3.19.29 \(\int \frac {(A+B x) (d+e x)^{3/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1829]

Optimal. Leaf size=313 \[ -\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac {e^2 (2 b B d-A b e-a B e) \sqrt {d+e x}}{64 b^3 (b d-a e)^2 (a+b x)^2}+\frac {3 e^3 (2 b B d-A b e-a B e) \sqrt {d+e x}}{128 b^3 (b d-a e)^3 (a+b x)}-\frac {(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}-\frac {3 e^4 (2 b B d-A b e-a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{7/2}} \]

[Out]

-1/8*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(3/2)/b^2/(-a*e+b*d)/(b*x+a)^4-1/5*(A*b-B*a)*(e*x+d)^(5/2)/b/(-a*e+b*d)/(b
*x+a)^5-3/128*e^4*(-A*b*e-B*a*e+2*B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(7/2)/(-a*e+b*d)^(7
/2)-1/16*e*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(1/2)/b^3/(-a*e+b*d)/(b*x+a)^3-1/64*e^2*(-A*b*e-B*a*e+2*B*b*d)*(e*x+
d)^(1/2)/b^3/(-a*e+b*d)^2/(b*x+a)^2+3/128*e^3*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(1/2)/b^3/(-a*e+b*d)^3/(b*x+a)

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Rubi [A]
time = 0.17, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 79, 43, 44, 65, 214} \begin {gather*} -\frac {3 e^4 (-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{7/2}}+\frac {3 e^3 \sqrt {d+e x} (-a B e-A b e+2 b B d)}{128 b^3 (a+b x) (b d-a e)^3}-\frac {e^2 \sqrt {d+e x} (-a B e-A b e+2 b B d)}{64 b^3 (a+b x)^2 (b d-a e)^2}-\frac {e \sqrt {d+e x} (-a B e-A b e+2 b B d)}{16 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{3/2} (-a B e-A b e+2 b B d)}{8 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{5/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(e*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/(b^3*(b*d - a*e)*(a + b*x)^3) - (e^2*(2*b*B*d - A*b*e - a*B*
e)*Sqrt[d + e*x])/(64*b^3*(b*d - a*e)^2*(a + b*x)^2) + (3*e^3*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/(128*b^
3*(b*d - a*e)^3*(a + b*x)) - ((2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(3/2))/(8*b^2*(b*d - a*e)*(a + b*x)^4) - ((A
*b - a*B)*(d + e*x)^(5/2))/(5*b*(b*d - a*e)*(a + b*x)^5) - (3*e^4*(2*b*B*d - A*b*e - a*B*e)*ArcTanh[(Sqrt[b]*S
qrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(7/2)*(b*d - a*e)^(7/2))

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 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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 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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

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 {(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^6} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(2 b B d-A b e-a B e) \int \frac {(d+e x)^{3/2}}{(a+b x)^5} \, dx}{2 b (b d-a e)}\\ &=-\frac {(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(3 e (2 b B d-A b e-a B e)) \int \frac {\sqrt {d+e x}}{(a+b x)^4} \, dx}{16 b^2 (b d-a e)}\\ &=-\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac {(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (e^2 (2 b B d-A b e-a B e)\right ) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{32 b^3 (b d-a e)}\\ &=-\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac {e^2 (2 b B d-A b e-a B e) \sqrt {d+e x}}{64 b^3 (b d-a e)^2 (a+b x)^2}-\frac {(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (3 e^3 (2 b B d-A b e-a B e)\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 b^3 (b d-a e)^2}\\ &=-\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac {e^2 (2 b B d-A b e-a B e) \sqrt {d+e x}}{64 b^3 (b d-a e)^2 (a+b x)^2}+\frac {3 e^3 (2 b B d-A b e-a B e) \sqrt {d+e x}}{128 b^3 (b d-a e)^3 (a+b x)}-\frac {(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (3 e^4 (2 b B d-A b e-a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^3 (b d-a e)^3}\\ &=-\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac {e^2 (2 b B d-A b e-a B e) \sqrt {d+e x}}{64 b^3 (b d-a e)^2 (a+b x)^2}+\frac {3 e^3 (2 b B d-A b e-a B e) \sqrt {d+e x}}{128 b^3 (b d-a e)^3 (a+b x)}-\frac {(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (3 e^3 (2 b B d-A b e-a B e)\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^3 (b d-a e)^3}\\ &=-\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac {e^2 (2 b B d-A b e-a B e) \sqrt {d+e x}}{64 b^3 (b d-a e)^2 (a+b x)^2}+\frac {3 e^3 (2 b B d-A b e-a B e) \sqrt {d+e x}}{128 b^3 (b d-a e)^3 (a+b x)}-\frac {(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}-\frac {3 e^4 (2 b B d-A b e-a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 2.71, size = 421, normalized size = 1.35 \begin {gather*} \frac {\frac {\sqrt {b} \sqrt {d+e x} \left (B \left (-15 a^5 e^4+10 a^4 b e^3 (2 d-7 e x)+2 a^3 b^2 e^2 \left (6 d^2+47 d e x-64 e^2 x^2\right )+10 b^5 d x \left (16 d^3+24 d^2 e x+2 d e^2 x^2-3 e^3 x^3\right )+2 a^2 b^3 e \left (-32 d^3+46 d^2 e x+233 d e^2 x^2+35 e^3 x^3\right )+a b^4 \left (32 d^4-336 d^3 e x-668 d^2 e^2 x^2-150 d e^3 x^3+15 e^4 x^4\right )\right )+A b \left (-15 a^4 e^4-10 a^3 b e^3 (d+7 e x)+2 a^2 b^2 e^2 \left (124 d^2+233 d e x+64 e^2 x^2\right )-2 a b^3 e \left (168 d^3+256 d^2 e x+23 d e^2 x^2-35 e^3 x^3\right )+b^4 \left (128 d^4+176 d^3 e x+8 d^2 e^2 x^2-10 d e^3 x^3+15 e^4 x^4\right )\right )\right )}{(-b d+a e)^3 (a+b x)^5}+\frac {15 e^4 (-2 b B d+A b e+a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{7/2}}}{640 b^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[b]*Sqrt[d + e*x]*(B*(-15*a^5*e^4 + 10*a^4*b*e^3*(2*d - 7*e*x) + 2*a^3*b^2*e^2*(6*d^2 + 47*d*e*x - 64*e^
2*x^2) + 10*b^5*d*x*(16*d^3 + 24*d^2*e*x + 2*d*e^2*x^2 - 3*e^3*x^3) + 2*a^2*b^3*e*(-32*d^3 + 46*d^2*e*x + 233*
d*e^2*x^2 + 35*e^3*x^3) + a*b^4*(32*d^4 - 336*d^3*e*x - 668*d^2*e^2*x^2 - 150*d*e^3*x^3 + 15*e^4*x^4)) + A*b*(
-15*a^4*e^4 - 10*a^3*b*e^3*(d + 7*e*x) + 2*a^2*b^2*e^2*(124*d^2 + 233*d*e*x + 64*e^2*x^2) - 2*a*b^3*e*(168*d^3
+ 256*d^2*e*x + 23*d*e^2*x^2 - 35*e^3*x^3) + b^4*(128*d^4 + 176*d^3*e*x + 8*d^2*e^2*x^2 - 10*d*e^3*x^3 + 15*e
^4*x^4))))/((-(b*d) + a*e)^3*(a + b*x)^5) + (15*e^4*(-2*b*B*d + A*b*e + a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/
Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(7/2))/(640*b^(7/2))

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Maple [A]
time = 0.95, size = 316, normalized size = 1.01

method	result	size
derivativedivides	\(2 e^{4} \left (\frac {\frac {3 \left (A b e +B a e -2 B b d \right ) b \left (e x +d \right )^{\frac {9}{2}}}{256 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {7 \left (A b e +B a e -2 B b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{128 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {\left (A b -B a \right ) e \left (e x +d \right )^{\frac {5}{2}}}{10 b \left (a e -b d \right )}-\frac {7 \left (A b e +B a e -2 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{128 b^{2}}-\frac {3 \left (A b e +B a e -2 B b d \right ) \left (a e -b d \right ) \sqrt {e x +d}}{256 b^{3}}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {3 \left (A b e +B a e -2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 b^{3} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \sqrt {b \left (a e -b d \right )}}\right )\)	\(316\)
default	\(2 e^{4} \left (\frac {\frac {3 \left (A b e +B a e -2 B b d \right ) b \left (e x +d \right )^{\frac {9}{2}}}{256 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {7 \left (A b e +B a e -2 B b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{128 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {\left (A b -B a \right ) e \left (e x +d \right )^{\frac {5}{2}}}{10 b \left (a e -b d \right )}-\frac {7 \left (A b e +B a e -2 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{128 b^{2}}-\frac {3 \left (A b e +B a e -2 B b d \right ) \left (a e -b d \right ) \sqrt {e x +d}}{256 b^{3}}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {3 \left (A b e +B a e -2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 b^{3} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \sqrt {b \left (a e -b d \right )}}\right )\)	\(316\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^4*((3/256*(A*b*e+B*a*e-2*B*b*d)*b/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)*(e*x+d)^(9/2)+7/128*(A*b*e
+B*a*e-2*B*b*d)/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(7/2)+1/10*(A*b-B*a)*e/b/(a*e-b*d)*(e*x+d)^(5/2)-7/128*(A*
b*e+B*a*e-2*B*b*d)/b^2*(e*x+d)^(3/2)-3/256*(A*b*e+B*a*e-2*B*b*d)*(a*e-b*d)/b^3*(e*x+d)^(1/2))/((e*x+d)*b+a*e-b
*d)^5+3/256*(A*b*e+B*a*e-2*B*b*d)/b^3/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/(b*(a*e-b*d))^(1/2)*arctan
(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(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. 1119 vs. \(2 (302) = 604\).
time = 0.80, size = 2253, normalized size = 7.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/1280*(15*sqrt(b^2*d - a*b*e)*((B*a^6 + A*a^5*b + (B*a*b^5 + A*b^6)*x^5 + 5*(B*a^2*b^4 + A*a*b^5)*x^4 + 10*
(B*a^3*b^3 + A*a^2*b^4)*x^3 + 10*(B*a^4*b^2 + A*a^3*b^3)*x^2 + 5*(B*a^5*b + A*a^4*b^2)*x)*e^5 - 2*(B*b^6*d*x^5
+ 5*B*a*b^5*d*x^4 + 10*B*a^2*b^4*d*x^3 + 10*B*a^3*b^3*d*x^2 + 5*B*a^4*b^2*d*x + B*a^5*b*d)*e^4)*log((2*b*d +
(b*x - a)*e - 2*sqrt(b^2*d - a*b*e)*sqrt(x*e + d))/(b*x + a)) + 2*(160*B*b^7*d^5*x + 32*(B*a*b^6 + 4*A*b^7)*d^
5 + (15*B*a^6*b + 15*A*a^5*b^2 - 15*(B*a^2*b^5 + A*a*b^6)*x^4 - 70*(B*a^3*b^4 + A*a^2*b^5)*x^3 + 128*(B*a^4*b^
3 - A*a^3*b^4)*x^2 + 70*(B*a^5*b^2 + A*a^4*b^3)*x)*e^5 + (15*(3*B*a*b^6 + A*b^7)*d*x^4 + 20*(11*B*a^2*b^5 + 4*
A*a*b^6)*d*x^3 - 6*(99*B*a^3*b^4 - 29*A*a^2*b^5)*d*x^2 - 4*(41*B*a^4*b^3 + 134*A*a^3*b^4)*d*x - 5*(7*B*a^5*b^2
+ A*a^4*b^3)*d)*e^4 - 2*(15*B*b^7*d^2*x^4 + 5*(17*B*a*b^6 + A*b^7)*d^2*x^3 - 27*(21*B*a^2*b^5 - A*a*b^6)*d^2*
x^2 - (B*a^3*b^4 + 489*A*a^2*b^5)*d^2*x - (4*B*a^4*b^3 - 129*A*a^3*b^4)*d^2)*e^3 + 4*(5*B*b^7*d^3*x^3 - (227*B
*a*b^6 - 2*A*b^7)*d^3*x^2 + (107*B*a^2*b^5 - 172*A*a*b^6)*d^3*x + (19*B*a^3*b^4 + 146*A*a^2*b^5)*d^3)*e^2 + 16
*(15*B*b^7*d^4*x^2 - (31*B*a*b^6 - 11*A*b^7)*d^4*x - (6*B*a^2*b^5 + 29*A*a*b^6)*d^4)*e)*sqrt(x*e + d))/(b^13*d
^4*x^5 + 5*a*b^12*d^4*x^4 + 10*a^2*b^11*d^4*x^3 + 10*a^3*b^10*d^4*x^2 + 5*a^4*b^9*d^4*x + a^5*b^8*d^4 + (a^4*b
^9*x^5 + 5*a^5*b^8*x^4 + 10*a^6*b^7*x^3 + 10*a^7*b^6*x^2 + 5*a^8*b^5*x + a^9*b^4)*e^4 - 4*(a^3*b^10*d*x^5 + 5*
a^4*b^9*d*x^4 + 10*a^5*b^8*d*x^3 + 10*a^6*b^7*d*x^2 + 5*a^7*b^6*d*x + a^8*b^5*d)*e^3 + 6*(a^2*b^11*d^2*x^5 + 5
*a^3*b^10*d^2*x^4 + 10*a^4*b^9*d^2*x^3 + 10*a^5*b^8*d^2*x^2 + 5*a^6*b^7*d^2*x + a^7*b^6*d^2)*e^2 - 4*(a*b^12*d
^3*x^5 + 5*a^2*b^11*d^3*x^4 + 10*a^3*b^10*d^3*x^3 + 10*a^4*b^9*d^3*x^2 + 5*a^5*b^8*d^3*x + a^6*b^7*d^3)*e), -1
/640*(15*sqrt(-b^2*d + a*b*e)*((B*a^6 + A*a^5*b + (B*a*b^5 + A*b^6)*x^5 + 5*(B*a^2*b^4 + A*a*b^5)*x^4 + 10*(B*
a^3*b^3 + A*a^2*b^4)*x^3 + 10*(B*a^4*b^2 + A*a^3*b^3)*x^2 + 5*(B*a^5*b + A*a^4*b^2)*x)*e^5 - 2*(B*b^6*d*x^5 +
5*B*a*b^5*d*x^4 + 10*B*a^2*b^4*d*x^3 + 10*B*a^3*b^3*d*x^2 + 5*B*a^4*b^2*d*x + B*a^5*b*d)*e^4)*arctan(sqrt(-b^2
*d + a*b*e)*sqrt(x*e + d)/(b*x*e + b*d)) + (160*B*b^7*d^5*x + 32*(B*a*b^6 + 4*A*b^7)*d^5 + (15*B*a^6*b + 15*A*
a^5*b^2 - 15*(B*a^2*b^5 + A*a*b^6)*x^4 - 70*(B*a^3*b^4 + A*a^2*b^5)*x^3 + 128*(B*a^4*b^3 - A*a^3*b^4)*x^2 + 70
*(B*a^5*b^2 + A*a^4*b^3)*x)*e^5 + (15*(3*B*a*b^6 + A*b^7)*d*x^4 + 20*(11*B*a^2*b^5 + 4*A*a*b^6)*d*x^3 - 6*(99*
B*a^3*b^4 - 29*A*a^2*b^5)*d*x^2 - 4*(41*B*a^4*b^3 + 134*A*a^3*b^4)*d*x - 5*(7*B*a^5*b^2 + A*a^4*b^3)*d)*e^4 -
2*(15*B*b^7*d^2*x^4 + 5*(17*B*a*b^6 + A*b^7)*d^2*x^3 - 27*(21*B*a^2*b^5 - A*a*b^6)*d^2*x^2 - (B*a^3*b^4 + 489*
A*a^2*b^5)*d^2*x - (4*B*a^4*b^3 - 129*A*a^3*b^4)*d^2)*e^3 + 4*(5*B*b^7*d^3*x^3 - (227*B*a*b^6 - 2*A*b^7)*d^3*x
^2 + (107*B*a^2*b^5 - 172*A*a*b^6)*d^3*x + (19*B*a^3*b^4 + 146*A*a^2*b^5)*d^3)*e^2 + 16*(15*B*b^7*d^4*x^2 - (3
1*B*a*b^6 - 11*A*b^7)*d^4*x - (6*B*a^2*b^5 + 29*A*a*b^6)*d^4)*e)*sqrt(x*e + d))/(b^13*d^4*x^5 + 5*a*b^12*d^4*x
^4 + 10*a^2*b^11*d^4*x^3 + 10*a^3*b^10*d^4*x^2 + 5*a^4*b^9*d^4*x + a^5*b^8*d^4 + (a^4*b^9*x^5 + 5*a^5*b^8*x^4
+ 10*a^6*b^7*x^3 + 10*a^7*b^6*x^2 + 5*a^8*b^5*x + a^9*b^4)*e^4 - 4*(a^3*b^10*d*x^5 + 5*a^4*b^9*d*x^4 + 10*a^5*
b^8*d*x^3 + 10*a^6*b^7*d*x^2 + 5*a^7*b^6*d*x + a^8*b^5*d)*e^3 + 6*(a^2*b^11*d^2*x^5 + 5*a^3*b^10*d^2*x^4 + 10*
a^4*b^9*d^2*x^3 + 10*a^5*b^8*d^2*x^2 + 5*a^6*b^7*d^2*x + a^7*b^6*d^2)*e^2 - 4*(a*b^12*d^3*x^5 + 5*a^2*b^11*d^3
*x^4 + 10*a^3*b^10*d^3*x^3 + 10*a^4*b^9*d^3*x^2 + 5*a^5*b^8*d^3*x + a^6*b^7*d^3)*e)]

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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((B*x+A)*(e*x+d)**(3/2)/(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. 814 vs. \(2 (302) = 604\).
time = 1.55, size = 814, normalized size = 2.60 \begin {gather*} \frac {3 \, {\left (2 \, B b d e^{4} - B a e^{5} - A b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e}} + \frac {30 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 140 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 140 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} - 30 \, \sqrt {x e + d} B b^{5} d^{5} e^{4} - 15 \, {\left (x e + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} - 15 \, {\left (x e + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 210 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 70 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} + 128 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} - 128 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} - 490 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} + 135 \, \sqrt {x e + d} B a b^{4} d^{4} e^{5} + 15 \, \sqrt {x e + d} A b^{5} d^{4} e^{5} - 70 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 70 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} - 256 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} + 256 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} + 630 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} - 240 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{6} - 60 \, \sqrt {x e + d} A a b^{4} d^{3} e^{6} + 128 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} - 128 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} - 350 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} - 210 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} + 210 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{7} + 90 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{7} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} - 90 \, \sqrt {x e + d} B a^{4} b d e^{8} - 60 \, \sqrt {x e + d} A a^{3} b^{2} d e^{8} + 15 \, \sqrt {x e + d} B a^{5} e^{9} + 15 \, \sqrt {x e + d} A a^{4} b e^{9}}{640 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/128*(2*B*b*d*e^4 - B*a*e^5 - A*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^3 - 3*a*b^5*d^2*e
+ 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*sqrt(-b^2*d + a*b*e)) + 1/640*(30*(x*e + d)^(9/2)*B*b^5*d*e^4 - 140*(x*e + d
)^(7/2)*B*b^5*d^2*e^4 + 140*(x*e + d)^(3/2)*B*b^5*d^4*e^4 - 30*sqrt(x*e + d)*B*b^5*d^5*e^4 - 15*(x*e + d)^(9/2
)*B*a*b^4*e^5 - 15*(x*e + d)^(9/2)*A*b^5*e^5 + 210*(x*e + d)^(7/2)*B*a*b^4*d*e^5 + 70*(x*e + d)^(7/2)*A*b^5*d*
e^5 + 128*(x*e + d)^(5/2)*B*a*b^4*d^2*e^5 - 128*(x*e + d)^(5/2)*A*b^5*d^2*e^5 - 490*(x*e + d)^(3/2)*B*a*b^4*d^
3*e^5 - 70*(x*e + d)^(3/2)*A*b^5*d^3*e^5 + 135*sqrt(x*e + d)*B*a*b^4*d^4*e^5 + 15*sqrt(x*e + d)*A*b^5*d^4*e^5
- 70*(x*e + d)^(7/2)*B*a^2*b^3*e^6 - 70*(x*e + d)^(7/2)*A*a*b^4*e^6 - 256*(x*e + d)^(5/2)*B*a^2*b^3*d*e^6 + 25
6*(x*e + d)^(5/2)*A*a*b^4*d*e^6 + 630*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^6 + 210*(x*e + d)^(3/2)*A*a*b^4*d^2*e^6
- 240*sqrt(x*e + d)*B*a^2*b^3*d^3*e^6 - 60*sqrt(x*e + d)*A*a*b^4*d^3*e^6 + 128*(x*e + d)^(5/2)*B*a^3*b^2*e^7 -
128*(x*e + d)^(5/2)*A*a^2*b^3*e^7 - 350*(x*e + d)^(3/2)*B*a^3*b^2*d*e^7 - 210*(x*e + d)^(3/2)*A*a^2*b^3*d*e^7
+ 210*sqrt(x*e + d)*B*a^3*b^2*d^2*e^7 + 90*sqrt(x*e + d)*A*a^2*b^3*d^2*e^7 + 70*(x*e + d)^(3/2)*B*a^4*b*e^8 +
70*(x*e + d)^(3/2)*A*a^3*b^2*e^8 - 90*sqrt(x*e + d)*B*a^4*b*d*e^8 - 60*sqrt(x*e + d)*A*a^3*b^2*d*e^8 + 15*sqr
t(x*e + d)*B*a^5*e^9 + 15*sqrt(x*e + d)*A*a^4*b*e^9)/((b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3
)*((x*e + d)*b - b*d + a*e)^5)

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Mupad [B]
time = 2.23, size = 535, normalized size = 1.71 \begin {gather*} \frac {\frac {7\,{\left (d+e\,x\right )}^{7/2}\,\left (A\,b\,e^5+B\,a\,e^5-2\,B\,b\,d\,e^4\right )}{64\,{\left (a\,e-b\,d\right )}^2}-\frac {7\,{\left (d+e\,x\right )}^{3/2}\,\left (A\,b\,e^5+B\,a\,e^5-2\,B\,b\,d\,e^4\right )}{64\,b^2}-\frac {3\,\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}\,\left (A\,b\,e^5+B\,a\,e^5-2\,B\,b\,d\,e^4\right )}{128\,b^3}+\frac {3\,b\,{\left (d+e\,x\right )}^{9/2}\,\left (A\,b\,e^5+B\,a\,e^5-2\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^3}+\frac {\left (A\,b\,e^5-B\,a\,e^5\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,b\,\left (a\,e-b\,d\right )}}{\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}+\frac {3\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^5+B\,a\,e^5-2\,B\,b\,d\,e^4\right )}\right )\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{128\,b^{7/2}\,{\left (a\,e-b\,d\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((7*(d + e*x)^(7/2)*(A*b*e^5 + B*a*e^5 - 2*B*b*d*e^4))/(64*(a*e - b*d)^2) - (7*(d + e*x)^(3/2)*(A*b*e^5 + B*a*
e^5 - 2*B*b*d*e^4))/(64*b^2) - (3*(a*e - b*d)*(d + e*x)^(1/2)*(A*b*e^5 + B*a*e^5 - 2*B*b*d*e^4))/(128*b^3) + (
3*b*(d + e*x)^(9/2)*(A*b*e^5 + B*a*e^5 - 2*B*b*d*e^4))/(128*(a*e - b*d)^3) + ((A*b*e^5 - B*a*e^5)*(d + e*x)^(5
/2))/(5*b*(a*e - b*d)))/((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) + (3*e^4*atan((b^(1/2)*e^4*(d + e*
x)^(1/2)*(A*b*e + B*a*e - 2*B*b*d))/((a*e - b*d)^(1/2)*(A*b*e^5 + B*a*e^5 - 2*B*b*d*e^4)))*(A*b*e + B*a*e - 2*
B*b*d))/(128*b^(7/2)*(a*e - b*d)^(7/2))

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