3.14.76 \(\int \frac {1}{(d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=266 \[ \frac {3003 b^{3/2} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{15/2}}-\frac {3003 b e^5}{128 \sqrt {d+e x} (b d-a e)^7}-\frac {1001 e^5}{128 (d+e x)^{3/2} (b d-a e)^6}-\frac {3003 e^4}{640 (a+b x) (d+e x)^{3/2} (b d-a e)^5}+\frac {429 e^3}{320 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}-\frac {143 e^2}{240 (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}+\frac {13 e}{40 (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)} \]

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Rubi [A]  time = 0.24, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \begin {gather*} \frac {3003 b^{3/2} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{15/2}}-\frac {3003 b e^5}{128 \sqrt {d+e x} (b d-a e)^7}-\frac {1001 e^5}{128 (d+e x)^{3/2} (b d-a e)^6}-\frac {3003 e^4}{640 (a+b x) (d+e x)^{3/2} (b d-a e)^5}+\frac {429 e^3}{320 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}-\frac {143 e^2}{240 (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}+\frac {13 e}{40 (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-1001*e^5)/(128*(b*d - a*e)^6*(d + e*x)^(3/2)) - 1/(5*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(3/2)) + (13*e)/(40*(
b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(3/2)) - (143*e^2)/(240*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(3/2)) + (429*e
^3)/(320*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(3/2)) - (3003*e^4)/(640*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(3/2))
- (3003*b*e^5)/(128*(b*d - a*e)^7*Sqrt[d + e*x]) + (3003*b^(3/2)*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d
- a*e]])/(128*(b*d - a*e)^(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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^6 (d+e x)^{5/2}} \, dx\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac {(13 e) \int \frac {1}{(a+b x)^5 (d+e x)^{5/2}} \, dx}{10 (b d-a e)}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac {\left (143 e^2\right ) \int \frac {1}{(a+b x)^4 (d+e x)^{5/2}} \, dx}{80 (b d-a e)^2}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac {\left (429 e^3\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{5/2}} \, dx}{160 (b d-a e)^3}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac {\left (3003 e^4\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{640 (b d-a e)^4}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac {\left (3003 e^5\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{256 (b d-a e)^5}\\ &=-\frac {1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac {\left (3003 b e^5\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^6}\\ &=-\frac {1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 \sqrt {d+e x}}-\frac {\left (3003 b^2 e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^7}\\ &=-\frac {1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 \sqrt {d+e x}}-\frac {\left (3003 b^2 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 )}{128 (b d-a e)^7}\\ &=-\frac {1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 \sqrt {d+e x}}+\frac {3003 b^{3/2} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{15/2}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 52, normalized size = 0.20 \begin {gather*} -\frac {2 e^5 \, _2F_1\left (-\frac {3}{2},6;-\frac {1}{2};-\frac {b (d+e x)}{a e-b d}\right )}{3 (d+e x)^{3/2} (a e-b d)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-2*e^5*Hypergeometric2F1[-3/2, 6, -1/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(3*(-(b*d) + a*e)^6*(d + e*x)^(3/2)
)

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IntegrateAlgebraic [B]  time = 2.55, size = 539, normalized size = 2.03 \begin {gather*} \frac {3003 b^{3/2} e^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 (b d-a e)^7 \sqrt {a e-b d}}-\frac {e^5 \left (1280 a^6 e^6-16640 a^5 b e^5 (d+e x)-7680 a^5 b d e^5+19200 a^4 b^2 d^2 e^4-137995 a^4 b^2 e^4 (d+e x)^2+83200 a^4 b^2 d e^4 (d+e x)-25600 a^3 b^3 d^3 e^3-166400 a^3 b^3 d^2 e^3 (d+e x)-338910 a^3 b^3 e^3 (d+e x)^3+551980 a^3 b^3 d e^3 (d+e x)^2+19200 a^2 b^4 d^4 e^2+166400 a^2 b^4 d^3 e^2 (d+e x)-827970 a^2 b^4 d^2 e^2 (d+e x)^2-384384 a^2 b^4 e^2 (d+e x)^4+1016730 a^2 b^4 d e^2 (d+e x)^3-7680 a b^5 d^5 e-83200 a b^5 d^4 e (d+e x)+551980 a b^5 d^3 e (d+e x)^2-1016730 a b^5 d^2 e (d+e x)^3-210210 a b^5 e (d+e x)^5+768768 a b^5 d e (d+e x)^4+1280 b^6 d^6+16640 b^6 d^5 (d+e x)-137995 b^6 d^4 (d+e x)^2+338910 b^6 d^3 (d+e x)^3-384384 b^6 d^2 (d+e x)^4-45045 b^6 (d+e x)^6+210210 b^6 d (d+e x)^5\right )}{1920 (d+e x)^{3/2} (b d-a e)^7 (-a e-b (d+e x)+b d)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

-1/1920*(e^5*(1280*b^6*d^6 - 7680*a*b^5*d^5*e + 19200*a^2*b^4*d^4*e^2 - 25600*a^3*b^3*d^3*e^3 + 19200*a^4*b^2*
d^2*e^4 - 7680*a^5*b*d*e^5 + 1280*a^6*e^6 + 16640*b^6*d^5*(d + e*x) - 83200*a*b^5*d^4*e*(d + e*x) + 166400*a^2
*b^4*d^3*e^2*(d + e*x) - 166400*a^3*b^3*d^2*e^3*(d + e*x) + 83200*a^4*b^2*d*e^4*(d + e*x) - 16640*a^5*b*e^5*(d
+ e*x) - 137995*b^6*d^4*(d + e*x)^2 + 551980*a*b^5*d^3*e*(d + e*x)^2 - 827970*a^2*b^4*d^2*e^2*(d + e*x)^2 + 5
51980*a^3*b^3*d*e^3*(d + e*x)^2 - 137995*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 - 384384*b^6*d^2*(d + e
*x)^4 + 768768*a*b^5*d*e*(d + e*x)^4 - 384384*a^2*b^4*e^2*(d + e*x)^4 + 210210*b^6*d*(d + e*x)^5 - 210210*a*b^
5*e*(d + e*x)^5 - 45045*b^6*(d + e*x)^6))/((b*d - a*e)^7*(d + e*x)^(3/2)*(b*d - a*e - b*(d + e*x))^5) + (3003*
b^(3/2)*e^5*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(128*(b*d - a*e)^7*Sqrt[-(b*d) + a
*e])

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fricas [B]  time = 0.53, size = 3244, normalized size = 12.20

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(45045*(b^6*e^7*x^7 + a^5*b*d^2*e^5 + (2*b^6*d*e^6 + 5*a*b^5*e^7)*x^6 + (b^6*d^2*e^5 + 10*a*b^5*d*e^6
+ 10*a^2*b^4*e^7)*x^5 + 5*(a*b^5*d^2*e^5 + 4*a^2*b^4*d*e^6 + 2*a^3*b^3*e^7)*x^4 + 5*(2*a^2*b^4*d^2*e^5 + 4*a^
3*b^3*d*e^6 + a^4*b^2*e^7)*x^3 + (10*a^3*b^3*d^2*e^5 + 10*a^4*b^2*d*e^6 + a^5*b*e^7)*x^2 + (5*a^4*b^2*d^2*e^5
+ 2*a^5*b*d*e^6)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e - 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a
*e)))/(b*x + a)) + 2*(45045*b^6*e^6*x^6 + 384*b^6*d^6 - 2928*a*b^5*d^5*e + 10024*a^2*b^4*d^4*e^2 - 21070*a^3*b
^3*d^3*e^3 + 35595*a^4*b^2*d^2*e^4 + 24320*a^5*b*d*e^5 - 1280*a^6*e^6 + 30030*(2*b^6*d*e^5 + 7*a*b^5*e^6)*x^5
+ 3003*(3*b^6*d^2*e^4 + 94*a*b^5*d*e^5 + 128*a^2*b^4*e^6)*x^4 - 858*(3*b^6*d^3*e^3 - 51*a*b^5*d^2*e^4 - 607*a^
2*b^4*d*e^5 - 395*a^3*b^3*e^6)*x^3 + 143*(8*b^6*d^4*e^2 - 86*a*b^5*d^3*e^3 + 588*a^2*b^4*d^2*e^4 + 3250*a^3*b^
3*d*e^5 + 965*a^4*b^2*e^6)*x^2 - 26*(24*b^6*d^5*e - 208*a*b^5*d^4*e^2 + 889*a^2*b^4*d^3*e^3 - 3045*a^3*b^3*d^2
*e^4 - 7415*a^4*b^2*d*e^5 - 640*a^5*b*e^6)*x)*sqrt(e*x + d))/(a^5*b^7*d^9 - 7*a^6*b^6*d^8*e + 21*a^7*b^5*d^7*e
^2 - 35*a^8*b^4*d^6*e^3 + 35*a^9*b^3*d^5*e^4 - 21*a^10*b^2*d^4*e^5 + 7*a^11*b*d^3*e^6 - a^12*d^2*e^7 + (b^12*d
^7*e^2 - 7*a*b^11*d^6*e^3 + 21*a^2*b^10*d^5*e^4 - 35*a^3*b^9*d^4*e^5 + 35*a^4*b^8*d^3*e^6 - 21*a^5*b^7*d^2*e^7
+ 7*a^6*b^6*d*e^8 - a^7*b^5*e^9)*x^7 + (2*b^12*d^8*e - 9*a*b^11*d^7*e^2 + 7*a^2*b^10*d^6*e^3 + 35*a^3*b^9*d^5
*e^4 - 105*a^4*b^8*d^4*e^5 + 133*a^5*b^7*d^3*e^6 - 91*a^6*b^6*d^2*e^7 + 33*a^7*b^5*d*e^8 - 5*a^8*b^4*e^9)*x^6
+ (b^12*d^9 + 3*a*b^11*d^8*e - 39*a^2*b^10*d^7*e^2 + 105*a^3*b^9*d^6*e^3 - 105*a^4*b^8*d^5*e^4 - 21*a^5*b^7*d^
4*e^5 + 147*a^6*b^6*d^3*e^6 - 141*a^7*b^5*d^2*e^7 + 60*a^8*b^4*d*e^8 - 10*a^9*b^3*e^9)*x^5 + 5*(a*b^11*d^9 - 3
*a^2*b^10*d^8*e - 5*a^3*b^9*d^7*e^2 + 35*a^4*b^8*d^6*e^3 - 63*a^5*b^7*d^5*e^4 + 49*a^6*b^6*d^4*e^5 - 7*a^7*b^5
*d^3*e^6 - 15*a^8*b^4*d^2*e^7 + 10*a^9*b^3*d*e^8 - 2*a^10*b^2*e^9)*x^4 + 5*(2*a^2*b^10*d^9 - 10*a^3*b^9*d^8*e
+ 15*a^4*b^8*d^7*e^2 + 7*a^5*b^7*d^6*e^3 - 49*a^6*b^6*d^5*e^4 + 63*a^7*b^5*d^4*e^5 - 35*a^8*b^4*d^3*e^6 + 5*a^
9*b^3*d^2*e^7 + 3*a^10*b^2*d*e^8 - a^11*b*e^9)*x^3 + (10*a^3*b^9*d^9 - 60*a^4*b^8*d^8*e + 141*a^5*b^7*d^7*e^2
- 147*a^6*b^6*d^6*e^3 + 21*a^7*b^5*d^5*e^4 + 105*a^8*b^4*d^4*e^5 - 105*a^9*b^3*d^3*e^6 + 39*a^10*b^2*d^2*e^7 -
3*a^11*b*d*e^8 - a^12*e^9)*x^2 + (5*a^4*b^8*d^9 - 33*a^5*b^7*d^8*e + 91*a^6*b^6*d^7*e^2 - 133*a^7*b^5*d^6*e^3
+ 105*a^8*b^4*d^5*e^4 - 35*a^9*b^3*d^4*e^5 - 7*a^10*b^2*d^3*e^6 + 9*a^11*b*d^2*e^7 - 2*a^12*d*e^8)*x), 1/1920
*(45045*(b^6*e^7*x^7 + a^5*b*d^2*e^5 + (2*b^6*d*e^6 + 5*a*b^5*e^7)*x^6 + (b^6*d^2*e^5 + 10*a*b^5*d*e^6 + 10*a^
2*b^4*e^7)*x^5 + 5*(a*b^5*d^2*e^5 + 4*a^2*b^4*d*e^6 + 2*a^3*b^3*e^7)*x^4 + 5*(2*a^2*b^4*d^2*e^5 + 4*a^3*b^3*d*
e^6 + a^4*b^2*e^7)*x^3 + (10*a^3*b^3*d^2*e^5 + 10*a^4*b^2*d*e^6 + a^5*b*e^7)*x^2 + (5*a^4*b^2*d^2*e^5 + 2*a^5*
b*d*e^6)*x)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (4504
5*b^6*e^6*x^6 + 384*b^6*d^6 - 2928*a*b^5*d^5*e + 10024*a^2*b^4*d^4*e^2 - 21070*a^3*b^3*d^3*e^3 + 35595*a^4*b^2
*d^2*e^4 + 24320*a^5*b*d*e^5 - 1280*a^6*e^6 + 30030*(2*b^6*d*e^5 + 7*a*b^5*e^6)*x^5 + 3003*(3*b^6*d^2*e^4 + 94
*a*b^5*d*e^5 + 128*a^2*b^4*e^6)*x^4 - 858*(3*b^6*d^3*e^3 - 51*a*b^5*d^2*e^4 - 607*a^2*b^4*d*e^5 - 395*a^3*b^3*
e^6)*x^3 + 143*(8*b^6*d^4*e^2 - 86*a*b^5*d^3*e^3 + 588*a^2*b^4*d^2*e^4 + 3250*a^3*b^3*d*e^5 + 965*a^4*b^2*e^6)
*x^2 - 26*(24*b^6*d^5*e - 208*a*b^5*d^4*e^2 + 889*a^2*b^4*d^3*e^3 - 3045*a^3*b^3*d^2*e^4 - 7415*a^4*b^2*d*e^5
- 640*a^5*b*e^6)*x)*sqrt(e*x + d))/(a^5*b^7*d^9 - 7*a^6*b^6*d^8*e + 21*a^7*b^5*d^7*e^2 - 35*a^8*b^4*d^6*e^3 +
35*a^9*b^3*d^5*e^4 - 21*a^10*b^2*d^4*e^5 + 7*a^11*b*d^3*e^6 - a^12*d^2*e^7 + (b^12*d^7*e^2 - 7*a*b^11*d^6*e^3
+ 21*a^2*b^10*d^5*e^4 - 35*a^3*b^9*d^4*e^5 + 35*a^4*b^8*d^3*e^6 - 21*a^5*b^7*d^2*e^7 + 7*a^6*b^6*d*e^8 - a^7*b
^5*e^9)*x^7 + (2*b^12*d^8*e - 9*a*b^11*d^7*e^2 + 7*a^2*b^10*d^6*e^3 + 35*a^3*b^9*d^5*e^4 - 105*a^4*b^8*d^4*e^5
+ 133*a^5*b^7*d^3*e^6 - 91*a^6*b^6*d^2*e^7 + 33*a^7*b^5*d*e^8 - 5*a^8*b^4*e^9)*x^6 + (b^12*d^9 + 3*a*b^11*d^8
*e - 39*a^2*b^10*d^7*e^2 + 105*a^3*b^9*d^6*e^3 - 105*a^4*b^8*d^5*e^4 - 21*a^5*b^7*d^4*e^5 + 147*a^6*b^6*d^3*e^
6 - 141*a^7*b^5*d^2*e^7 + 60*a^8*b^4*d*e^8 - 10*a^9*b^3*e^9)*x^5 + 5*(a*b^11*d^9 - 3*a^2*b^10*d^8*e - 5*a^3*b^
9*d^7*e^2 + 35*a^4*b^8*d^6*e^3 - 63*a^5*b^7*d^5*e^4 + 49*a^6*b^6*d^4*e^5 - 7*a^7*b^5*d^3*e^6 - 15*a^8*b^4*d^2*
e^7 + 10*a^9*b^3*d*e^8 - 2*a^10*b^2*e^9)*x^4 + 5*(2*a^2*b^10*d^9 - 10*a^3*b^9*d^8*e + 15*a^4*b^8*d^7*e^2 + 7*a
^5*b^7*d^6*e^3 - 49*a^6*b^6*d^5*e^4 + 63*a^7*b^5*d^4*e^5 - 35*a^8*b^4*d^3*e^6 + 5*a^9*b^3*d^2*e^7 + 3*a^10*b^2
*d*e^8 - a^11*b*e^9)*x^3 + (10*a^3*b^9*d^9 - 60*a^4*b^8*d^8*e + 141*a^5*b^7*d^7*e^2 - 147*a^6*b^6*d^6*e^3 + 21
*a^7*b^5*d^5*e^4 + 105*a^8*b^4*d^4*e^5 - 105*a^9*b^3*d^3*e^6 + 39*a^10*b^2*d^2*e^7 - 3*a^11*b*d*e^8 - a^12*e^9
)*x^2 + (5*a^4*b^8*d^9 - 33*a^5*b^7*d^8*e + 91*a^6*b^6*d^7*e^2 - 133*a^7*b^5*d^6*e^3 + 105*a^8*b^4*d^5*e^4 - 3
5*a^9*b^3*d^4*e^5 - 7*a^10*b^2*d^3*e^6 + 9*a^11*b*d^2*e^7 - 2*a^12*d*e^8)*x)]

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giac [B]  time = 0.28, size = 637, normalized size = 2.39 \begin {gather*} -\frac {3003 \, b^{2} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{5}}{128 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, {\left (18 \, {\left (x e + d\right )} b e^{5} + b d e^{5} - a e^{6}\right )}}{3 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} - \frac {22005 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{6} e^{5} - 96290 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{6} d e^{5} + 160384 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} d^{2} e^{5} - 121310 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} d^{3} e^{5} + 35595 \, \sqrt {x e + d} b^{6} d^{4} e^{5} + 96290 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{5} e^{6} - 320768 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{5} d e^{6} + 363930 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{5} d^{2} e^{6} - 142380 \, \sqrt {x e + d} a b^{5} d^{3} e^{6} + 160384 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{4} e^{7} - 363930 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{4} d e^{7} + 213570 \, \sqrt {x e + d} a^{2} b^{4} d^{2} e^{7} + 121310 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{3} e^{8} - 142380 \, \sqrt {x e + d} a^{3} b^{3} d e^{8} + 35595 \, \sqrt {x e + d} a^{4} b^{2} e^{9}}{1920 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-3003/128*b^2*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2
- 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*sqrt(-b^2*d + a*b*e)
) - 2/3*(18*(x*e + d)*b*e^5 + b*d*e^5 - a*e^6)/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4
*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*(x*e + d)^(3/2)) - 1/1920*(22005*(x*
e + d)^(9/2)*b^6*e^5 - 96290*(x*e + d)^(7/2)*b^6*d*e^5 + 160384*(x*e + d)^(5/2)*b^6*d^2*e^5 - 121310*(x*e + d)
^(3/2)*b^6*d^3*e^5 + 35595*sqrt(x*e + d)*b^6*d^4*e^5 + 96290*(x*e + d)^(7/2)*a*b^5*e^6 - 320768*(x*e + d)^(5/2
)*a*b^5*d*e^6 + 363930*(x*e + d)^(3/2)*a*b^5*d^2*e^6 - 142380*sqrt(x*e + d)*a*b^5*d^3*e^6 + 160384*(x*e + d)^(
5/2)*a^2*b^4*e^7 - 363930*(x*e + d)^(3/2)*a^2*b^4*d*e^7 + 213570*sqrt(x*e + d)*a^2*b^4*d^2*e^7 + 121310*(x*e +
d)^(3/2)*a^3*b^3*e^8 - 142380*sqrt(x*e + d)*a^3*b^3*d*e^8 + 35595*sqrt(x*e + d)*a^4*b^2*e^9)/((b^7*d^7 - 7*a*
b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6
- a^7*e^7)*((x*e + d)*b - b*d + a*e)^5)

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maple [B]  time = 0.08, size = 668, normalized size = 2.51 \begin {gather*} \frac {2373 \sqrt {e x +d}\, a^{4} b^{2} e^{9}}{128 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}-\frac {2373 \sqrt {e x +d}\, a^{3} b^{3} d \,e^{8}}{32 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}+\frac {7119 \sqrt {e x +d}\, a^{2} b^{4} d^{2} e^{7}}{64 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}-\frac {2373 \sqrt {e x +d}\, a \,b^{5} d^{3} e^{6}}{32 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}+\frac {2373 \sqrt {e x +d}\, b^{6} d^{4} e^{5}}{128 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}+\frac {12131 \left (e x +d \right )^{\frac {3}{2}} a^{3} b^{3} e^{8}}{192 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}-\frac {12131 \left (e x +d \right )^{\frac {3}{2}} a^{2} b^{4} d \,e^{7}}{64 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}+\frac {12131 \left (e x +d \right )^{\frac {3}{2}} a \,b^{5} d^{2} e^{6}}{64 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}-\frac {12131 \left (e x +d \right )^{\frac {3}{2}} b^{6} d^{3} e^{5}}{192 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}+\frac {1253 \left (e x +d \right )^{\frac {5}{2}} a^{2} b^{4} e^{7}}{15 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}-\frac {2506 \left (e x +d \right )^{\frac {5}{2}} a \,b^{5} d \,e^{6}}{15 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}+\frac {1253 \left (e x +d \right )^{\frac {5}{2}} b^{6} d^{2} e^{5}}{15 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}+\frac {9629 \left (e x +d \right )^{\frac {7}{2}} a \,b^{5} e^{6}}{192 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}-\frac {9629 \left (e x +d \right )^{\frac {7}{2}} b^{6} d \,e^{5}}{192 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}+\frac {1467 \left (e x +d \right )^{\frac {9}{2}} b^{6} e^{5}}{128 \left (a e -b d \right )^{7} \left (b e x +a e \right )^{5}}+\frac {3003 b^{2} e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a e -b d \right )^{7} \sqrt {\left (a e -b d \right ) b}}+\frac {12 b \,e^{5}}{\left (a e -b d \right )^{7} \sqrt {e x +d}}-\frac {2 e^{5}}{3 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1467/128*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)+9629/192*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(7
/2)*a-9629/192*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d+1253/15*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*
x+d)^(5/2)*a^2-2506/15*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a*d+1253/15*e^5/(a*e-b*d)^7*b^6/(b*e*x+
a*e)^5*(e*x+d)^(5/2)*d^2+12131/192*e^8/(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3-12131/64*e^7/(a*e-b*d)^
7*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^2*d+12131/64*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^2-12131/1
92*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^3+2373/128*e^9/(a*e-b*d)^7*b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2
)*a^4-2373/32*e^8/(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^3*d+7119/64*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*
(e*x+d)^(1/2)*a^2*d^2-2373/32*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^3+2373/128*e^5/(a*e-b*d)^7*b
^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^4+3003/128*e^5/(a*e-b*d)^7*b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e
-b*d)*b)^(1/2)*b)-2/3*e^5/(a*e-b*d)^6/(e*x+d)^(3/2)+12*e^5/(a*e-b*d)^7*b/(e*x+d)^(1/2)

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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(1/(e*x+d)^(5/2)/(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 = 1.26, size = 555, normalized size = 2.09 \begin {gather*} \frac {\frac {27599\,b^2\,e^5\,{\left (d+e\,x\right )}^2}{384\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,e^5}{3\,\left (a\,e-b\,d\right )}+\frac {11297\,b^3\,e^5\,{\left (d+e\,x\right )}^3}{64\,{\left (a\,e-b\,d\right )}^4}+\frac {1001\,b^4\,e^5\,{\left (d+e\,x\right )}^4}{5\,{\left (a\,e-b\,d\right )}^5}+\frac {7007\,b^5\,e^5\,{\left (d+e\,x\right )}^5}{64\,{\left (a\,e-b\,d\right )}^6}+\frac {3003\,b^6\,e^5\,{\left (d+e\,x\right )}^6}{128\,{\left (a\,e-b\,d\right )}^7}+\frac {26\,b\,e^5\,\left (d+e\,x\right )}{3\,{\left (a\,e-b\,d\right )}^2}}{{\left (d+e\,x\right )}^{3/2}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )-{\left (d+e\,x\right )}^{7/2}\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+{\left (d+e\,x\right )}^{5/2}\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )+b^5\,{\left (d+e\,x\right )}^{13/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{11/2}+{\left (d+e\,x\right )}^{9/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}+\frac {3003\,b^{3/2}\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}{{\left (a\,e-b\,d\right )}^{15/2}}\right )}{128\,{\left (a\,e-b\,d\right )}^{15/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)

[Out]

((27599*b^2*e^5*(d + e*x)^2)/(384*(a*e - b*d)^3) - (2*e^5)/(3*(a*e - b*d)) + (11297*b^3*e^5*(d + e*x)^3)/(64*(
a*e - b*d)^4) + (1001*b^4*e^5*(d + e*x)^4)/(5*(a*e - b*d)^5) + (7007*b^5*e^5*(d + e*x)^5)/(64*(a*e - b*d)^6) +
(3003*b^6*e^5*(d + e*x)^6)/(128*(a*e - b*d)^7) + (26*b*e^5*(d + e*x))/(3*(a*e - b*d)^2))/((d + e*x)^(3/2)*(a^
5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) - (d + e*x)^(7/2)*(
10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*e^2 - 30*a*b^4*d^2*e) + (d + e*x)^(5/2)*(5*b^5*d^4 + 5*a^4*b*e^4 -
20*a^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 - 20*a*b^4*d^3*e) + b^5*(d + e*x)^(13/2) - (5*b^5*d - 5*a*b^4*e)*(d + e*
x)^(11/2) + (d + e*x)^(9/2)*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e)) + (3003*b^(3/2)*e^5*atan((b^(1/2)*(d
+ e*x)^(1/2)*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35*a^3*b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d
^2*e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e^6))/(a*e - b*d)^(15/2)))/(128*(a*e - b*d)^(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(1/(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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