \left (b e x +a e \right )^{5} b}-\frac {2373 \left (e x +d \right )^{\frac {9}{2}} a^{2} e^{7}}{128 \left (b e x +a e \right )^{5} b^{3}}+\frac {2373 \left (e x +d \right )^{\frac {9}{2}} a d \,e^{6}}{64 \left (b e x +a e \right )^{5} b^{2}}-\frac {2373 \left (e x +d \right )^{\frac {9}{2}} d^{2} e^{5}}{128 \left (b e x +a e \right )^{5} b}+\frac {3003 a^{2} e^{7} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}\, b^{7}}-\frac {3003 a d \,e^{6} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{64 \sqrt {\left (a e -b d \right ) b}\, b^{6}}+\frac {3003 d^{2} e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}\, b^{5}}-\frac {12 \sqrt {e x +d}\, a \,e^{6}}{b^{7}}+\frac {12 \sqrt {e x +d}\, d \,e^{5}}{b^{6}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} e^{5}}{3 b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-3003/64*e^6/b^6/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a*d+48145/192*e^9/b^5/(b*e*x+
a*e)^5*(e*x+d)^(3/2)*a^4*d-48145/96*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3*d^2+48145/96*e^7/b^3/(b*e*x+a*e)^5
*(e*x+d)^(3/2)*a^2*d^3-48145/192*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^4*a+4401/64*e^10/b^6/(b*e*x+a*e)^5*(e*x
+d)^(1/2)*a^5*d-22005/128*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^4*d^2+7335/32*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(1
/2)*a^3*d^3+2373/64*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(9/2)*a*d+5012/15*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^3*d-
22005/128*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^2*d^4+4401/64*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^5+12131/
64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^2*d-12131/64*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a*d^2-2506/5*e^7/b^3
/(b*e*x+a*e)^5*(e*x+d)^(5/2)*d^2*a^2+5012/15*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a*d^3-12*e^6/b^7*a*(e*x+d)^(1
/2)+12*e^5/b^6*(e*x+d)^(1/2)*d+2/3*e^5*(e*x+d)^(3/2)/b^6+3003/128*e^7/b^7/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(
1/2)/((a*e-b*d)*b)^(1/2)*b)*a^2-2373/128*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(9/2)*a^2-1253/15*e^5/b/(b*e*x+a*e)^5*(
e*x+d)^(5/2)*d^4+9629/192*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^5-1467/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^6
+12131/192*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d^3-9629/192*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^5-2373/128*e^
5/b/(b*e*x+a*e)^5*(e*x+d)^(9/2)*d^2-1253/15*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^4-1467/128*e^11/b^7/(b*e*x+a
*e)^5*(e*x+d)^(1/2)*a^6+3003/128*e^5/b^5/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*d^2-1
2131/192*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^3

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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)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^3,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?

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mupad [B]  time = 0.81, size = 711, normalized size = 3.17 \begin {gather*} \frac {2\,e^5\,{\left (d+e\,x\right )}^{3/2}}{3\,b^6}-\frac {{\left (d+e\,x\right )}^{9/2}\,\left (\frac {2373\,a^2\,b^4\,e^7}{128}-\frac {2373\,a\,b^5\,d\,e^6}{64}+\frac {2373\,b^6\,d^2\,e^5}{128}\right )+{\left (d+e\,x\right )}^{7/2}\,\left (\frac {12131\,a^3\,b^3\,e^8}{192}-\frac {12131\,a^2\,b^4\,d\,e^7}{64}+\frac {12131\,a\,b^5\,d^2\,e^6}{64}-\frac {12131\,b^6\,d^3\,e^5}{192}\right )+\sqrt {d+e\,x}\,\left (\frac {1467\,a^6\,e^{11}}{128}-\frac {4401\,a^5\,b\,d\,e^{10}}{64}+\frac {22005\,a^4\,b^2\,d^2\,e^9}{128}-\frac {7335\,a^3\,b^3\,d^3\,e^8}{32}+\frac {22005\,a^2\,b^4\,d^4\,e^7}{128}-\frac {4401\,a\,b^5\,d^5\,e^6}{64}+\frac {1467\,b^6\,d^6\,e^5}{128}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {1253\,a^4\,b^2\,e^9}{15}-\frac {5012\,a^3\,b^3\,d\,e^8}{15}+\frac {2506\,a^2\,b^4\,d^2\,e^7}{5}-\frac {5012\,a\,b^5\,d^3\,e^6}{15}+\frac {1253\,b^6\,d^4\,e^5}{15}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {9629\,a^5\,b\,e^{10}}{192}-\frac {48145\,a^4\,b^2\,d\,e^9}{192}+\frac {48145\,a^3\,b^3\,d^2\,e^8}{96}-\frac {48145\,a^2\,b^4\,d^3\,e^7}{96}+\frac {48145\,a\,b^5\,d^4\,e^6}{192}-\frac {9629\,b^6\,d^5\,e^5}{192}\right )}{\left (d+e\,x\right )\,\left (5\,a^4\,b^8\,e^4-20\,a^3\,b^9\,d\,e^3+30\,a^2\,b^{10}\,d^2\,e^2-20\,a\,b^{11}\,d^3\,e+5\,b^{12}\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^9\,e^3+30\,a^2\,b^{10}\,d\,e^2-30\,a\,b^{11}\,d^2\,e+10\,b^{12}\,d^3\right )+b^{12}\,{\left (d+e\,x\right )}^5-\left (5\,b^{12}\,d-5\,a\,b^{11}\,e\right )\,{\left (d+e\,x\right )}^4-b^{12}\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^{10}\,e^2-20\,a\,b^{11}\,d\,e+10\,b^{12}\,d^2\right )+a^5\,b^7\,e^5-5\,a^4\,b^8\,d\,e^4-10\,a^2\,b^{10}\,d^3\,e^2+10\,a^3\,b^9\,d^2\,e^3+5\,a\,b^{11}\,d^4\,e}+\frac {2\,e^5\,\left (6\,b^6\,d-6\,a\,b^5\,e\right )\,\sqrt {d+e\,x}}{b^{12}}+\frac {3003\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^5\,{\left (a\,e-b\,d\right )}^{3/2}\,\sqrt {d+e\,x}}{a^2\,e^7-2\,a\,b\,d\,e^6+b^2\,d^2\,e^5}\right )\,{\left (a\,e-b\,d\right )}^{3/2}}{128\,b^{15/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(13/2)/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

(2*e^5*(d + e*x)^(3/2))/(3*b^6) - ((d + e*x)^(9/2)*((2373*a^2*b^4*e^7)/128 + (2373*b^6*d^2*e^5)/128 - (2373*a*
b^5*d*e^6)/64) + (d + e*x)^(7/2)*((12131*a^3*b^3*e^8)/192 - (12131*b^6*d^3*e^5)/192 + (12131*a*b^5*d^2*e^6)/64
- (12131*a^2*b^4*d*e^7)/64) + (d + e*x)^(1/2)*((1467*a^6*e^11)/128 + (1467*b^6*d^6*e^5)/128 - (4401*a*b^5*d^5
*e^6)/64 + (22005*a^2*b^4*d^4*e^7)/128 - (7335*a^3*b^3*d^3*e^8)/32 + (22005*a^4*b^2*d^2*e^9)/128 - (4401*a^5*b
*d*e^10)/64) + (d + e*x)^(5/2)*((1253*a^4*b^2*e^9)/15 + (1253*b^6*d^4*e^5)/15 - (5012*a*b^5*d^3*e^6)/15 - (501
2*a^3*b^3*d*e^8)/15 + (2506*a^2*b^4*d^2*e^7)/5) + (d + e*x)^(3/2)*((9629*a^5*b*e^10)/192 - (9629*b^6*d^5*e^5)/
192 + (48145*a*b^5*d^4*e^6)/192 - (48145*a^4*b^2*d*e^9)/192 - (48145*a^2*b^4*d^3*e^7)/96 + (48145*a^3*b^3*d^2*
e^8)/96))/((d + e*x)*(5*b^12*d^4 + 5*a^4*b^8*e^4 - 20*a^3*b^9*d*e^3 + 30*a^2*b^10*d^2*e^2 - 20*a*b^11*d^3*e) -
(d + e*x)^2*(10*b^12*d^3 - 10*a^3*b^9*e^3 + 30*a^2*b^10*d*e^2 - 30*a*b^11*d^2*e) + b^12*(d + e*x)^5 - (5*b^12
*d - 5*a*b^11*e)*(d + e*x)^4 - b^12*d^5 + (d + e*x)^3*(10*b^12*d^2 + 10*a^2*b^10*e^2 - 20*a*b^11*d*e) + a^5*b^
7*e^5 - 5*a^4*b^8*d*e^4 - 10*a^2*b^10*d^3*e^2 + 10*a^3*b^9*d^2*e^3 + 5*a*b^11*d^4*e) + (2*e^5*(6*b^6*d - 6*a*b
^5*e)*(d + e*x)^(1/2))/b^12 + (3003*e^5*atan((b^(1/2)*e^5*(a*e - b*d)^(3/2)*(d + e*x)^(1/2))/(a^2*e^7 + b^2*d^
2*e^5 - 2*a*b*d*e^6))*(a*e - b*d)^(3/2))/(128*b^(15/2))

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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)**(13/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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