3.17.75 \(\int \frac {1}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\) [1675]

Optimal. Leaf size=295 \[ -\frac {9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 (d+e x)^{3/2}}-\frac {9009 b^2 e^5}{128 (b d-a e)^8 \sqrt {d+e x}}+\frac {9009 b^{5/2} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}} \]

[Out]

-9009/640*e^5/(-a*e+b*d)^6/(e*x+d)^(5/2)-1/5/(-a*e+b*d)/(b*x+a)^5/(e*x+d)^(5/2)+3/8*e/(-a*e+b*d)^2/(b*x+a)^4/(
e*x+d)^(5/2)-13/16*e^2/(-a*e+b*d)^3/(b*x+a)^3/(e*x+d)^(5/2)+143/64*e^3/(-a*e+b*d)^4/(b*x+a)^2/(e*x+d)^(5/2)-12
87/128*e^4/(-a*e+b*d)^5/(b*x+a)/(e*x+d)^(5/2)-3003/128*b*e^5/(-a*e+b*d)^7/(e*x+d)^(3/2)+9009/128*b^(5/2)*e^5*a
rctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(17/2)-9009/128*b^2*e^5/(-a*e+b*d)^8/(e*x+d)^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 44, 53, 65, 214} \begin {gather*} \frac {9009 b^{5/2} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}}-\frac {9009 b^2 e^5}{128 \sqrt {d+e x} (b d-a e)^8}-\frac {3003 b e^5}{128 (d+e x)^{3/2} (b d-a e)^7}-\frac {9009 e^5}{640 (d+e x)^{5/2} (b d-a e)^6}-\frac {1287 e^4}{128 (a+b x) (d+e x)^{5/2} (b d-a e)^5}+\frac {143 e^3}{64 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}-\frac {13 e^2}{16 (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}+\frac {3 e}{8 (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac {1}{5 (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-9009*e^5)/(640*(b*d - a*e)^6*(d + e*x)^(5/2)) - 1/(5*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(5/2)) + (3*e)/(8*(b*
d - a*e)^2*(a + b*x)^4*(d + e*x)^(5/2)) - (13*e^2)/(16*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(5/2)) + (143*e^3)/
(64*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(5/2)) - (1287*e^4)/(128*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(5/2)) - (3
003*b*e^5)/(128*(b*d - a*e)^7*(d + e*x)^(3/2)) - (9009*b^2*e^5)/(128*(b*d - a*e)^8*Sqrt[d + e*x]) + (9009*b^(5
/2)*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(17/2))

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 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]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^6 (d+e x)^{7/2}} \, dx\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {(3 e) \int \frac {1}{(a+b x)^5 (d+e x)^{7/2}} \, dx}{2 (b d-a e)}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {\left (39 e^2\right ) \int \frac {1}{(a+b x)^4 (d+e x)^{7/2}} \, dx}{16 (b d-a e)^2}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {\left (143 e^3\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{7/2}} \, dx}{32 (b d-a e)^3}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {\left (1287 e^4\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{7/2}} \, dx}{128 (b d-a e)^4}\\ &=-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac {\left (9009 e^5\right ) \int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx}{256 (b d-a e)^5}\\ &=-\frac {9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac {\left (9009 b e^5\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{256 (b d-a e)^6}\\ &=-\frac {9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 (d+e x)^{3/2}}-\frac {\left (9009 b^2 e^5\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^7}\\ &=-\frac {9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 (d+e x)^{3/2}}-\frac {9009 b^2 e^5}{128 (b d-a e)^8 \sqrt {d+e x}}-\frac {\left (9009 b^3 e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^8}\\ &=-\frac {9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 (d+e x)^{3/2}}-\frac {9009 b^2 e^5}{128 (b d-a e)^8 \sqrt {d+e x}}-\frac {\left (9009 b^3 e^4\right ) \text {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)^8}\\ &=-\frac {9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac {1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac {3003 b e^5}{128 (b d-a e)^7 (d+e x)^{3/2}}-\frac {9009 b^2 e^5}{128 (b d-a e)^8 \sqrt {d+e x}}+\frac {9009 b^{5/2} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}}\\ \end {align*}

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Mathematica [A]
time = 2.30, size = 443, normalized size = 1.50 \begin {gather*} \frac {1}{640} \left (\frac {-256 a^7 e^7+256 a^6 b e^6 (12 d+5 e x)-256 a^5 b^2 e^5 \left (116 d^2+160 d e x+65 e^2 x^2\right )-5 a^4 b^3 e^4 \left (5327 d^3+45677 d^2 e x+66157 d e^2 x^2+27599 e^3 x^3\right )-10 a^3 b^4 e^3 \left (-1211 d^4+5810 d^3 e x+54392 d^2 e^2 x^2+80366 d e^3 x^3+33891 e^4 x^4\right )-2 a^2 b^5 e^2 \left (2324 d^5-6545 d^4 e x+30485 d^3 e^2 x^2+302445 d^2 e^3 x^3+452595 d e^4 x^4+192192 e^5 x^5\right )-2 a b^6 e \left (-568 d^6+1240 d^5 e x-3445 d^4 e^2 x^2+15730 d^3 e^3 x^3+163020 d^2 e^4 x^4+246246 d e^5 x^5+105105 e^6 x^6\right )-b^7 \left (128 d^7-240 d^6 e x+520 d^5 e^2 x^2-1430 d^4 e^3 x^3+6435 d^3 e^4 x^4+69069 d^2 e^5 x^5+105105 d e^6 x^6+45045 e^7 x^7\right )}{(b d-a e)^8 (a+b x)^5 (d+e x)^{5/2}}-\frac {45045 b^{5/2} e^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{17/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

((-256*a^7*e^7 + 256*a^6*b*e^6*(12*d + 5*e*x) - 256*a^5*b^2*e^5*(116*d^2 + 160*d*e*x + 65*e^2*x^2) - 5*a^4*b^3
*e^4*(5327*d^3 + 45677*d^2*e*x + 66157*d*e^2*x^2 + 27599*e^3*x^3) - 10*a^3*b^4*e^3*(-1211*d^4 + 5810*d^3*e*x +
54392*d^2*e^2*x^2 + 80366*d*e^3*x^3 + 33891*e^4*x^4) - 2*a^2*b^5*e^2*(2324*d^5 - 6545*d^4*e*x + 30485*d^3*e^2
*x^2 + 302445*d^2*e^3*x^3 + 452595*d*e^4*x^4 + 192192*e^5*x^5) - 2*a*b^6*e*(-568*d^6 + 1240*d^5*e*x - 3445*d^4
*e^2*x^2 + 15730*d^3*e^3*x^3 + 163020*d^2*e^4*x^4 + 246246*d*e^5*x^5 + 105105*e^6*x^6) - b^7*(128*d^7 - 240*d^
6*e*x + 520*d^5*e^2*x^2 - 1430*d^4*e^3*x^3 + 6435*d^3*e^4*x^4 + 69069*d^2*e^5*x^5 + 105105*d*e^6*x^6 + 45045*e
^7*x^7))/((b*d - a*e)^8*(a + b*x)^5*(d + e*x)^(5/2)) - (45045*b^(5/2)*e^5*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[
-(b*d) + a*e]])/(-(b*d) + a*e)^(17/2))/640

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Maple [A]
time = 0.79, size = 314, normalized size = 1.06

method	result	size
derivativedivides	\(2 e^{5} \left (-\frac {b^{3} \left (\frac {\frac {3633 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}+\frac {7837 \left (a e -b d \right ) b^{3} \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (\frac {1001}{10} a^{2} b^{2} e^{2}-\frac {1001}{5} a \,b^{3} d e +\frac {1001}{10} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {9443}{128} a^{3} b \,e^{3}-\frac {28329}{128} a^{2} b^{2} d \,e^{2}+\frac {28329}{128} d^{2} e a \,b^{3}-\frac {9443}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {5327}{256} e^{4} a^{4}-\frac {5327}{64} a^{3} b d \,e^{3}+\frac {15981}{128} a^{2} b^{2} d^{2} e^{2}-\frac {5327}{64} a \,b^{3} d^{3} e +\frac {5327}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {9009 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{8}}-\frac {1}{5 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {5}{2}}}-\frac {21 b^{2}}{\left (a e -b d \right )^{8} \sqrt {e x +d}}+\frac {2 b}{\left (a e -b d \right )^{7} \left (e x +d \right )^{\frac {3}{2}}}\right )\)	\(314\)
default	\(2 e^{5} \left (-\frac {b^{3} \left (\frac {\frac {3633 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}+\frac {7837 \left (a e -b d \right ) b^{3} \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (\frac {1001}{10} a^{2} b^{2} e^{2}-\frac {1001}{5} a \,b^{3} d e +\frac {1001}{10} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {9443}{128} a^{3} b \,e^{3}-\frac {28329}{128} a^{2} b^{2} d \,e^{2}+\frac {28329}{128} d^{2} e a \,b^{3}-\frac {9443}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {5327}{256} e^{4} a^{4}-\frac {5327}{64} a^{3} b d \,e^{3}+\frac {15981}{128} a^{2} b^{2} d^{2} e^{2}-\frac {5327}{64} a \,b^{3} d^{3} e +\frac {5327}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {9009 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{8}}-\frac {1}{5 \left (a e -b d \right )^{6} \left (e x +d \right )^{\frac {5}{2}}}-\frac {21 b^{2}}{\left (a e -b d \right )^{8} \sqrt {e x +d}}+\frac {2 b}{\left (a e -b d \right )^{7} \left (e x +d \right )^{\frac {3}{2}}}\right )\)	\(314\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^5*(-1/(a*e-b*d)^8*b^3*((3633/256*b^4*(e*x+d)^(9/2)+7837/128*(a*e-b*d)*b^3*(e*x+d)^(7/2)+(1001/10*a^2*b^2*e
^2-1001/5*a*b^3*d*e+1001/10*b^4*d^2)*(e*x+d)^(5/2)+(9443/128*a^3*b*e^3-28329/128*a^2*b^2*d*e^2+28329/128*d^2*e
*a*b^3-9443/128*b^4*d^3)*(e*x+d)^(3/2)+(5327/256*e^4*a^4-5327/64*a^3*b*d*e^3+15981/128*a^2*b^2*d^2*e^2-5327/64
*a*b^3*d^3*e+5327/256*b^4*d^4)*(e*x+d)^(1/2))/((e*x+d)*b+a*e-b*d)^5+9009/256/(b*(a*e-b*d))^(1/2)*arctan(b*(e*x
+d)^(1/2)/(b*(a*e-b*d))^(1/2)))-1/5/(a*e-b*d)^6/(e*x+d)^(5/2)-21/(a*e-b*d)^8*b^2/(e*x+d)^(1/2)+2/(a*e-b*d)^7*b
/(e*x+d)^(3/2))

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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)^(7/2)/(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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2053 vs. \(2 (264) = 528\).
time = 3.67, size = 4117, normalized size = 13.96 \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)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(45045*((b^7*x^8 + 5*a*b^6*x^7 + 10*a^2*b^5*x^6 + 10*a^3*b^4*x^5 + 5*a^4*b^3*x^4 + a^5*b^2*x^3)*e^8 +
3*(b^7*d*x^7 + 5*a*b^6*d*x^6 + 10*a^2*b^5*d*x^5 + 10*a^3*b^4*d*x^4 + 5*a^4*b^3*d*x^3 + a^5*b^2*d*x^2)*e^7 + 3*
(b^7*d^2*x^6 + 5*a*b^6*d^2*x^5 + 10*a^2*b^5*d^2*x^4 + 10*a^3*b^4*d^2*x^3 + 5*a^4*b^3*d^2*x^2 + a^5*b^2*d^2*x)*
e^6 + (b^7*d^3*x^5 + 5*a*b^6*d^3*x^4 + 10*a^2*b^5*d^3*x^3 + 10*a^3*b^4*d^3*x^2 + 5*a^4*b^3*d^3*x + a^5*b^2*d^3
)*e^5)*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*(128*b^7*d^7 + (45045*b^7*x^7 + 210210*a*b^6*x^6 + 384384*a^2*b^5*x^5 + 338910*a^3*b^4*x^4 + 137995*a^
4*b^3*x^3 + 16640*a^5*b^2*x^2 - 1280*a^6*b*x + 256*a^7)*e^7 + (105105*b^7*d*x^6 + 492492*a*b^6*d*x^5 + 905190*
a^2*b^5*d*x^4 + 803660*a^3*b^4*d*x^3 + 330785*a^4*b^3*d*x^2 + 40960*a^5*b^2*d*x - 3072*a^6*b*d)*e^6 + (69069*b
^7*d^2*x^5 + 326040*a*b^6*d^2*x^4 + 604890*a^2*b^5*d^2*x^3 + 543920*a^3*b^4*d^2*x^2 + 228385*a^4*b^3*d^2*x + 2
9696*a^5*b^2*d^2)*e^5 + 5*(1287*b^7*d^3*x^4 + 6292*a*b^6*d^3*x^3 + 12194*a^2*b^5*d^3*x^2 + 11620*a^3*b^4*d^3*x
+ 5327*a^4*b^3*d^3)*e^4 - 10*(143*b^7*d^4*x^3 + 689*a*b^6*d^4*x^2 + 1309*a^2*b^5*d^4*x + 1211*a^3*b^4*d^4)*e^
3 + 8*(65*b^7*d^5*x^2 + 310*a*b^6*d^5*x + 581*a^2*b^5*d^5)*e^2 - 16*(15*b^7*d^6*x + 71*a*b^6*d^6)*e)*sqrt(x*e
+ d))/(b^13*d^11*x^5 + 5*a*b^12*d^11*x^4 + 10*a^2*b^11*d^11*x^3 + 10*a^3*b^10*d^11*x^2 + 5*a^4*b^9*d^11*x + a^
5*b^8*d^11 + (a^8*b^5*x^8 + 5*a^9*b^4*x^7 + 10*a^10*b^3*x^6 + 10*a^11*b^2*x^5 + 5*a^12*b*x^4 + a^13*x^3)*e^11
- (8*a^7*b^6*d*x^8 + 37*a^8*b^5*d*x^7 + 65*a^9*b^4*d*x^6 + 50*a^10*b^3*d*x^5 + 10*a^11*b^2*d*x^4 - 7*a^12*b*d*
x^3 - 3*a^13*d*x^2)*e^10 + (28*a^6*b^7*d^2*x^8 + 116*a^7*b^6*d^2*x^7 + 163*a^8*b^5*d^2*x^6 + 55*a^9*b^4*d^2*x^
5 - 70*a^10*b^3*d^2*x^4 - 62*a^11*b^2*d^2*x^3 - 9*a^12*b*d^2*x^2 + 3*a^13*d^2*x)*e^9 - (56*a^5*b^8*d^3*x^8 + 1
96*a^6*b^7*d^3*x^7 + 164*a^7*b^6*d^3*x^6 - 161*a^8*b^5*d^3*x^5 - 325*a^9*b^4*d^3*x^4 - 134*a^10*b^3*d^3*x^3 +
26*a^11*b^2*d^3*x^2 + 19*a^12*b*d^3*x - a^13*d^3)*e^8 + 2*(35*a^4*b^9*d^4*x^8 + 91*a^5*b^8*d^4*x^7 - 28*a^6*b^
7*d^4*x^6 - 284*a^7*b^6*d^4*x^5 - 265*a^8*b^5*d^4*x^4 - 5*a^9*b^4*d^4*x^3 + 86*a^10*b^3*d^4*x^2 + 22*a^11*b^2*
d^4*x - 4*a^12*b*d^4)*e^7 - 14*(4*a^3*b^10*d^5*x^8 + 5*a^4*b^9*d^5*x^7 - 23*a^5*b^8*d^5*x^6 - 52*a^6*b^7*d^5*x
^5 - 20*a^7*b^6*d^5*x^4 + 29*a^8*b^5*d^5*x^3 + 25*a^9*b^4*d^5*x^2 + 2*a^10*b^3*d^5*x - 2*a^11*b^2*d^5)*e^6 + 1
4*(2*a^2*b^11*d^6*x^8 - 2*a^3*b^10*d^6*x^7 - 25*a^4*b^9*d^6*x^6 - 29*a^5*b^8*d^6*x^5 + 20*a^6*b^7*d^6*x^4 + 52
*a^7*b^6*d^6*x^3 + 23*a^8*b^5*d^6*x^2 - 5*a^9*b^4*d^6*x - 4*a^10*b^3*d^6)*e^5 - 2*(4*a*b^12*d^7*x^8 - 22*a^2*b
^11*d^7*x^7 - 86*a^3*b^10*d^7*x^6 + 5*a^4*b^9*d^7*x^5 + 265*a^5*b^8*d^7*x^4 + 284*a^6*b^7*d^7*x^3 + 28*a^7*b^6
*d^7*x^2 - 91*a^8*b^5*d^7*x - 35*a^9*b^4*d^7)*e^4 + (b^13*d^8*x^8 - 19*a*b^12*d^8*x^7 - 26*a^2*b^11*d^8*x^6 +
134*a^3*b^10*d^8*x^5 + 325*a^4*b^9*d^8*x^4 + 161*a^5*b^8*d^8*x^3 - 164*a^6*b^7*d^8*x^2 - 196*a^7*b^6*d^8*x - 5
6*a^8*b^5*d^8)*e^3 + (3*b^13*d^9*x^7 - 9*a*b^12*d^9*x^6 - 62*a^2*b^11*d^9*x^5 - 70*a^3*b^10*d^9*x^4 + 55*a^4*b
^9*d^9*x^3 + 163*a^5*b^8*d^9*x^2 + 116*a^6*b^7*d^9*x + 28*a^7*b^6*d^9)*e^2 + (3*b^13*d^10*x^6 + 7*a*b^12*d^10*
x^5 - 10*a^2*b^11*d^10*x^4 - 50*a^3*b^10*d^10*x^3 - 65*a^4*b^9*d^10*x^2 - 37*a^5*b^8*d^10*x - 8*a^6*b^7*d^10)*
e), 1/640*(45045*((b^7*x^8 + 5*a*b^6*x^7 + 10*a^2*b^5*x^6 + 10*a^3*b^4*x^5 + 5*a^4*b^3*x^4 + a^5*b^2*x^3)*e^8
+ 3*(b^7*d*x^7 + 5*a*b^6*d*x^6 + 10*a^2*b^5*d*x^5 + 10*a^3*b^4*d*x^4 + 5*a^4*b^3*d*x^3 + a^5*b^2*d*x^2)*e^7 +
3*(b^7*d^2*x^6 + 5*a*b^6*d^2*x^5 + 10*a^2*b^5*d^2*x^4 + 10*a^3*b^4*d^2*x^3 + 5*a^4*b^3*d^2*x^2 + a^5*b^2*d^2*x
)*e^6 + (b^7*d^3*x^5 + 5*a*b^6*d^3*x^4 + 10*a^2*b^5*d^3*x^3 + 10*a^3*b^4*d^3*x^2 + 5*a^4*b^3*d^3*x + a^5*b^2*d
^3)*e^5)*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)) - (128*b^7
*d^7 + (45045*b^7*x^7 + 210210*a*b^6*x^6 + 384384*a^2*b^5*x^5 + 338910*a^3*b^4*x^4 + 137995*a^4*b^3*x^3 + 1664
0*a^5*b^2*x^2 - 1280*a^6*b*x + 256*a^7)*e^7 + (105105*b^7*d*x^6 + 492492*a*b^6*d*x^5 + 905190*a^2*b^5*d*x^4 +
803660*a^3*b^4*d*x^3 + 330785*a^4*b^3*d*x^2 + 40960*a^5*b^2*d*x - 3072*a^6*b*d)*e^6 + (69069*b^7*d^2*x^5 + 326
040*a*b^6*d^2*x^4 + 604890*a^2*b^5*d^2*x^3 + 543920*a^3*b^4*d^2*x^2 + 228385*a^4*b^3*d^2*x + 29696*a^5*b^2*d^2
)*e^5 + 5*(1287*b^7*d^3*x^4 + 6292*a*b^6*d^3*x^3 + 12194*a^2*b^5*d^3*x^2 + 11620*a^3*b^4*d^3*x + 5327*a^4*b^3*
d^3)*e^4 - 10*(143*b^7*d^4*x^3 + 689*a*b^6*d^4*x^2 + 1309*a^2*b^5*d^4*x + 1211*a^3*b^4*d^4)*e^3 + 8*(65*b^7*d^
5*x^2 + 310*a*b^6*d^5*x + 581*a^2*b^5*d^5)*e^2 - 16*(15*b^7*d^6*x + 71*a*b^6*d^6)*e)*sqrt(x*e + d))/(b^13*d^11
*x^5 + 5*a*b^12*d^11*x^4 + 10*a^2*b^11*d^11*x^3 + 10*a^3*b^10*d^11*x^2 + 5*a^4*b^9*d^11*x + a^5*b^8*d^11 + (a^
8*b^5*x^8 + 5*a^9*b^4*x^7 + 10*a^10*b^3*x^6 + 10*a^11*b^2*x^5 + 5*a^12*b*x^4 + a^13*x^3)*e^11 - (8*a^7*b^6*d*x
^8 + 37*a^8*b^5*d*x^7 + 65*a^9*b^4*d*x^6 + 50*a^10*b^3*d*x^5 + 10*a^11*b^2*d*x^4 - 7*a^12*b*d*x^3 - 3*a^13*d*x
^2)*e^10 + (28*a^6*b^7*d^2*x^8 + 116*a^7*b^6*d^...

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Sympy [F(-1)] Timed out
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)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (264) = 528\).
time = 1.12, size = 884, normalized size = 3.00 \begin {gather*} -\frac {9009 \, b^{3} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{5}}{128 \, {\left (b^{8} d^{8} - 8 \, a b^{7} d^{7} e + 28 \, a^{2} b^{6} d^{6} e^{2} - 56 \, a^{3} b^{5} d^{5} e^{3} + 70 \, a^{4} b^{4} d^{4} e^{4} - 56 \, a^{5} b^{3} d^{3} e^{5} + 28 \, a^{6} b^{2} d^{2} e^{6} - 8 \, a^{7} b d e^{7} + a^{8} e^{8}\right )} \sqrt {-b^{2} d + a b e}} - \frac {45045 \, {\left (x e + d\right )}^{7} b^{7} e^{5} - 210210 \, {\left (x e + d\right )}^{6} b^{7} d e^{5} + 384384 \, {\left (x e + d\right )}^{5} b^{7} d^{2} e^{5} - 338910 \, {\left (x e + d\right )}^{4} b^{7} d^{3} e^{5} + 137995 \, {\left (x e + d\right )}^{3} b^{7} d^{4} e^{5} - 16640 \, {\left (x e + d\right )}^{2} b^{7} d^{5} e^{5} - 1280 \, {\left (x e + d\right )} b^{7} d^{6} e^{5} - 256 \, b^{7} d^{7} e^{5} + 210210 \, {\left (x e + d\right )}^{6} a b^{6} e^{6} - 768768 \, {\left (x e + d\right )}^{5} a b^{6} d e^{6} + 1016730 \, {\left (x e + d\right )}^{4} a b^{6} d^{2} e^{6} - 551980 \, {\left (x e + d\right )}^{3} a b^{6} d^{3} e^{6} + 83200 \, {\left (x e + d\right )}^{2} a b^{6} d^{4} e^{6} + 7680 \, {\left (x e + d\right )} a b^{6} d^{5} e^{6} + 1792 \, a b^{6} d^{6} e^{6} + 384384 \, {\left (x e + d\right )}^{5} a^{2} b^{5} e^{7} - 1016730 \, {\left (x e + d\right )}^{4} a^{2} b^{5} d e^{7} + 827970 \, {\left (x e + d\right )}^{3} a^{2} b^{5} d^{2} e^{7} - 166400 \, {\left (x e + d\right )}^{2} a^{2} b^{5} d^{3} e^{7} - 19200 \, {\left (x e + d\right )} a^{2} b^{5} d^{4} e^{7} - 5376 \, a^{2} b^{5} d^{5} e^{7} + 338910 \, {\left (x e + d\right )}^{4} a^{3} b^{4} e^{8} - 551980 \, {\left (x e + d\right )}^{3} a^{3} b^{4} d e^{8} + 166400 \, {\left (x e + d\right )}^{2} a^{3} b^{4} d^{2} e^{8} + 25600 \, {\left (x e + d\right )} a^{3} b^{4} d^{3} e^{8} + 8960 \, a^{3} b^{4} d^{4} e^{8} + 137995 \, {\left (x e + d\right )}^{3} a^{4} b^{3} e^{9} - 83200 \, {\left (x e + d\right )}^{2} a^{4} b^{3} d e^{9} - 19200 \, {\left (x e + d\right )} a^{4} b^{3} d^{2} e^{9} - 8960 \, a^{4} b^{3} d^{3} e^{9} + 16640 \, {\left (x e + d\right )}^{2} a^{5} b^{2} e^{10} + 7680 \, {\left (x e + d\right )} a^{5} b^{2} d e^{10} + 5376 \, a^{5} b^{2} d^{2} e^{10} - 1280 \, {\left (x e + d\right )} a^{6} b e^{11} - 1792 \, a^{6} b d e^{11} + 256 \, a^{7} e^{12}}{640 \, {\left (b^{8} d^{8} - 8 \, a b^{7} d^{7} e + 28 \, a^{2} b^{6} d^{6} e^{2} - 56 \, a^{3} b^{5} d^{5} e^{3} + 70 \, a^{4} b^{4} d^{4} e^{4} - 56 \, a^{5} b^{3} d^{3} e^{5} + 28 \, a^{6} b^{2} d^{2} e^{6} - 8 \, a^{7} b d e^{7} + a^{8} e^{8}\right )} {\left ({\left (x e + d\right )}^{\frac {3}{2}} b - \sqrt {x e + d} b d + \sqrt {x e + d} a e\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-9009/128*b^3*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^8*d^8 - 8*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2
- 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8)
*sqrt(-b^2*d + a*b*e)) - 1/640*(45045*(x*e + d)^7*b^7*e^5 - 210210*(x*e + d)^6*b^7*d*e^5 + 384384*(x*e + d)^5*
b^7*d^2*e^5 - 338910*(x*e + d)^4*b^7*d^3*e^5 + 137995*(x*e + d)^3*b^7*d^4*e^5 - 16640*(x*e + d)^2*b^7*d^5*e^5
- 1280*(x*e + d)*b^7*d^6*e^5 - 256*b^7*d^7*e^5 + 210210*(x*e + d)^6*a*b^6*e^6 - 768768*(x*e + d)^5*a*b^6*d*e^6
+ 1016730*(x*e + d)^4*a*b^6*d^2*e^6 - 551980*(x*e + d)^3*a*b^6*d^3*e^6 + 83200*(x*e + d)^2*a*b^6*d^4*e^6 + 76
80*(x*e + d)*a*b^6*d^5*e^6 + 1792*a*b^6*d^6*e^6 + 384384*(x*e + d)^5*a^2*b^5*e^7 - 1016730*(x*e + d)^4*a^2*b^5
*d*e^7 + 827970*(x*e + d)^3*a^2*b^5*d^2*e^7 - 166400*(x*e + d)^2*a^2*b^5*d^3*e^7 - 19200*(x*e + d)*a^2*b^5*d^4
*e^7 - 5376*a^2*b^5*d^5*e^7 + 338910*(x*e + d)^4*a^3*b^4*e^8 - 551980*(x*e + d)^3*a^3*b^4*d*e^8 + 166400*(x*e
+ d)^2*a^3*b^4*d^2*e^8 + 25600*(x*e + d)*a^3*b^4*d^3*e^8 + 8960*a^3*b^4*d^4*e^8 + 137995*(x*e + d)^3*a^4*b^3*e
^9 - 83200*(x*e + d)^2*a^4*b^3*d*e^9 - 19200*(x*e + d)*a^4*b^3*d^2*e^9 - 8960*a^4*b^3*d^3*e^9 + 16640*(x*e + d
)^2*a^5*b^2*e^10 + 7680*(x*e + d)*a^5*b^2*d*e^10 + 5376*a^5*b^2*d^2*e^10 - 1280*(x*e + d)*a^6*b*e^11 - 1792*a^
6*b*d*e^11 + 256*a^7*e^12)/((b^8*d^8 - 8*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^
4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8)*((x*e + d)^(3/2)*b - sqrt(x*e + d)*
b*d + sqrt(x*e + d)*a*e)^5)

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Mupad [B]
time = 1.51, size = 594, normalized size = 2.01 \begin {gather*} -\frac {\frac {2\,e^5}{5\,\left (a\,e-b\,d\right )}+\frac {26\,b^2\,e^5\,{\left (d+e\,x\right )}^2}{{\left (a\,e-b\,d\right )}^3}+\frac {27599\,b^3\,e^5\,{\left (d+e\,x\right )}^3}{128\,{\left (a\,e-b\,d\right )}^4}+\frac {33891\,b^4\,e^5\,{\left (d+e\,x\right )}^4}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {3003\,b^5\,e^5\,{\left (d+e\,x\right )}^5}{5\,{\left (a\,e-b\,d\right )}^6}+\frac {21021\,b^6\,e^5\,{\left (d+e\,x\right )}^6}{64\,{\left (a\,e-b\,d\right )}^7}+\frac {9009\,b^7\,e^5\,{\left (d+e\,x\right )}^7}{128\,{\left (a\,e-b\,d\right )}^8}-\frac {2\,b\,e^5\,\left (d+e\,x\right )}{{\left (a\,e-b\,d\right )}^2}}{{\left (d+e\,x\right )}^{5/2}\,\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 )}^{9/2}\,\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 )}^{7/2}\,\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 )}^{15/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{13/2}+{\left (d+e\,x\right )}^{11/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}-\frac {9009\,b^{5/2}\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^8\,e^8-8\,a^7\,b\,d\,e^7+28\,a^6\,b^2\,d^2\,e^6-56\,a^5\,b^3\,d^3\,e^5+70\,a^4\,b^4\,d^4\,e^4-56\,a^3\,b^5\,d^5\,e^3+28\,a^2\,b^6\,d^6\,e^2-8\,a\,b^7\,d^7\,e+b^8\,d^8\right )}{{\left (a\,e-b\,d\right )}^{17/2}}\right )}{128\,{\left (a\,e-b\,d\right )}^{17/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)

[Out]

- ((2*e^5)/(5*(a*e - b*d)) + (26*b^2*e^5*(d + e*x)^2)/(a*e - b*d)^3 + (27599*b^3*e^5*(d + e*x)^3)/(128*(a*e -
b*d)^4) + (33891*b^4*e^5*(d + e*x)^4)/(64*(a*e - b*d)^5) + (3003*b^5*e^5*(d + e*x)^5)/(5*(a*e - b*d)^6) + (210
21*b^6*e^5*(d + e*x)^6)/(64*(a*e - b*d)^7) + (9009*b^7*e^5*(d + e*x)^7)/(128*(a*e - b*d)^8) - (2*b*e^5*(d + e*
x))/(a*e - b*d)^2)/((d + e*x)^(5/2)*(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)^(9/2)*(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)^(7/2)*(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)
^(15/2) - (5*b^5*d - 5*a*b^4*e)*(d + e*x)^(13/2) + (d + e*x)^(11/2)*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*
e)) - (9009*b^(5/2)*e^5*atan((b^(1/2)*(d + e*x)^(1/2)*(a^8*e^8 + b^8*d^8 + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5
*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a*b^7*d^7*e - 8*a^7*b*d*e^7))/(a*e - b
*d)^(17/2)))/(128*(a*e - b*d)^(17/2))

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