3.18.29 \(\int \frac {1}{(d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1729]

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

[Out]

231/64*e^3/(-a*e+b*d)^4/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2)-1/4/(-a*e+b*d)/(b*x+a)^3/(e*x+d)^(3/2)/((b*x+a)^2)^(1/
2)+11/24*e/(-a*e+b*d)^2/(b*x+a)^2/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2)-33/32*e^2/(-a*e+b*d)^3/(b*x+a)/(e*x+d)^(3/2)
/((b*x+a)^2)^(1/2)+385/64*e^4*(b*x+a)/(-a*e+b*d)^5/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2)-1155/64*b^(3/2)*e^4*(b*x+a)
*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(13/2)/((b*x+a)^2)^(1/2)+1155/64*b*e^4*(b*x+a)/(-a
*e+b*d)^6/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 44, 53, 65, 214} \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 {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}} \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 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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 65

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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 660

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 ) \text {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 [A]
time = 1.38, size = 301, normalized size = 0.79 \begin {gather*} \frac {e^4 (a+b x)^5 \left (\frac {-128 a^5 e^5+128 a^4 b e^4 (16 d+11 e x)+a^3 b^2 e^3 \left (2295 d^2+12782 d e x+9207 e^2 x^2\right )+a^2 b^3 e^2 \left (-1030 d^3+3795 d^2 e x+22968 d e^2 x^2+16863 e^3 x^3\right )+a b^4 e \left (328 d^4-748 d^3 e x+2673 d^2 e^2 x^2+17094 d e^3 x^3+12705 e^4 x^4\right )+b^5 \left (-48 d^5+88 d^4 e x-198 d^3 e^2 x^2+693 d^2 e^3 x^3+4620 d e^4 x^4+3465 e^5 x^5\right )}{e^4 (b d-a e)^6 (a+b x)^4 (d+e x)^{3/2}}+\frac {3465 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{13/2}}\right )}{192 \left ((a+b x)^2\right )^{5/2}} \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]

(e^4*(a + b*x)^5*((-128*a^5*e^5 + 128*a^4*b*e^4*(16*d + 11*e*x) + a^3*b^2*e^3*(2295*d^2 + 12782*d*e*x + 9207*e
^2*x^2) + a^2*b^3*e^2*(-1030*d^3 + 3795*d^2*e*x + 22968*d*e^2*x^2 + 16863*e^3*x^3) + a*b^4*e*(328*d^4 - 748*d^
3*e*x + 2673*d^2*e^2*x^2 + 17094*d*e^3*x^3 + 12705*e^4*x^4) + b^5*(-48*d^5 + 88*d^4*e*x - 198*d^3*e^2*x^2 + 69
3*d^2*e^3*x^3 + 4620*d*e^4*x^4 + 3465*e^5*x^5))/(e^4*(b*d - a*e)^6*(a + b*x)^4*(d + e*x)^(3/2)) + (3465*b^(3/2
)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(13/2)))/(192*((a + b*x)^2)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(762\) vs. \(2(268)=536\).
time = 0.80, size = 763, normalized size = 2.00

method	result	size
default	\(\frac {\left (2673 \sqrt {b \left (a e -b d \right )}\, a \,b^{4} d^{2} e^{3} x^{2}+3795 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{3} d^{2} e^{3} x -748 \sqrt {b \left (a e -b d \right )}\, a \,b^{4} d^{3} e^{2} x +13860 \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{5} e^{4} x^{3}+20790 \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{4} e^{4} x^{2}+12705 \sqrt {b \left (a e -b d \right )}\, a \,b^{4} e^{5} x^{4}+4620 \sqrt {b \left (a e -b d \right )}\, b^{5} d \,e^{4} x^{4}+3465 \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{4} b^{2} e^{4}+16863 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{3} e^{5} x^{3}+9207 \sqrt {b \left (a e -b d \right )}\, a^{3} b^{2} e^{5} x^{2}+1408 \sqrt {b \left (a e -b d \right )}\, a^{4} b \,e^{5} x -48 \sqrt {b \left (a e -b d \right )}\, b^{5} d^{5}+13860 \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3} b^{3} e^{4} x +17094 \sqrt {b \left (a e -b d \right )}\, a \,b^{4} d \,e^{4} x^{3}-128 \sqrt {b \left (a e -b d \right )}\, a^{5} e^{5}+22968 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{3} d \,e^{4} x^{2}+12782 \sqrt {b \left (a e -b d \right )}\, a^{3} b^{2} d \,e^{4} x +693 \sqrt {b \left (a e -b d \right )}\, b^{5} d^{2} e^{3} x^{3}-198 \sqrt {b \left (a e -b d \right )}\, b^{5} d^{3} e^{2} x^{2}+88 \sqrt {b \left (a e -b d \right )}\, b^{5} d^{4} e x +2295 \sqrt {b \left (a e -b d \right )}\, a^{3} b^{2} d^{2} e^{3}-1030 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{3} d^{3} e^{2}+328 \sqrt {b \left (a e -b d \right )}\, a \,b^{4} d^{4} e +3465 \sqrt {b \left (a e -b d \right )}\, b^{5} e^{5} x^{5}+2048 \sqrt {b \left (a e -b d \right )}\, a^{4} b d \,e^{4}+3465 \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{6} e^{4} x^{4}\right ) \left (b x +a \right )}{192 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} \left (a e -b d \right )^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(763\)

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,method=_RETURNVERBOSE)

[Out]

1/192*(2673*(b*(a*e-b*d))^(1/2)*a*b^4*d^2*e^3*x^2+3795*(b*(a*e-b*d))^(1/2)*a^2*b^3*d^2*e^3*x-748*(b*(a*e-b*d))
^(1/2)*a*b^4*d^3*e^2*x+13860*(e*x+d)^(3/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a*b^5*e^4*x^3+20790*(e*
x+d)^(3/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^2*b^4*e^4*x^2+12705*(b*(a*e-b*d))^(1/2)*a*b^4*e^5*x^4
+4620*(b*(a*e-b*d))^(1/2)*b^5*d*e^4*x^4+3465*(e*x+d)^(3/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^4*b^2
*e^4+16863*(b*(a*e-b*d))^(1/2)*a^2*b^3*e^5*x^3+9207*(b*(a*e-b*d))^(1/2)*a^3*b^2*e^5*x^2+1408*(b*(a*e-b*d))^(1/
2)*a^4*b*e^5*x-48*(b*(a*e-b*d))^(1/2)*b^5*d^5+13860*(e*x+d)^(3/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*
a^3*b^3*e^4*x+17094*(b*(a*e-b*d))^(1/2)*a*b^4*d*e^4*x^3-128*(b*(a*e-b*d))^(1/2)*a^5*e^5+22968*(b*(a*e-b*d))^(1
/2)*a^2*b^3*d*e^4*x^2+12782*(b*(a*e-b*d))^(1/2)*a^3*b^2*d*e^4*x+693*(b*(a*e-b*d))^(1/2)*b^5*d^2*e^3*x^3-198*(b
*(a*e-b*d))^(1/2)*b^5*d^3*e^2*x^2+88*(b*(a*e-b*d))^(1/2)*b^5*d^4*e*x+2295*(b*(a*e-b*d))^(1/2)*a^3*b^2*d^2*e^3-
1030*(b*(a*e-b*d))^(1/2)*a^2*b^3*d^3*e^2+328*(b*(a*e-b*d))^(1/2)*a*b^4*d^4*e+3465*(b*(a*e-b*d))^(1/2)*b^5*e^5*
x^5+2048*(b*(a*e-b*d))^(1/2)*a^4*b*d*e^4+3465*(e*x+d)^(3/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*b^6*e^
4*x^4)*(b*x+a)/(b*(a*e-b*d))^(1/2)/(e*x+d)^(3/2)/(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*} \text {Failed to integrate} \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)*(x*e + d)^(5/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1203 vs. \(2 (279) = 558\).
time = 2.74, size = 2418, normalized size = 6.35 \begin {gather*} \text {Too large to display} \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="fricas")

[Out]

[1/384*(3465*((b^5*x^6 + 4*a*b^4*x^5 + 6*a^2*b^3*x^4 + 4*a^3*b^2*x^3 + a^4*b*x^2)*e^6 + 2*(b^5*d*x^5 + 4*a*b^4
*d*x^4 + 6*a^2*b^3*d*x^3 + 4*a^3*b^2*d*x^2 + a^4*b*d*x)*e^5 + (b^5*d^2*x^4 + 4*a*b^4*d^2*x^3 + 6*a^2*b^3*d^2*x
^2 + 4*a^3*b^2*d^2*x + a^4*b*d^2)*e^4)*sqrt(b/(b*d - a*e))*log((2*b*d - 2*(b*d - a*e)*sqrt(x*e + d)*sqrt(b/(b*
d - a*e)) + (b*x - a)*e)/(b*x + a)) - 2*(48*b^5*d^5 - (3465*b^5*x^5 + 12705*a*b^4*x^4 + 16863*a^2*b^3*x^3 + 92
07*a^3*b^2*x^2 + 1408*a^4*b*x - 128*a^5)*e^5 - 2*(2310*b^5*d*x^4 + 8547*a*b^4*d*x^3 + 11484*a^2*b^3*d*x^2 + 63
91*a^3*b^2*d*x + 1024*a^4*b*d)*e^4 - 3*(231*b^5*d^2*x^3 + 891*a*b^4*d^2*x^2 + 1265*a^2*b^3*d^2*x + 765*a^3*b^2
*d^2)*e^3 + 2*(99*b^5*d^3*x^2 + 374*a*b^4*d^3*x + 515*a^2*b^3*d^3)*e^2 - 8*(11*b^5*d^4*x + 41*a*b^4*d^4)*e)*sq
rt(x*e + d))/(b^10*d^8*x^4 + 4*a*b^9*d^8*x^3 + 6*a^2*b^8*d^8*x^2 + 4*a^3*b^7*d^8*x + a^4*b^6*d^8 + (a^6*b^4*x^
6 + 4*a^7*b^3*x^5 + 6*a^8*b^2*x^4 + 4*a^9*b*x^3 + a^10*x^2)*e^8 - 2*(3*a^5*b^5*d*x^6 + 11*a^6*b^4*d*x^5 + 14*a
^7*b^3*d*x^4 + 6*a^8*b^2*d*x^3 - a^9*b*d*x^2 - a^10*d*x)*e^7 + (15*a^4*b^6*d^2*x^6 + 48*a^5*b^5*d^2*x^5 + 43*a
^6*b^4*d^2*x^4 - 8*a^7*b^3*d^2*x^3 - 27*a^8*b^2*d^2*x^2 - 8*a^9*b*d^2*x + a^10*d^2)*e^6 - 2*(10*a^3*b^7*d^3*x^
6 + 25*a^4*b^6*d^3*x^5 + 3*a^5*b^5*d^3*x^4 - 38*a^6*b^4*d^3*x^3 - 32*a^7*b^3*d^3*x^2 - 3*a^8*b^2*d^3*x + 3*a^9
*b*d^3)*e^5 + 5*(3*a^2*b^8*d^4*x^6 + 4*a^3*b^7*d^4*x^5 - 11*a^4*b^6*d^4*x^4 - 24*a^5*b^5*d^4*x^3 - 11*a^6*b^4*
d^4*x^2 + 4*a^7*b^3*d^4*x + 3*a^8*b^2*d^4)*e^4 - 2*(3*a*b^9*d^5*x^6 - 3*a^2*b^8*d^5*x^5 - 32*a^3*b^7*d^5*x^4 -
38*a^4*b^6*d^5*x^3 + 3*a^5*b^5*d^5*x^2 + 25*a^6*b^4*d^5*x + 10*a^7*b^3*d^5)*e^3 + (b^10*d^6*x^6 - 8*a*b^9*d^6
*x^5 - 27*a^2*b^8*d^6*x^4 - 8*a^3*b^7*d^6*x^3 + 43*a^4*b^6*d^6*x^2 + 48*a^5*b^5*d^6*x + 15*a^6*b^4*d^6)*e^2 +
2*(b^10*d^7*x^5 + a*b^9*d^7*x^4 - 6*a^2*b^8*d^7*x^3 - 14*a^3*b^7*d^7*x^2 - 11*a^4*b^6*d^7*x - 3*a^5*b^5*d^7)*e
), -1/192*(3465*((b^5*x^6 + 4*a*b^4*x^5 + 6*a^2*b^3*x^4 + 4*a^3*b^2*x^3 + a^4*b*x^2)*e^6 + 2*(b^5*d*x^5 + 4*a*
b^4*d*x^4 + 6*a^2*b^3*d*x^3 + 4*a^3*b^2*d*x^2 + a^4*b*d*x)*e^5 + (b^5*d^2*x^4 + 4*a*b^4*d^2*x^3 + 6*a^2*b^3*d^
2*x^2 + 4*a^3*b^2*d^2*x + a^4*b*d^2)*e^4)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(x*e + d)*sqrt(-b/(b*d
- a*e))/(b*x*e + b*d)) + (48*b^5*d^5 - (3465*b^5*x^5 + 12705*a*b^4*x^4 + 16863*a^2*b^3*x^3 + 9207*a^3*b^2*x^2
+ 1408*a^4*b*x - 128*a^5)*e^5 - 2*(2310*b^5*d*x^4 + 8547*a*b^4*d*x^3 + 11484*a^2*b^3*d*x^2 + 6391*a^3*b^2*d*x
+ 1024*a^4*b*d)*e^4 - 3*(231*b^5*d^2*x^3 + 891*a*b^4*d^2*x^2 + 1265*a^2*b^3*d^2*x + 765*a^3*b^2*d^2)*e^3 + 2*(
99*b^5*d^3*x^2 + 374*a*b^4*d^3*x + 515*a^2*b^3*d^3)*e^2 - 8*(11*b^5*d^4*x + 41*a*b^4*d^4)*e)*sqrt(x*e + d))/(b
^10*d^8*x^4 + 4*a*b^9*d^8*x^3 + 6*a^2*b^8*d^8*x^2 + 4*a^3*b^7*d^8*x + a^4*b^6*d^8 + (a^6*b^4*x^6 + 4*a^7*b^3*x
^5 + 6*a^8*b^2*x^4 + 4*a^9*b*x^3 + a^10*x^2)*e^8 - 2*(3*a^5*b^5*d*x^6 + 11*a^6*b^4*d*x^5 + 14*a^7*b^3*d*x^4 +
6*a^8*b^2*d*x^3 - a^9*b*d*x^2 - a^10*d*x)*e^7 + (15*a^4*b^6*d^2*x^6 + 48*a^5*b^5*d^2*x^5 + 43*a^6*b^4*d^2*x^4
- 8*a^7*b^3*d^2*x^3 - 27*a^8*b^2*d^2*x^2 - 8*a^9*b*d^2*x + a^10*d^2)*e^6 - 2*(10*a^3*b^7*d^3*x^6 + 25*a^4*b^6*
d^3*x^5 + 3*a^5*b^5*d^3*x^4 - 38*a^6*b^4*d^3*x^3 - 32*a^7*b^3*d^3*x^2 - 3*a^8*b^2*d^3*x + 3*a^9*b*d^3)*e^5 + 5
*(3*a^2*b^8*d^4*x^6 + 4*a^3*b^7*d^4*x^5 - 11*a^4*b^6*d^4*x^4 - 24*a^5*b^5*d^4*x^3 - 11*a^6*b^4*d^4*x^2 + 4*a^7
*b^3*d^4*x + 3*a^8*b^2*d^4)*e^4 - 2*(3*a*b^9*d^5*x^6 - 3*a^2*b^8*d^5*x^5 - 32*a^3*b^7*d^5*x^4 - 38*a^4*b^6*d^5
*x^3 + 3*a^5*b^5*d^5*x^2 + 25*a^6*b^4*d^5*x + 10*a^7*b^3*d^5)*e^3 + (b^10*d^6*x^6 - 8*a*b^9*d^6*x^5 - 27*a^2*b
^8*d^6*x^4 - 8*a^3*b^7*d^6*x^3 + 43*a^4*b^6*d^6*x^2 + 48*a^5*b^5*d^6*x + 15*a^6*b^4*d^6)*e^2 + 2*(b^10*d^7*x^5
+ a*b^9*d^7*x^4 - 6*a^2*b^8*d^7*x^3 - 14*a^3*b^7*d^7*x^2 - 11*a^4*b^6*d^7*x - 3*a^5*b^5*d^7)*e)]

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

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 626 vs. \(2 (279) = 558\).
time = 1.07, size = 626, normalized size = 1.64 \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 (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\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 (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\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 (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\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(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x +
a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) - 20*a^3*b^3*d^3*e^3*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 6*a
^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))*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(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) - 20*a^3*b^3*d^3*e^
3*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 6*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))*(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 + 6
885*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*s
qrt(x*e + d)*a^3*b^2*e^7)/((b^6*d^6*sgn(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a
) - 20*a^3*b^3*d^3*e^3*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 6*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*s
gn(b*x + a))*((x*e + d)*b - b*d + a*e)^4)

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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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