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

Optimal. Leaf size=381 \[ \frac {1155 b e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^6}+\frac {385 e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}+\frac {231 e^3}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}-\frac {33 e^2}{32 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac {11 e}{24 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac {1155 b^{3/2} e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{13/2}} \]

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Rubi [A]  time = 0.24, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {646, 51, 63, 208} \begin {gather*} \frac {1155 b e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^6}+\frac {385 e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}+\frac {231 e^3}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}-\frac {33 e^2}{32 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}-\frac {1155 b^{3/2} e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{13/2}}+\frac {11 e}{24 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (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)^(5/2)),x]

[Out]

(231*e^3)/(64*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(4*(b*d - a*e)*(a + b*x)^3*(d +
e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (11*e)/(24*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2]) - (33*e^2)/(32*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (38
5*e^4*(a + b*x))/(64*(b*d - a*e)^5*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (1155*b*e^4*(a + b*x))/(64
*(b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (1155*b^(3/2)*e^4*(a + b*x)*ArcTanh[(Sqrt[b]*Sqr
t[d + e*x])/Sqrt[b*d - a*e]])/(64*(b*d - a*e)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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]

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^5 (d+e x)^{5/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (11 b^3 e \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^4 (d+e x)^{5/2}} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (33 b^2 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 (d+e x)^{5/2}} \, dx}{16 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (231 b e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 (d+e x)^{5/2}} \, dx}{64 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {231 e^3}{64 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{5/2}} \, dx}{128 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {231 e^3}{64 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x)}{64 (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 b e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{128 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {231 e^3}{64 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x)}{64 (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1155 b e^4 (a+b x)}{64 (b d-a e)^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 b^2 e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {231 e^3}{64 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x)}{64 (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1155 b e^4 (a+b x)}{64 (b d-a e)^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 b^2 e^3 \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {231 e^3}{64 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2}{32 (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x)}{64 (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1155 b e^4 (a+b x)}{64 (b d-a e)^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1155 b^{3/2} e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 67, normalized size = 0.18 \begin {gather*} \frac {2 e^4 (a+b x) \, _2F_1\left (-\frac {3}{2},5;-\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{3 \sqrt {(a+b x)^2} (d+e x)^{3/2} (b d-a e)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e^4*(a + b*x)*Hypergeometric2F1[-3/2, 5, -1/2, (b*(d + e*x))/(b*d - a*e)])/(3*(b*d - a*e)^5*Sqrt[(a + b*x)^
2]*(d + e*x)^(3/2))

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IntegrateAlgebraic [A]  time = 65.84, size = 438, normalized size = 1.15 \begin {gather*} \frac {(-a e-b e x) \left (\frac {e^4 \left (128 a^5 e^5-1408 a^4 b e^4 (d+e x)-640 a^4 b d e^4+1280 a^3 b^2 d^2 e^3-9207 a^3 b^2 e^3 (d+e x)^2+5632 a^3 b^2 d e^3 (d+e x)-1280 a^2 b^3 d^3 e^2-8448 a^2 b^3 d^2 e^2 (d+e x)-16863 a^2 b^3 e^2 (d+e x)^3+27621 a^2 b^3 d e^2 (d+e x)^2+640 a b^4 d^4 e+5632 a b^4 d^3 e (d+e x)-27621 a b^4 d^2 e (d+e x)^2-12705 a b^4 e (d+e x)^4+33726 a b^4 d e (d+e x)^3-128 b^5 d^5-1408 b^5 d^4 (d+e x)+9207 b^5 d^3 (d+e x)^2-16863 b^5 d^2 (d+e x)^3-3465 b^5 (d+e x)^5+12705 b^5 d (d+e x)^4\right )}{192 (d+e x)^{3/2} (b d-a e)^6 (-a e-b (d+e x)+b d)^4}+\frac {1155 b^{3/2} e^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 (a e-b d)^{13/2}}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-(a*e) - b*e*x)*((e^4*(-128*b^5*d^5 + 640*a*b^4*d^4*e - 1280*a^2*b^3*d^3*e^2 + 1280*a^3*b^2*d^2*e^3 - 640*a^
4*b*d*e^4 + 128*a^5*e^5 - 1408*b^5*d^4*(d + e*x) + 5632*a*b^4*d^3*e*(d + e*x) - 8448*a^2*b^3*d^2*e^2*(d + e*x)
+ 5632*a^3*b^2*d*e^3*(d + e*x) - 1408*a^4*b*e^4*(d + e*x) + 9207*b^5*d^3*(d + e*x)^2 - 27621*a*b^4*d^2*e*(d +
e*x)^2 + 27621*a^2*b^3*d*e^2*(d + e*x)^2 - 9207*a^3*b^2*e^3*(d + e*x)^2 - 16863*b^5*d^2*(d + e*x)^3 + 33726*a
*b^4*d*e*(d + e*x)^3 - 16863*a^2*b^3*e^2*(d + e*x)^3 + 12705*b^5*d*(d + e*x)^4 - 12705*a*b^4*e*(d + e*x)^4 - 3
465*b^5*(d + e*x)^5))/(192*(b*d - a*e)^6*(d + e*x)^(3/2)*(b*d - a*e - b*(d + e*x))^4) + (1155*b^(3/2)*e^4*ArcT
an[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(64*(-(b*d) + a*e)^(13/2))))/(e*Sqrt[(a*e + b*e*x)
^2/e^2])

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fricas [B]  time = 0.49, size = 2494, normalized size = 6.55

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)^(5/2),x, algorithm="fricas")

[Out]

[1/384*(3465*(b^5*e^6*x^6 + a^4*b*d^2*e^4 + 2*(b^5*d*e^5 + 2*a*b^4*e^6)*x^5 + (b^5*d^2*e^4 + 8*a*b^4*d*e^5 + 6
*a^2*b^3*e^6)*x^4 + 4*(a*b^4*d^2*e^4 + 3*a^2*b^3*d*e^5 + a^3*b^2*e^6)*x^3 + (6*a^2*b^3*d^2*e^4 + 8*a^3*b^2*d*e
^5 + a^4*b*e^6)*x^2 + 2*(2*a^3*b^2*d^2*e^4 + a^4*b*d*e^5)*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*(3465*b^5*e^5*x^5 - 48*b^5*d^5 + 328*a*b^4*d^4*e
- 1030*a^2*b^3*d^3*e^2 + 2295*a^3*b^2*d^2*e^3 + 2048*a^4*b*d*e^4 - 128*a^5*e^5 + 1155*(4*b^5*d*e^4 + 11*a*b^4
*e^5)*x^4 + 231*(3*b^5*d^2*e^3 + 74*a*b^4*d*e^4 + 73*a^2*b^3*e^5)*x^3 - 99*(2*b^5*d^3*e^2 - 27*a*b^4*d^2*e^3 -
232*a^2*b^3*d*e^4 - 93*a^3*b^2*e^5)*x^2 + 11*(8*b^5*d^4*e - 68*a*b^4*d^3*e^2 + 345*a^2*b^3*d^2*e^3 + 1162*a^3
*b^2*d*e^4 + 128*a^4*b*e^5)*x)*sqrt(e*x + d))/(a^4*b^6*d^8 - 6*a^5*b^5*d^7*e + 15*a^6*b^4*d^6*e^2 - 20*a^7*b^3
*d^5*e^3 + 15*a^8*b^2*d^4*e^4 - 6*a^9*b*d^3*e^5 + a^10*d^2*e^6 + (b^10*d^6*e^2 - 6*a*b^9*d^5*e^3 + 15*a^2*b^8*
d^4*e^4 - 20*a^3*b^7*d^3*e^5 + 15*a^4*b^6*d^2*e^6 - 6*a^5*b^5*d*e^7 + a^6*b^4*e^8)*x^6 + 2*(b^10*d^7*e - 4*a*b
^9*d^6*e^2 + 3*a^2*b^8*d^5*e^3 + 10*a^3*b^7*d^4*e^4 - 25*a^4*b^6*d^3*e^5 + 24*a^5*b^5*d^2*e^6 - 11*a^6*b^4*d*e
^7 + 2*a^7*b^3*e^8)*x^5 + (b^10*d^8 + 2*a*b^9*d^7*e - 27*a^2*b^8*d^6*e^2 + 64*a^3*b^7*d^5*e^3 - 55*a^4*b^6*d^4
*e^4 - 6*a^5*b^5*d^3*e^5 + 43*a^6*b^4*d^2*e^6 - 28*a^7*b^3*d*e^7 + 6*a^8*b^2*e^8)*x^4 + 4*(a*b^9*d^8 - 3*a^2*b
^8*d^7*e - 2*a^3*b^7*d^6*e^2 + 19*a^4*b^6*d^5*e^3 - 30*a^5*b^5*d^4*e^4 + 19*a^6*b^4*d^3*e^5 - 2*a^7*b^3*d^2*e^
6 - 3*a^8*b^2*d*e^7 + a^9*b*e^8)*x^3 + (6*a^2*b^8*d^8 - 28*a^3*b^7*d^7*e + 43*a^4*b^6*d^6*e^2 - 6*a^5*b^5*d^5*
e^3 - 55*a^6*b^4*d^4*e^4 + 64*a^7*b^3*d^3*e^5 - 27*a^8*b^2*d^2*e^6 + 2*a^9*b*d*e^7 + a^10*e^8)*x^2 + 2*(2*a^3*
b^7*d^8 - 11*a^4*b^6*d^7*e + 24*a^5*b^5*d^6*e^2 - 25*a^6*b^4*d^5*e^3 + 10*a^7*b^3*d^4*e^4 + 3*a^8*b^2*d^3*e^5
- 4*a^9*b*d^2*e^6 + a^10*d*e^7)*x), -1/192*(3465*(b^5*e^6*x^6 + a^4*b*d^2*e^4 + 2*(b^5*d*e^5 + 2*a*b^4*e^6)*x^
5 + (b^5*d^2*e^4 + 8*a*b^4*d*e^5 + 6*a^2*b^3*e^6)*x^4 + 4*(a*b^4*d^2*e^4 + 3*a^2*b^3*d*e^5 + a^3*b^2*e^6)*x^3
+ (6*a^2*b^3*d^2*e^4 + 8*a^3*b^2*d*e^5 + a^4*b*e^6)*x^2 + 2*(2*a^3*b^2*d^2*e^4 + a^4*b*d*e^5)*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)) - (3465*b^5*e^5*x^5 - 48*b^5*d^5
+ 328*a*b^4*d^4*e - 1030*a^2*b^3*d^3*e^2 + 2295*a^3*b^2*d^2*e^3 + 2048*a^4*b*d*e^4 - 128*a^5*e^5 + 1155*(4*b^
5*d*e^4 + 11*a*b^4*e^5)*x^4 + 231*(3*b^5*d^2*e^3 + 74*a*b^4*d*e^4 + 73*a^2*b^3*e^5)*x^3 - 99*(2*b^5*d^3*e^2 -
27*a*b^4*d^2*e^3 - 232*a^2*b^3*d*e^4 - 93*a^3*b^2*e^5)*x^2 + 11*(8*b^5*d^4*e - 68*a*b^4*d^3*e^2 + 345*a^2*b^3*
d^2*e^3 + 1162*a^3*b^2*d*e^4 + 128*a^4*b*e^5)*x)*sqrt(e*x + d))/(a^4*b^6*d^8 - 6*a^5*b^5*d^7*e + 15*a^6*b^4*d^
6*e^2 - 20*a^7*b^3*d^5*e^3 + 15*a^8*b^2*d^4*e^4 - 6*a^9*b*d^3*e^5 + a^10*d^2*e^6 + (b^10*d^6*e^2 - 6*a*b^9*d^5
*e^3 + 15*a^2*b^8*d^4*e^4 - 20*a^3*b^7*d^3*e^5 + 15*a^4*b^6*d^2*e^6 - 6*a^5*b^5*d*e^7 + a^6*b^4*e^8)*x^6 + 2*(
b^10*d^7*e - 4*a*b^9*d^6*e^2 + 3*a^2*b^8*d^5*e^3 + 10*a^3*b^7*d^4*e^4 - 25*a^4*b^6*d^3*e^5 + 24*a^5*b^5*d^2*e^
6 - 11*a^6*b^4*d*e^7 + 2*a^7*b^3*e^8)*x^5 + (b^10*d^8 + 2*a*b^9*d^7*e - 27*a^2*b^8*d^6*e^2 + 64*a^3*b^7*d^5*e^
3 - 55*a^4*b^6*d^4*e^4 - 6*a^5*b^5*d^3*e^5 + 43*a^6*b^4*d^2*e^6 - 28*a^7*b^3*d*e^7 + 6*a^8*b^2*e^8)*x^4 + 4*(a
*b^9*d^8 - 3*a^2*b^8*d^7*e - 2*a^3*b^7*d^6*e^2 + 19*a^4*b^6*d^5*e^3 - 30*a^5*b^5*d^4*e^4 + 19*a^6*b^4*d^3*e^5
- 2*a^7*b^3*d^2*e^6 - 3*a^8*b^2*d*e^7 + a^9*b*e^8)*x^3 + (6*a^2*b^8*d^8 - 28*a^3*b^7*d^7*e + 43*a^4*b^6*d^6*e^
2 - 6*a^5*b^5*d^5*e^3 - 55*a^6*b^4*d^4*e^4 + 64*a^7*b^3*d^3*e^5 - 27*a^8*b^2*d^2*e^6 + 2*a^9*b*d*e^7 + a^10*e^
8)*x^2 + 2*(2*a^3*b^7*d^8 - 11*a^4*b^6*d^7*e + 24*a^5*b^5*d^6*e^2 - 25*a^6*b^4*d^5*e^3 + 10*a^7*b^3*d^4*e^4 +
3*a^8*b^2*d^3*e^5 - 4*a^9*b*d^2*e^6 + a^10*d*e^7)*x)]

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giac [B]  time = 0.56, size = 962, normalized size = 2.52 \begin {gather*} \frac {1155 \, b^{2} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{4}}{64 \, {\left (b^{6} d^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{6} e^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )} b e^{4} + b d e^{4} - a e^{5}\right )}}{3 \, {\left (b^{6} d^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{6} e^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} {\left (x e + d\right )}^{\frac {3}{2}}} + \frac {1545 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} e^{4} - 5153 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d e^{4} + 5855 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{2} e^{4} - 2295 \, \sqrt {x e + d} b^{5} d^{3} e^{4} + 5153 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} e^{5} - 11710 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d e^{5} + 6885 \, \sqrt {x e + d} a b^{4} d^{2} e^{5} + 5855 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} e^{6} - 6885 \, \sqrt {x e + d} a^{2} b^{3} d e^{6} + 2295 \, \sqrt {x e + d} a^{3} b^{2} e^{7}}{192 \, {\left (b^{6} d^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{6} e^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \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)^(5/2),x, algorithm="giac")

[Out]

1155/64*b^2*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^6*d^6*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*
a*b^5*d^5*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^2*b^4*d^4*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 20*a^
3*b^3*d^3*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^4*b^2*d^2*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a
^5*b*d*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^6*e^6*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(-b^2*d + a*b*
e)) + 2/3*(15*(x*e + d)*b*e^4 + b*d*e^4 - a*e^5)/((b^6*d^6*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a*b^5*d^5*e*
sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^2*b^4*d^4*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 20*a^3*b^3*d^3*e^
3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^4*b^2*d^2*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a^5*b*d*e^5*s
gn((x*e + d)*b*e - b*d*e + a*e^2) + a^6*e^6*sgn((x*e + d)*b*e - b*d*e + a*e^2))*(x*e + d)^(3/2)) + 1/192*(1545
*(x*e + d)^(7/2)*b^5*e^4 - 5153*(x*e + d)^(5/2)*b^5*d*e^4 + 5855*(x*e + d)^(3/2)*b^5*d^2*e^4 - 2295*sqrt(x*e +
d)*b^5*d^3*e^4 + 5153*(x*e + d)^(5/2)*a*b^4*e^5 - 11710*(x*e + d)^(3/2)*a*b^4*d*e^5 + 6885*sqrt(x*e + d)*a*b^
4*d^2*e^5 + 5855*(x*e + d)^(3/2)*a^2*b^3*e^6 - 6885*sqrt(x*e + d)*a^2*b^3*d*e^6 + 2295*sqrt(x*e + d)*a^3*b^2*e
^7)/((b^6*d^6*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a*b^5*d^5*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^2*b
^4*d^4*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 20*a^3*b^3*d^3*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^4
*b^2*d^2*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a^5*b*d*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^6*e^6*s
gn((x*e + d)*b*e - b*d*e + a*e^2))*((x*e + d)*b - b*d + a*e)^4)

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maple [B]  time = 0.08, size = 763, normalized size = 2.00 \begin {gather*} \frac {\left (3465 \left (e x +d \right )^{\frac {3}{2}} b^{6} e^{4} x^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+3465 \sqrt {\left (a e -b d \right ) b}\, b^{5} e^{5} x^{5}+13860 \left (e x +d \right )^{\frac {3}{2}} a \,b^{5} e^{4} x^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+12705 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} e^{5} x^{4}+4620 \sqrt {\left (a e -b d \right ) b}\, b^{5} d \,e^{4} x^{4}+20790 \left (e x +d \right )^{\frac {3}{2}} a^{2} b^{4} e^{4} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+16863 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{3} e^{5} x^{3}+17094 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} d \,e^{4} x^{3}+693 \sqrt {\left (a e -b d \right ) b}\, b^{5} d^{2} e^{3} x^{3}+13860 \left (e x +d \right )^{\frac {3}{2}} a^{3} b^{3} e^{4} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+9207 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{2} e^{5} x^{2}+22968 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{3} d \,e^{4} x^{2}+2673 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} d^{2} e^{3} x^{2}-198 \sqrt {\left (a e -b d \right ) b}\, b^{5} d^{3} e^{2} x^{2}+3465 \left (e x +d \right )^{\frac {3}{2}} a^{4} b^{2} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+1408 \sqrt {\left (a e -b d \right ) b}\, a^{4} b \,e^{5} x +12782 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{2} d \,e^{4} x +3795 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{3} d^{2} e^{3} x -748 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} d^{3} e^{2} x +88 \sqrt {\left (a e -b d \right ) b}\, b^{5} d^{4} e x -128 \sqrt {\left (a e -b d \right ) b}\, a^{5} e^{5}+2048 \sqrt {\left (a e -b d \right ) b}\, a^{4} b d \,e^{4}+2295 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{2} d^{2} e^{3}-1030 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{3} d^{3} e^{2}+328 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} d^{4} e -48 \sqrt {\left (a e -b d \right ) b}\, b^{5} d^{5}\right ) \left (b x +a \right )}{192 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{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)^(5/2),x)

[Out]

1/192*(3795*((a*e-b*d)*b)^(1/2)*x*a^2*b^3*d^2*e^3+3465*((a*e-b*d)*b)^(1/2)*x^5*b^5*e^5-128*((a*e-b*d)*b)^(1/2)
*a^5*e^5-48*((a*e-b*d)*b)^(1/2)*b^5*d^5+13860*(e*x+d)^(3/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a*
b^5*e^4+20790*(e*x+d)^(3/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^2*b^4*e^4+3465*(e*x+d)^(3/2)*arc
tan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^4*b^2*e^4+16863*((a*e-b*d)*b)^(1/2)*x^3*a^2*b^3*e^5+9207*((a*e-b*d)
*b)^(1/2)*x^2*a^3*b^2*e^5+1408*((a*e-b*d)*b)^(1/2)*x*a^4*b*e^5+693*((a*e-b*d)*b)^(1/2)*x^3*b^5*d^2*e^3-198*((a
*e-b*d)*b)^(1/2)*x^2*b^5*d^3*e^2+88*((a*e-b*d)*b)^(1/2)*x*b^5*d^4*e+3465*(e*x+d)^(3/2)*arctan((e*x+d)^(1/2)/((
a*e-b*d)*b)^(1/2)*b)*x^4*b^6*e^4+12705*((a*e-b*d)*b)^(1/2)*x^4*a*b^4*e^5+4620*((a*e-b*d)*b)^(1/2)*x^4*b^5*d*e^
4-1030*((a*e-b*d)*b)^(1/2)*a^2*b^3*d^3*e^2+328*((a*e-b*d)*b)^(1/2)*a*b^4*d^4*e-748*((a*e-b*d)*b)^(1/2)*x*a*b^4
*d^3*e^2+2673*((a*e-b*d)*b)^(1/2)*x^2*a*b^4*d^2*e^3+22968*((a*e-b*d)*b)^(1/2)*x^2*a^2*b^3*d*e^4+12782*((a*e-b*
d)*b)^(1/2)*x*a^3*b^2*d*e^4+13860*(e*x+d)^(3/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^3*b^3*e^4+1709
4*((a*e-b*d)*b)^(1/2)*x^3*a*b^4*d*e^4+2048*((a*e-b*d)*b)^(1/2)*a^4*b*d*e^4+2295*((a*e-b*d)*b)^(1/2)*a^3*b^2*d^
2*e^3)*(b*x+a)/(e*x+d)^(3/2)/((a*e-b*d)*b)^(1/2)/(a*e-b*d)^6/((b*x+a)^2)^(5/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \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)^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \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)^(5/2)),x)

[Out]

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

[Out]

Timed out

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