3.17.7 \(\int \frac {(A+B x) (d+e x)^{7/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=313 \[ -\frac {7 e^4 (-9 a B e-A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} (b d-a e)^{3/2}}-\frac {7 e^3 \sqrt {d+e x} (-9 a B e-A b e+10 b B d)}{128 b^5 (a+b x) (b d-a e)}-\frac {7 e^2 (d+e x)^{3/2} (-9 a B e-A b e+10 b B d)}{192 b^4 (a+b x)^2 (b d-a e)}-\frac {7 e (d+e x)^{5/2} (-9 a B e-A b e+10 b B d)}{240 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{7/2} (-9 a B e-A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

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Rubi [A]  time = 0.26, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 78, 47, 63, 208} \begin {gather*} -\frac {7 e^2 (d+e x)^{3/2} (-9 a B e-A b e+10 b B d)}{192 b^4 (a+b x)^2 (b d-a e)}-\frac {7 e^3 \sqrt {d+e x} (-9 a B e-A b e+10 b B d)}{128 b^5 (a+b x) (b d-a e)}-\frac {7 e^4 (-9 a B e-A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} (b d-a e)^{3/2}}-\frac {(d+e x)^{7/2} (-9 a B e-A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {7 e (d+e x)^{5/2} (-9 a B e-A b e+10 b B d)}{240 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{9/2} (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)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(-7*e^3*(10*b*B*d - A*b*e - 9*a*B*e)*Sqrt[d + e*x])/(128*b^5*(b*d - a*e)*(a + b*x)) - (7*e^2*(10*b*B*d - A*b*e
- 9*a*B*e)*(d + e*x)^(3/2))/(192*b^4*(b*d - a*e)*(a + b*x)^2) - (7*e*(10*b*B*d - A*b*e - 9*a*B*e)*(d + e*x)^(
5/2))/(240*b^3*(b*d - a*e)*(a + b*x)^3) - ((10*b*B*d - A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(40*b^2*(b*d - a*e)*(
a + b*x)^4) - ((A*b - a*B)*(d + e*x)^(9/2))/(5*b*(b*d - a*e)*(a + b*x)^5) - (7*e^4*(10*b*B*d - A*b*e - 9*a*B*e
)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(11/2)*(b*d - a*e)^(3/2))

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 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 78

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] || LtQ
[p, n]))))

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 {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^6} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(10 b B d-A b e-9 a B e) \int \frac {(d+e x)^{7/2}}{(a+b x)^5} \, dx}{10 b (b d-a e)}\\ &=-\frac {(10 b B d-A b e-9 a B e) (d+e x)^{7/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(7 e (10 b B d-A b e-9 a B e)) \int \frac {(d+e x)^{5/2}}{(a+b x)^4} \, dx}{80 b^2 (b d-a e)}\\ &=-\frac {7 e (10 b B d-A b e-9 a B e) (d+e x)^{5/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-A b e-9 a B e) (d+e x)^{7/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (7 e^2 (10 b B d-A b e-9 a B e)\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^3} \, dx}{96 b^3 (b d-a e)}\\ &=-\frac {7 e^2 (10 b B d-A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) (a+b x)^2}-\frac {7 e (10 b B d-A b e-9 a B e) (d+e x)^{5/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-A b e-9 a B e) (d+e x)^{7/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (7 e^3 (10 b B d-A b e-9 a B e)\right ) \int \frac {\sqrt {d+e x}}{(a+b x)^2} \, dx}{128 b^4 (b d-a e)}\\ &=-\frac {7 e^3 (10 b B d-A b e-9 a B e) \sqrt {d+e x}}{128 b^5 (b d-a e) (a+b x)}-\frac {7 e^2 (10 b B d-A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) (a+b x)^2}-\frac {7 e (10 b B d-A b e-9 a B e) (d+e x)^{5/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-A b e-9 a B e) (d+e x)^{7/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (7 e^4 (10 b B d-A b e-9 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^5 (b d-a e)}\\ &=-\frac {7 e^3 (10 b B d-A b e-9 a B e) \sqrt {d+e x}}{128 b^5 (b d-a e) (a+b x)}-\frac {7 e^2 (10 b B d-A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) (a+b x)^2}-\frac {7 e (10 b B d-A b e-9 a B e) (d+e x)^{5/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-A b e-9 a B e) (d+e x)^{7/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (7 e^3 (10 b B d-A b e-9 a B e)\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 (b d-a e)}\\ &=-\frac {7 e^3 (10 b B d-A b e-9 a B e) \sqrt {d+e x}}{128 b^5 (b d-a e) (a+b x)}-\frac {7 e^2 (10 b B d-A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) (a+b x)^2}-\frac {7 e (10 b B d-A b e-9 a B e) (d+e x)^{5/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-A b e-9 a B e) (d+e x)^{7/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}-\frac {7 e^4 (10 b B d-A b e-9 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} (b d-a e)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.95, size = 260, normalized size = 0.83 \begin {gather*} \frac {\frac {(a+b x) (9 a B e+A b e-10 b B d) \left (48 b^4 (d+e x)^4 \sqrt {a e-b d}+56 b^3 e (a+b x) (d+e x)^3 \sqrt {a e-b d}+70 b^2 e^2 (a+b x)^2 (d+e x)^2 \sqrt {a e-b d}-105 \sqrt {b} e^4 (a+b x)^4 \sqrt {d+e x} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {a e-b d}}\right )+105 b e^3 (a+b x)^3 (d+e x) \sqrt {a e-b d}\right )}{\sqrt {a e-b d}}-384 b^5 (d+e x)^5 (A b-a B)}{1920 b^6 (a+b x)^5 \sqrt {d+e x} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-384*b^5*(A*b - a*B)*(d + e*x)^5 + ((-10*b*B*d + A*b*e + 9*a*B*e)*(a + b*x)*(105*b*e^3*Sqrt[-(b*d) + a*e]*(a
+ b*x)^3*(d + e*x) + 70*b^2*e^2*Sqrt[-(b*d) + a*e]*(a + b*x)^2*(d + e*x)^2 + 56*b^3*e*Sqrt[-(b*d) + a*e]*(a +
b*x)*(d + e*x)^3 + 48*b^4*Sqrt[-(b*d) + a*e]*(d + e*x)^4 - 105*Sqrt[b]*e^4*(a + b*x)^4*Sqrt[d + e*x]*ArcTan[(S
qrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]]))/Sqrt[-(b*d) + a*e])/(1920*b^6*(b*d - a*e)*(a + b*x)^5*Sqrt[d + e*x
])

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IntegrateAlgebraic [B]  time = 4.93, size = 676, normalized size = 2.16 \begin {gather*} -\frac {e^4 \sqrt {d+e x} \left (945 a^5 B e^5+105 a^4 A b e^5+4410 a^4 b B e^4 (d+e x)-4830 a^4 b B d e^4+490 a^3 A b^2 e^4 (d+e x)-420 a^3 A b^2 d e^4+9870 a^3 b^2 B d^2 e^3+8064 a^3 b^2 B e^3 (d+e x)^2-18130 a^3 b^2 B d e^3 (d+e x)+630 a^2 A b^3 d^2 e^3+896 a^2 A b^3 e^3 (d+e x)^2-1470 a^2 A b^3 d e^3 (d+e x)-10080 a^2 b^3 B d^3 e^2+27930 a^2 b^3 B d^2 e^2 (d+e x)+7110 a^2 b^3 B e^2 (d+e x)^3-25088 a^2 b^3 B d e^2 (d+e x)^2-420 a A b^4 d^3 e^2+1470 a A b^4 d^2 e^2 (d+e x)+790 a A b^4 e^2 (d+e x)^3-1792 a A b^4 d e^2 (d+e x)^2+5145 a b^4 B d^4 e-19110 a b^4 B d^3 e (d+e x)+25984 a b^4 B d^2 e (d+e x)^2+2895 a b^4 B e (d+e x)^4-15010 a b^4 B d e (d+e x)^3+105 A b^5 d^4 e-490 A b^5 d^3 e (d+e x)+896 A b^5 d^2 e (d+e x)^2-105 A b^5 e (d+e x)^4-790 A b^5 d e (d+e x)^3-1050 b^5 B d^5+4900 b^5 B d^4 (d+e x)-8960 b^5 B d^3 (d+e x)^2+7900 b^5 B d^2 (d+e x)^3-2790 b^5 B d (d+e x)^4\right )}{1920 b^5 (b d-a e) (-a e-b (d+e x)+b d)^5}-\frac {7 \left (-9 a B e^5-A b e^5+10 b B d e^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 b^{11/2} (b d-a e) \sqrt {a e-b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^(7/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-1/1920*(e^4*Sqrt[d + e*x]*(-1050*b^5*B*d^5 + 105*A*b^5*d^4*e + 5145*a*b^4*B*d^4*e - 420*a*A*b^4*d^3*e^2 - 100
80*a^2*b^3*B*d^3*e^2 + 630*a^2*A*b^3*d^2*e^3 + 9870*a^3*b^2*B*d^2*e^3 - 420*a^3*A*b^2*d*e^4 - 4830*a^4*b*B*d*e
^4 + 105*a^4*A*b*e^5 + 945*a^5*B*e^5 + 4900*b^5*B*d^4*(d + e*x) - 490*A*b^5*d^3*e*(d + e*x) - 19110*a*b^4*B*d^
3*e*(d + e*x) + 1470*a*A*b^4*d^2*e^2*(d + e*x) + 27930*a^2*b^3*B*d^2*e^2*(d + e*x) - 1470*a^2*A*b^3*d*e^3*(d +
e*x) - 18130*a^3*b^2*B*d*e^3*(d + e*x) + 490*a^3*A*b^2*e^4*(d + e*x) + 4410*a^4*b*B*e^4*(d + e*x) - 8960*b^5*
B*d^3*(d + e*x)^2 + 896*A*b^5*d^2*e*(d + e*x)^2 + 25984*a*b^4*B*d^2*e*(d + e*x)^2 - 1792*a*A*b^4*d*e^2*(d + e*
x)^2 - 25088*a^2*b^3*B*d*e^2*(d + e*x)^2 + 896*a^2*A*b^3*e^3*(d + e*x)^2 + 8064*a^3*b^2*B*e^3*(d + e*x)^2 + 79
00*b^5*B*d^2*(d + e*x)^3 - 790*A*b^5*d*e*(d + e*x)^3 - 15010*a*b^4*B*d*e*(d + e*x)^3 + 790*a*A*b^4*e^2*(d + e*
x)^3 + 7110*a^2*b^3*B*e^2*(d + e*x)^3 - 2790*b^5*B*d*(d + e*x)^4 - 105*A*b^5*e*(d + e*x)^4 + 2895*a*b^4*B*e*(d
+ e*x)^4))/(b^5*(b*d - a*e)*(b*d - a*e - b*(d + e*x))^5) - (7*(10*b*B*d*e^4 - A*b*e^5 - 9*a*B*e^5)*ArcTan[(Sq
rt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(128*b^(11/2)*(b*d - a*e)*Sqrt[-(b*d) + a*e])

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fricas [B]  time = 0.49, size = 2041, normalized size = 6.52

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/3840*(105*(10*B*a^5*b*d*e^4 - (9*B*a^6 + A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (9*B*a*b^5 + A*b^6)*e^5)*x^5 + 5*
(10*B*a*b^5*d*e^4 - (9*B*a^2*b^4 + A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (9*B*a^3*b^3 + A*a^2*b^4)*e^5)
*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (9*B*a^4*b^2 + A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (9*B*a^5*b + A*a^
4*b^2)*e^5)*x)*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*(96*(B*a*b^6 + 4*A*b^7)*d^5 + 16*(2*B*a^2*b^5 - 27*A*a*b^6)*d^4*e + 4*(17*B*a^3*b^4 - 2*A*a^2*b^5)*d^3*e^2
+ 14*(16*B*a^4*b^3 - A*a^3*b^4)*d^2*e^3 - 35*(39*B*a^5*b^2 + A*a^4*b^3)*d*e^4 + 105*(9*B*a^6*b + A*a^5*b^2)*e
^5 + 15*(186*B*b^7*d^2*e^3 - (379*B*a*b^6 - 7*A*b^7)*d*e^4 + (193*B*a^2*b^5 - 7*A*a*b^6)*e^5)*x^4 + 10*(326*B*
b^7*d^3*e^2 + (17*B*a*b^6 + 121*A*b^7)*d^2*e^3 - 2*(527*B*a^2*b^5 + 100*A*a*b^6)*d*e^4 + 79*(9*B*a^3*b^4 + A*a
^2*b^5)*e^5)*x^3 + 2*(1000*B*b^7*d^4*e - 2*(81*B*a*b^6 - 526*A*b^7)*d^3*e^2 + 3*(347*B*a^2*b^5 - 447*A*a*b^6)*
d^2*e^3 - (5911*B*a^3*b^4 + 159*A*a^2*b^5)*d*e^4 + 448*(9*B*a^4*b^3 + A*a^3*b^4)*e^5)*x^2 + 2*(240*B*b^7*d^5 +
8*(7*B*a*b^6 + 93*A*b^7)*d^4*e + 2*(81*B*a^2*b^5 - 436*A*a*b^6)*d^3*e^2 + 3*(181*B*a^3*b^4 - 11*A*a^2*b^5)*d^
2*e^3 - 14*(229*B*a^4*b^3 + 6*A*a^3*b^4)*d*e^4 + 245*(9*B*a^5*b^2 + A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8
*d^2 - 2*a^6*b^7*d*e + a^7*b^6*e^2 + (b^13*d^2 - 2*a*b^12*d*e + a^2*b^11*e^2)*x^5 + 5*(a*b^12*d^2 - 2*a^2*b^11
*d*e + a^3*b^10*e^2)*x^4 + 10*(a^2*b^11*d^2 - 2*a^3*b^10*d*e + a^4*b^9*e^2)*x^3 + 10*(a^3*b^10*d^2 - 2*a^4*b^9
*d*e + a^5*b^8*e^2)*x^2 + 5*(a^4*b^9*d^2 - 2*a^5*b^8*d*e + a^6*b^7*e^2)*x), 1/1920*(105*(10*B*a^5*b*d*e^4 - (9
*B*a^6 + A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (9*B*a*b^5 + A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 - (9*B*a^2*b^4 +
A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (9*B*a^3*b^3 + A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (9*
B*a^4*b^2 + A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (9*B*a^5*b + A*a^4*b^2)*e^5)*x)*sqrt(-b^2*d + a*b*e)
*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) - (96*(B*a*b^6 + 4*A*b^7)*d^5 + 16*(2*B*a^2*b^5 - 27
*A*a*b^6)*d^4*e + 4*(17*B*a^3*b^4 - 2*A*a^2*b^5)*d^3*e^2 + 14*(16*B*a^4*b^3 - A*a^3*b^4)*d^2*e^3 - 35*(39*B*a^
5*b^2 + A*a^4*b^3)*d*e^4 + 105*(9*B*a^6*b + A*a^5*b^2)*e^5 + 15*(186*B*b^7*d^2*e^3 - (379*B*a*b^6 - 7*A*b^7)*d
*e^4 + (193*B*a^2*b^5 - 7*A*a*b^6)*e^5)*x^4 + 10*(326*B*b^7*d^3*e^2 + (17*B*a*b^6 + 121*A*b^7)*d^2*e^3 - 2*(52
7*B*a^2*b^5 + 100*A*a*b^6)*d*e^4 + 79*(9*B*a^3*b^4 + A*a^2*b^5)*e^5)*x^3 + 2*(1000*B*b^7*d^4*e - 2*(81*B*a*b^6
- 526*A*b^7)*d^3*e^2 + 3*(347*B*a^2*b^5 - 447*A*a*b^6)*d^2*e^3 - (5911*B*a^3*b^4 + 159*A*a^2*b^5)*d*e^4 + 448
*(9*B*a^4*b^3 + A*a^3*b^4)*e^5)*x^2 + 2*(240*B*b^7*d^5 + 8*(7*B*a*b^6 + 93*A*b^7)*d^4*e + 2*(81*B*a^2*b^5 - 43
6*A*a*b^6)*d^3*e^2 + 3*(181*B*a^3*b^4 - 11*A*a^2*b^5)*d^2*e^3 - 14*(229*B*a^4*b^3 + 6*A*a^3*b^4)*d*e^4 + 245*(
9*B*a^5*b^2 + A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d^2 - 2*a^6*b^7*d*e + a^7*b^6*e^2 + (b^13*d^2 - 2*a*b
^12*d*e + a^2*b^11*e^2)*x^5 + 5*(a*b^12*d^2 - 2*a^2*b^11*d*e + a^3*b^10*e^2)*x^4 + 10*(a^2*b^11*d^2 - 2*a^3*b^
10*d*e + a^4*b^9*e^2)*x^3 + 10*(a^3*b^10*d^2 - 2*a^4*b^9*d*e + a^5*b^8*e^2)*x^2 + 5*(a^4*b^9*d^2 - 2*a^5*b^8*d
*e + a^6*b^7*e^2)*x)]

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giac [B]  time = 0.30, size = 781, normalized size = 2.50 \begin {gather*} \frac {7 \, {\left (10 \, B b d e^{4} - 9 \, 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 - a b^{5} e\right )} \sqrt {-b^{2} d + a b e}} - \frac {2790 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 7900 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 8960 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} - 4900 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} + 1050 \, \sqrt {x e + d} B b^{5} d^{5} e^{4} - 2895 \, {\left (x e + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} + 105 \, {\left (x e + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 15010 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 790 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} - 25984 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} - 896 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} + 19110 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} + 490 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} - 5145 \, \sqrt {x e + d} B a b^{4} d^{4} e^{5} - 105 \, \sqrt {x e + d} A b^{5} d^{4} e^{5} - 7110 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 790 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} + 25088 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} + 1792 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} - 27930 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} - 1470 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} + 10080 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{6} + 420 \, \sqrt {x e + d} A a b^{4} d^{3} e^{6} - 8064 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} - 896 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} + 18130 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} + 1470 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} - 9870 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{7} - 630 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{7} - 4410 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} - 490 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} + 4830 \, \sqrt {x e + d} B a^{4} b d e^{8} + 420 \, \sqrt {x e + d} A a^{3} b^{2} d e^{8} - 945 \, \sqrt {x e + d} B a^{5} e^{9} - 105 \, \sqrt {x e + d} A a^{4} b e^{9}}{1920 \, {\left (b^{6} d - a b^{5} e\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)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

7/128*(10*B*b*d*e^4 - 9*B*a*e^5 - A*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d - a*b^5*e)*sqr
t(-b^2*d + a*b*e)) - 1/1920*(2790*(x*e + d)^(9/2)*B*b^5*d*e^4 - 7900*(x*e + d)^(7/2)*B*b^5*d^2*e^4 + 8960*(x*e
+ d)^(5/2)*B*b^5*d^3*e^4 - 4900*(x*e + d)^(3/2)*B*b^5*d^4*e^4 + 1050*sqrt(x*e + d)*B*b^5*d^5*e^4 - 2895*(x*e
+ d)^(9/2)*B*a*b^4*e^5 + 105*(x*e + d)^(9/2)*A*b^5*e^5 + 15010*(x*e + d)^(7/2)*B*a*b^4*d*e^5 + 790*(x*e + d)^(
7/2)*A*b^5*d*e^5 - 25984*(x*e + d)^(5/2)*B*a*b^4*d^2*e^5 - 896*(x*e + d)^(5/2)*A*b^5*d^2*e^5 + 19110*(x*e + d)
^(3/2)*B*a*b^4*d^3*e^5 + 490*(x*e + d)^(3/2)*A*b^5*d^3*e^5 - 5145*sqrt(x*e + d)*B*a*b^4*d^4*e^5 - 105*sqrt(x*e
+ d)*A*b^5*d^4*e^5 - 7110*(x*e + d)^(7/2)*B*a^2*b^3*e^6 - 790*(x*e + d)^(7/2)*A*a*b^4*e^6 + 25088*(x*e + d)^(
5/2)*B*a^2*b^3*d*e^6 + 1792*(x*e + d)^(5/2)*A*a*b^4*d*e^6 - 27930*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^6 - 1470*(x*
e + d)^(3/2)*A*a*b^4*d^2*e^6 + 10080*sqrt(x*e + d)*B*a^2*b^3*d^3*e^6 + 420*sqrt(x*e + d)*A*a*b^4*d^3*e^6 - 806
4*(x*e + d)^(5/2)*B*a^3*b^2*e^7 - 896*(x*e + d)^(5/2)*A*a^2*b^3*e^7 + 18130*(x*e + d)^(3/2)*B*a^3*b^2*d*e^7 +
1470*(x*e + d)^(3/2)*A*a^2*b^3*d*e^7 - 9870*sqrt(x*e + d)*B*a^3*b^2*d^2*e^7 - 630*sqrt(x*e + d)*A*a^2*b^3*d^2*
e^7 - 4410*(x*e + d)^(3/2)*B*a^4*b*e^8 - 490*(x*e + d)^(3/2)*A*a^3*b^2*e^8 + 4830*sqrt(x*e + d)*B*a^4*b*d*e^8
+ 420*sqrt(x*e + d)*A*a^3*b^2*d*e^8 - 945*sqrt(x*e + d)*B*a^5*e^9 - 105*sqrt(x*e + d)*A*a^4*b*e^9)/((b^6*d - a
*b^5*e)*((x*e + d)*b - b*d + a*e)^5)

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maple [B]  time = 0.07, size = 959, normalized size = 3.06

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-79/192*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(7/2)*A+7/128*e^5/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(9/2)*A+21/128*e^7/(b*e*
x+a*e)^5/b^3*(e*x+d)^(1/2)*A*a^2*d-35/64*e^4/(a*e-b*d)/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)
*b)^(1/2)*b)*B*d+49/96*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/2)*A*a*d+343/48*e^6/(b*e*x+a*e)^5/b^3*(e*x+d)^(3/2)*B*
a^2*d-1421/192*e^5/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/2)*B*a*d^2+259/128*e^7/(b*e*x+a*e)^5/b^4*(e*x+d)^(1/2)*B*d*a^3
-399/128*e^6/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*B*a^2*d^2-21/128*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(1/2)*A*d^2*a-193/
128*e^5/(b*e*x+a*e)^5/(a*e-b*d)/b*(e*x+d)^(9/2)*a*B+63/128*e^5/(a*e-b*d)/b^5/((a*e-b*d)*b)^(1/2)*arctan((e*x+d
)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a*B+273/128*e^5/(b*e*x+a*e)^5/b^2*(e*x+d)^(1/2)*B*a*d^3+133/15*e^5/(b*e*x+a*e)^
5/b^2*(e*x+d)^(5/2)*B*a*d-63/128*e^8/(b*e*x+a*e)^5/b^5*(e*x+d)^(1/2)*B*a^4-7/15*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^
(5/2)*A*a+7/128*e^5/(a*e-b*d)/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A+7/15*e^5/(
b*e*x+a*e)^5/b*(e*x+d)^(5/2)*A*d-14/3*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(5/2)*B*d^2+245/96*e^4/(b*e*x+a*e)^5/b*(e*x+
d)^(3/2)*B*d^3-35/64*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*B*d^4+93/64*e^4/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(9/2)*B
*d+395/96*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(7/2)*B*d-237/64*e^5/(b*e*x+a*e)^5/b^2*(e*x+d)^(7/2)*a*B-21/5*e^6/(b*e*x
+a*e)^5/b^3*(e*x+d)^(5/2)*B*a^2-49/192*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(3/2)*A*a^2-147/64*e^7/(b*e*x+a*e)^5/b^4*
(e*x+d)^(3/2)*B*a^3-49/192*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(3/2)*A*d^2-7/128*e^8/(b*e*x+a*e)^5/b^4*(e*x+d)^(1/2)*A
*a^3+7/128*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*A*d^3

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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)^(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(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 = 0.44, size = 594, normalized size = 1.90 \begin {gather*} \frac {7\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (A\,b\,e+9\,B\,a\,e-10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}\right )\,\left (A\,b\,e+9\,B\,a\,e-10\,B\,b\,d\right )}{128\,b^{11/2}\,{\left (a\,e-b\,d\right )}^{3/2}}-\frac {\frac {79\,{\left (d+e\,x\right )}^{7/2}\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{192\,b^2}+\frac {7\,\sqrt {d+e\,x}\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{128\,b^5}+\frac {7\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{15\,b^3}-\frac {{\left (d+e\,x\right )}^{9/2}\,\left (7\,A\,b\,e^5-193\,B\,a\,e^5+186\,B\,b\,d\,e^4\right )}{128\,b\,\left (a\,e-b\,d\right )}+\frac {49\,{\left (d+e\,x\right )}^{3/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{192\,b^4}}{\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7*e^4*atan((b^(1/2)*e^4*(d + e*x)^(1/2)*(A*b*e + 9*B*a*e - 10*B*b*d))/((a*e - b*d)^(1/2)*(A*b*e^5 + 9*B*a*e^5
- 10*B*b*d*e^4)))*(A*b*e + 9*B*a*e - 10*B*b*d))/(128*b^(11/2)*(a*e - b*d)^(3/2)) - ((79*(d + e*x)^(7/2)*(A*b*
e^5 + 9*B*a*e^5 - 10*B*b*d*e^4))/(192*b^2) + (7*(d + e*x)^(1/2)*(A*b*e^5 + 9*B*a*e^5 - 10*B*b*d*e^4)*(a^3*e^3
- b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*b*d*e^2))/(128*b^5) + (7*(a*e - b*d)*(d + e*x)^(5/2)*(A*b*e^5 + 9*B*a*e^5 -
10*B*b*d*e^4))/(15*b^3) - ((d + e*x)^(9/2)*(7*A*b*e^5 - 193*B*a*e^5 + 186*B*b*d*e^4))/(128*b*(a*e - b*d)) + (4
9*(d + e*x)^(3/2)*(a^2*e^2 + b^2*d^2 - 2*a*b*d*e)*(A*b*e^5 + 9*B*a*e^5 - 10*B*b*d*e^4))/(192*b^4))/((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 \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)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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