3.1667 \(\int \frac {(d+e x)^{9/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=178 \[ -\frac {63 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} \sqrt {b d-a e}}-\frac {63 e^4 \sqrt {d+e x}}{128 b^5 (a+b x)}-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5} \]

[Out]

-21/64*e^3*(e*x+d)^(3/2)/b^4/(b*x+a)^2-21/80*e^2*(e*x+d)^(5/2)/b^3/(b*x+a)^3-9/40*e*(e*x+d)^(7/2)/b^2/(b*x+a)^
4-1/5*(e*x+d)^(9/2)/b/(b*x+a)^5-63/128*e^5*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(11/2)/(-a*e+b*d)
^(1/2)-63/128*e^4*(e*x+d)^(1/2)/b^5/(b*x+a)

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Rubi [A]  time = 0.09, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 47, 63, 208} \[ -\frac {63 e^4 \sqrt {d+e x}}{128 b^5 (a+b x)}-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {63 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} \sqrt {b d-a e}}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(-63*e^4*Sqrt[d + e*x])/(128*b^5*(a + b*x)) - (21*e^3*(d + e*x)^(3/2))/(64*b^4*(a + b*x)^2) - (21*e^2*(d + e*x
)^(5/2))/(80*b^3*(a + b*x)^3) - (9*e*(d + e*x)^(7/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(9/2)/(5*b*(a + b*x)^5)
- (63*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(11/2)*Sqrt[b*d - a*e])

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^{9/2}}{(a+b x)^6} \, dx\\ &=-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac {(9 e) \int \frac {(d+e x)^{7/2}}{(a+b x)^5} \, dx}{10 b}\\ &=-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac {\left (63 e^2\right ) \int \frac {(d+e x)^{5/2}}{(a+b x)^4} \, dx}{80 b^2}\\ &=-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac {\left (21 e^3\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^3} \, dx}{32 b^3}\\ &=-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac {\left (63 e^4\right ) \int \frac {\sqrt {d+e x}}{(a+b x)^2} \, dx}{128 b^4}\\ &=-\frac {63 e^4 \sqrt {d+e x}}{128 b^5 (a+b x)}-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac {\left (63 e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^5}\\ &=-\frac {63 e^4 \sqrt {d+e x}}{128 b^5 (a+b x)}-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac {\left (63 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^5}\\ &=-\frac {63 e^4 \sqrt {d+e x}}{128 b^5 (a+b x)}-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}-\frac {63 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} \sqrt {b d-a e}}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 178, normalized size = 1.00 \[ \frac {63 e^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {a e-b d}}\right )}{128 b^{11/2} \sqrt {a e-b d}}-\frac {63 e^4 \sqrt {d+e x}}{128 b^5 (a+b x)}-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(-63*e^4*Sqrt[d + e*x])/(128*b^5*(a + b*x)) - (21*e^3*(d + e*x)^(3/2))/(64*b^4*(a + b*x)^2) - (21*e^2*(d + e*x
)^(5/2))/(80*b^3*(a + b*x)^3) - (9*e*(d + e*x)^(7/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(9/2)/(5*b*(a + b*x)^5)
+ (63*e^5*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(11/2)*Sqrt[-(b*d) + a*e])

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fricas [B]  time = 0.94, size = 1003, normalized size = 5.63 \[ \left [\frac {315 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (128 \, b^{6} d^{5} + 16 \, a b^{5} d^{4} e + 24 \, a^{2} b^{4} d^{3} e^{2} + 42 \, a^{3} b^{3} d^{2} e^{3} + 105 \, a^{4} b^{2} d e^{4} - 315 \, a^{5} b e^{5} + 965 \, {\left (b^{6} d e^{4} - a b^{5} e^{5}\right )} x^{4} + 10 \, {\left (149 \, b^{6} d^{2} e^{3} + 88 \, a b^{5} d e^{4} - 237 \, a^{2} b^{4} e^{5}\right )} x^{3} + 6 \, {\left (228 \, b^{6} d^{3} e^{2} + 61 \, a b^{5} d^{2} e^{3} + 159 \, a^{2} b^{4} d e^{4} - 448 \, a^{3} b^{3} e^{5}\right )} x^{2} + 2 \, {\left (328 \, b^{6} d^{4} e + 56 \, a b^{5} d^{3} e^{2} + 99 \, a^{2} b^{4} d^{2} e^{3} + 252 \, a^{3} b^{3} d e^{4} - 735 \, a^{4} b^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{1280 \, {\left (a^{5} b^{7} d - a^{6} b^{6} e + {\left (b^{12} d - a b^{11} e\right )} x^{5} + 5 \, {\left (a b^{11} d - a^{2} b^{10} e\right )} x^{4} + 10 \, {\left (a^{2} b^{10} d - a^{3} b^{9} e\right )} x^{3} + 10 \, {\left (a^{3} b^{9} d - a^{4} b^{8} e\right )} x^{2} + 5 \, {\left (a^{4} b^{8} d - a^{5} b^{7} e\right )} x\right )}}, \frac {315 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (128 \, b^{6} d^{5} + 16 \, a b^{5} d^{4} e + 24 \, a^{2} b^{4} d^{3} e^{2} + 42 \, a^{3} b^{3} d^{2} e^{3} + 105 \, a^{4} b^{2} d e^{4} - 315 \, a^{5} b e^{5} + 965 \, {\left (b^{6} d e^{4} - a b^{5} e^{5}\right )} x^{4} + 10 \, {\left (149 \, b^{6} d^{2} e^{3} + 88 \, a b^{5} d e^{4} - 237 \, a^{2} b^{4} e^{5}\right )} x^{3} + 6 \, {\left (228 \, b^{6} d^{3} e^{2} + 61 \, a b^{5} d^{2} e^{3} + 159 \, a^{2} b^{4} d e^{4} - 448 \, a^{3} b^{3} e^{5}\right )} x^{2} + 2 \, {\left (328 \, b^{6} d^{4} e + 56 \, a b^{5} d^{3} e^{2} + 99 \, a^{2} b^{4} d^{2} e^{3} + 252 \, a^{3} b^{3} d e^{4} - 735 \, a^{4} b^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{640 \, {\left (a^{5} b^{7} d - a^{6} b^{6} e + {\left (b^{12} d - a b^{11} e\right )} x^{5} + 5 \, {\left (a b^{11} d - a^{2} b^{10} e\right )} x^{4} + 10 \, {\left (a^{2} b^{10} d - a^{3} b^{9} e\right )} x^{3} + 10 \, {\left (a^{3} b^{9} d - a^{4} b^{8} e\right )} x^{2} + 5 \, {\left (a^{4} b^{8} d - a^{5} b^{7} e\right )} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(315*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^
5)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) - 2*(128*b^6
*d^5 + 16*a*b^5*d^4*e + 24*a^2*b^4*d^3*e^2 + 42*a^3*b^3*d^2*e^3 + 105*a^4*b^2*d*e^4 - 315*a^5*b*e^5 + 965*(b^6
*d*e^4 - a*b^5*e^5)*x^4 + 10*(149*b^6*d^2*e^3 + 88*a*b^5*d*e^4 - 237*a^2*b^4*e^5)*x^3 + 6*(228*b^6*d^3*e^2 + 6
1*a*b^5*d^2*e^3 + 159*a^2*b^4*d*e^4 - 448*a^3*b^3*e^5)*x^2 + 2*(328*b^6*d^4*e + 56*a*b^5*d^3*e^2 + 99*a^2*b^4*
d^2*e^3 + 252*a^3*b^3*d*e^4 - 735*a^4*b^2*e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d - a^6*b^6*e + (b^12*d - a*b^11*e)*
x^5 + 5*(a*b^11*d - a^2*b^10*e)*x^4 + 10*(a^2*b^10*d - a^3*b^9*e)*x^3 + 10*(a^3*b^9*d - a^4*b^8*e)*x^2 + 5*(a^
4*b^8*d - a^5*b^7*e)*x), 1/640*(315*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 +
5*a^4*b*e^5*x + a^5*e^5)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) - (128
*b^6*d^5 + 16*a*b^5*d^4*e + 24*a^2*b^4*d^3*e^2 + 42*a^3*b^3*d^2*e^3 + 105*a^4*b^2*d*e^4 - 315*a^5*b*e^5 + 965*
(b^6*d*e^4 - a*b^5*e^5)*x^4 + 10*(149*b^6*d^2*e^3 + 88*a*b^5*d*e^4 - 237*a^2*b^4*e^5)*x^3 + 6*(228*b^6*d^3*e^2
+ 61*a*b^5*d^2*e^3 + 159*a^2*b^4*d*e^4 - 448*a^3*b^3*e^5)*x^2 + 2*(328*b^6*d^4*e + 56*a*b^5*d^3*e^2 + 99*a^2*
b^4*d^2*e^3 + 252*a^3*b^3*d*e^4 - 735*a^4*b^2*e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d - a^6*b^6*e + (b^12*d - a*b^11
*e)*x^5 + 5*(a*b^11*d - a^2*b^10*e)*x^4 + 10*(a^2*b^10*d - a^3*b^9*e)*x^3 + 10*(a^3*b^9*d - a^4*b^8*e)*x^2 + 5
*(a^4*b^8*d - a^5*b^7*e)*x)]

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giac [B]  time = 0.24, size = 334, normalized size = 1.88 \[ \frac {63 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{5}}{128 \, \sqrt {-b^{2} d + a b e} b^{5}} - \frac {965 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{4} e^{5} - 2370 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} d e^{5} + 2688 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{5} - 1470 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{5} + 315 \, \sqrt {x e + d} b^{4} d^{4} e^{5} + 2370 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{3} e^{6} - 5376 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} d e^{6} + 4410 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{6} - 1260 \, \sqrt {x e + d} a b^{3} d^{3} e^{6} + 2688 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{7} - 4410 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{7} + 1890 \, \sqrt {x e + d} a^{2} b^{2} d^{2} e^{7} + 1470 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b e^{8} - 1260 \, \sqrt {x e + d} a^{3} b d e^{8} + 315 \, \sqrt {x e + d} a^{4} e^{9}}{640 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

63/128*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/(sqrt(-b^2*d + a*b*e)*b^5) - 1/640*(965*(x*e + d)^(9/2
)*b^4*e^5 - 2370*(x*e + d)^(7/2)*b^4*d*e^5 + 2688*(x*e + d)^(5/2)*b^4*d^2*e^5 - 1470*(x*e + d)^(3/2)*b^4*d^3*e
^5 + 315*sqrt(x*e + d)*b^4*d^4*e^5 + 2370*(x*e + d)^(7/2)*a*b^3*e^6 - 5376*(x*e + d)^(5/2)*a*b^3*d*e^6 + 4410*
(x*e + d)^(3/2)*a*b^3*d^2*e^6 - 1260*sqrt(x*e + d)*a*b^3*d^3*e^6 + 2688*(x*e + d)^(5/2)*a^2*b^2*e^7 - 4410*(x*
e + d)^(3/2)*a^2*b^2*d*e^7 + 1890*sqrt(x*e + d)*a^2*b^2*d^2*e^7 + 1470*(x*e + d)^(3/2)*a^3*b*e^8 - 1260*sqrt(x
*e + d)*a^3*b*d*e^8 + 315*sqrt(x*e + d)*a^4*e^9)/(((x*e + d)*b - b*d + a*e)^5*b^5)

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maple [B]  time = 0.07, size = 463, normalized size = 2.60 \[ -\frac {63 \sqrt {e x +d}\, a^{4} e^{9}}{128 \left (b e x +a e \right )^{5} b^{5}}+\frac {63 \sqrt {e x +d}\, a^{3} d \,e^{8}}{32 \left (b e x +a e \right )^{5} b^{4}}-\frac {189 \sqrt {e x +d}\, a^{2} d^{2} e^{7}}{64 \left (b e x +a e \right )^{5} b^{3}}+\frac {63 \sqrt {e x +d}\, a \,d^{3} e^{6}}{32 \left (b e x +a e \right )^{5} b^{2}}-\frac {63 \sqrt {e x +d}\, d^{4} e^{5}}{128 \left (b e x +a e \right )^{5} b}-\frac {147 \left (e x +d \right )^{\frac {3}{2}} a^{3} e^{8}}{64 \left (b e x +a e \right )^{5} b^{4}}+\frac {441 \left (e x +d \right )^{\frac {3}{2}} a^{2} d \,e^{7}}{64 \left (b e x +a e \right )^{5} b^{3}}-\frac {441 \left (e x +d \right )^{\frac {3}{2}} a \,d^{2} e^{6}}{64 \left (b e x +a e \right )^{5} b^{2}}+\frac {147 \left (e x +d \right )^{\frac {3}{2}} d^{3} e^{5}}{64 \left (b e x +a e \right )^{5} b}-\frac {21 \left (e x +d \right )^{\frac {5}{2}} a^{2} e^{7}}{5 \left (b e x +a e \right )^{5} b^{3}}+\frac {42 \left (e x +d \right )^{\frac {5}{2}} a d \,e^{6}}{5 \left (b e x +a e \right )^{5} b^{2}}-\frac {21 \left (e x +d \right )^{\frac {5}{2}} d^{2} e^{5}}{5 \left (b e x +a e \right )^{5} b}-\frac {237 \left (e x +d \right )^{\frac {7}{2}} a \,e^{6}}{64 \left (b e x +a e \right )^{5} b^{2}}+\frac {237 \left (e x +d \right )^{\frac {7}{2}} d \,e^{5}}{64 \left (b e x +a e \right )^{5} b}-\frac {193 \left (e x +d \right )^{\frac {9}{2}} e^{5}}{128 \left (b e x +a e \right )^{5} b}+\frac {63 e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}\, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-193/128*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(9/2)-237/64*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(7/2)*a+237/64*e^5/(b*e*x+a*e)
^5/b*(e*x+d)^(7/2)*d-21/5*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(5/2)*a^2+42/5*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(5/2)*a*d
-21/5*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(5/2)*d^2-147/64*e^8/(b*e*x+a*e)^5/b^4*(e*x+d)^(3/2)*a^3+441/64*e^7/(b*e*x+a
*e)^5/b^3*(e*x+d)^(3/2)*a^2*d-441/64*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/2)*a*d^2+147/64*e^5/(b*e*x+a*e)^5/b*(e*x
+d)^(3/2)*d^3-63/128*e^9/(b*e*x+a*e)^5/b^5*(e*x+d)^(1/2)*a^4+63/32*e^8/(b*e*x+a*e)^5/b^4*(e*x+d)^(1/2)*a^3*d-1
89/64*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*a^2*d^2+63/32*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(1/2)*a*d^3-63/128*e^5/(
b*e*x+a*e)^5/b*(e*x+d)^(1/2)*d^4+63/128*e^5/b^5/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b
)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d positive or negative?

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mupad [B]  time = 0.74, size = 480, normalized size = 2.70 \[ \frac {63\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,b^{11/2}\,\sqrt {a\,e-b\,d}}-\frac {\frac {193\,e^5\,{\left (d+e\,x\right )}^{9/2}}{128\,b}+\frac {63\,e^5\,\sqrt {d+e\,x}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{128\,b^5}+\frac {21\,e^5\,{\left (d+e\,x\right )}^{5/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{5\,b^3}+\frac {147\,e^5\,{\left (d+e\,x\right )}^{3/2}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{64\,b^4}+\frac {237\,e^5\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{7/2}}{64\,b^2}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(9/2)/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

(63*e^5*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(128*b^(11/2)*(a*e - b*d)^(1/2)) - ((193*e^5*(d + e
*x)^(9/2))/(128*b) + (63*e^5*(d + e*x)^(1/2)*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*
d*e^3))/(128*b^5) + (21*e^5*(d + e*x)^(5/2)*(a^2*e^2 + b^2*d^2 - 2*a*b*d*e))/(5*b^3) + (147*e^5*(d + e*x)^(3/2
)*(a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*b*d*e^2))/(64*b^4) + (237*e^5*(a*e - b*d)*(d + e*x)^(7/2))/(64*b^
2))/((d + e*x)*(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) - (d + e*x)^
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) + b^5*(d + e*x)^5 - (5*b^5*d - 5*a*b^4*e)*
(d + e*x)^4 + a^5*e^5 - b^5*d^5 + (d + e*x)^3*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e) - 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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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