3.18.58 \(\int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^3} \, dx\) [1758]

Optimal. Leaf size=274 \[ \frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) \sqrt {d+e x}}{4 b^5}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}-\frac {7 e (b d-a e)^{3/2} (4 b B d+5 A b e-9 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2}} \]

[Out]

7/12*e*(5*A*b*e-9*B*a*e+4*B*b*d)*(e*x+d)^(3/2)/b^4+7/20*e*(5*A*b*e-9*B*a*e+4*B*b*d)*(e*x+d)^(5/2)/b^3/(-a*e+b*
d)-1/4*(5*A*b*e-9*B*a*e+4*B*b*d)*(e*x+d)^(7/2)/b^2/(-a*e+b*d)/(b*x+a)-1/2*(A*b-B*a)*(e*x+d)^(9/2)/b/(-a*e+b*d)
/(b*x+a)^2-7/4*e*(-a*e+b*d)^(3/2)*(5*A*b*e-9*B*a*e+4*B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^
(11/2)+7/4*e*(-a*e+b*d)*(5*A*b*e-9*B*a*e+4*B*b*d)*(e*x+d)^(1/2)/b^5

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Rubi [A]
time = 0.16, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 43, 52, 65, 214} \begin {gather*} -\frac {7 e (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2}}+\frac {7 e \sqrt {d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5}+\frac {7 e (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4}+\frac {7 e (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 (b d-a e)}-\frac {(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*e*(b*d - a*e)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*Sqrt[d + e*x])/(4*b^5) + (7*e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(d
+ e*x)^(3/2))/(12*b^4) + (7*e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(d + e*x)^(5/2))/(20*b^3*(b*d - a*e)) - ((4*b*B*d
+ 5*A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*(a + b*x)) - ((A*b - a*B)*(d + e*x)^(9/2))/(2*b*(b*d
- a*e)*(a + b*x)^2) - (7*e*(b*d - a*e)^(3/2)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqr
t[b*d - a*e]])/(4*b^(11/2))

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 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 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)^{7/2}}{(a+b x)^3} \, dx &=-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(4 b B d+5 A b e-9 a B e) \int \frac {(d+e x)^{7/2}}{(a+b x)^2} \, dx}{4 b (b d-a e)}\\ &=-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(7 e (4 b B d+5 A b e-9 a B e)) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{8 b^2 (b d-a e)}\\ &=\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(7 e (4 b B d+5 A b e-9 a B e)) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{8 b^3}\\ &=\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {(7 e (b d-a e) (4 b B d+5 A b e-9 a B e)) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{8 b^4}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) \sqrt {d+e x}}{4 b^5}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {\left (7 e (b d-a e)^2 (4 b B d+5 A b e-9 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{8 b^5}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) \sqrt {d+e x}}{4 b^5}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac {\left (7 (b d-a e)^2 (4 b B d+5 A b e-9 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 )}{4 b^5}\\ &=\frac {7 e (b d-a e) (4 b B d+5 A b e-9 a B e) \sqrt {d+e x}}{4 b^5}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac {7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac {(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac {(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}-\frac {7 e (b d-a e)^{3/2} (4 b B d+5 A b e-9 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.97, size = 309, normalized size = 1.13 \begin {gather*} \frac {\sqrt {d+e x} \left (-5 A b \left (105 a^3 e^3+35 a^2 b e^2 (-4 d+5 e x)+7 a b^2 e \left (3 d^2-34 d e x+8 e^2 x^2\right )+b^3 \left (6 d^3+39 d^2 e x-80 d e^2 x^2-8 e^3 x^3\right )\right )+B \left (945 a^4 e^3+105 a^3 b e^2 (-16 d+15 e x)+7 a^2 b^2 e \left (107 d^2-406 d e x+72 e^2 x^2\right )+4 b^4 x \left (-15 d^3+116 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )-a b^3 \left (30 d^3-1303 d^2 e x+944 d e^2 x^2+72 e^3 x^3\right )\right )\right )}{60 b^5 (a+b x)^2}+\frac {7 e (-b d+a e)^{3/2} (4 b B d+5 A b e-9 a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{4 b^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(-5*A*b*(105*a^3*e^3 + 35*a^2*b*e^2*(-4*d + 5*e*x) + 7*a*b^2*e*(3*d^2 - 34*d*e*x + 8*e^2*x^2) +
b^3*(6*d^3 + 39*d^2*e*x - 80*d*e^2*x^2 - 8*e^3*x^3)) + B*(945*a^4*e^3 + 105*a^3*b*e^2*(-16*d + 15*e*x) + 7*a^
2*b^2*e*(107*d^2 - 406*d*e*x + 72*e^2*x^2) + 4*b^4*x*(-15*d^3 + 116*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3) - a*b^
3*(30*d^3 - 1303*d^2*e*x + 944*d*e^2*x^2 + 72*e^3*x^3))))/(60*b^5*(a + b*x)^2) + (7*e*(-(b*d) + a*e)^(3/2)*(4*
b*B*d + 5*A*b*e - 9*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(4*b^(11/2))

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Maple [A]
time = 0.14, size = 467, normalized size = 1.70 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*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

2*e*(-1/b^5*(-1/5*b^2*B*(e*x+d)^(5/2)-1/3*A*b^2*e*(e*x+d)^(3/2)+B*a*b*e*(e*x+d)^(3/2)-2/3*B*b^2*d*(e*x+d)^(3/2
)+3*A*a*b*e^2*(e*x+d)^(1/2)-3*A*b^2*d*e*(e*x+d)^(1/2)-6*B*a^2*e^2*(e*x+d)^(1/2)+9*B*a*b*d*e*(e*x+d)^(1/2)-3*B*
b^2*d^2*(e*x+d)^(1/2))+1/b^5*(((-13/8*A*a^2*b^2*e^3+13/4*A*a*b^3*d*e^2-13/8*A*b^4*d^2*e+17/8*B*a^3*b*e^3-19/4*
B*a^2*b^2*d*e^2+25/8*B*a*b^3*d^2*e-1/2*B*b^4*d^3)*(e*x+d)^(3/2)+(-11/8*A*a^3*b*e^4+33/8*A*a^2*b^2*d*e^3-33/8*A
*a*b^3*d^2*e^2+11/8*A*b^4*d^3*e+15/8*B*a^4*e^4-49/8*B*a^3*b*d*e^3+57/8*B*a^2*b^2*d^2*e^2-27/8*B*a*b^3*d^3*e+1/
2*B*b^4*d^4)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^2+7/8*(5*A*a^2*b*e^3-10*A*a*b^2*d*e^2+5*A*b^3*d^2*e-9*B*a^3*e^
3+22*B*a^2*b*d*e^2-17*B*a*b^2*d^2*e+4*B*b^3*d^3)/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(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)^(7/2)/(b*x+a)^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 [A]
time = 1.18, size = 1020, normalized size = 3.72 \begin {gather*} \left [\frac {105 \, {\left ({\left (9 \, B a^{4} - 5 \, A a^{3} b + {\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 2 \, {\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} e^{3} - {\left ({\left (13 \, B a b^{3} - 5 \, A b^{4}\right )} d x^{2} + 2 \, {\left (13 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} d x + {\left (13 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} d\right )} e^{2} + 4 \, {\left (B b^{4} d^{2} x^{2} + 2 \, B a b^{3} d^{2} x + B a^{2} b^{2} d^{2}\right )} e\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {2 \, b d - 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (60 \, B b^{4} d^{3} x + 30 \, {\left (B a b^{3} + A b^{4}\right )} d^{3} - {\left (24 \, B b^{4} x^{4} + 945 \, B a^{4} - 525 \, A a^{3} b - 8 \, {\left (9 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 56 \, {\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 175 \, {\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} e^{3} - 2 \, {\left (64 \, B b^{4} d x^{3} - 8 \, {\left (59 \, B a b^{3} - 25 \, A b^{4}\right )} d x^{2} - 7 \, {\left (203 \, B a^{2} b^{2} - 85 \, A a b^{3}\right )} d x - 70 \, {\left (12 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} d\right )} e^{2} - {\left (464 \, B b^{4} d^{2} x^{2} + {\left (1303 \, B a b^{3} - 195 \, A b^{4}\right )} d^{2} x + 7 \, {\left (107 \, B a^{2} b^{2} - 15 \, A a b^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{120 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac {105 \, {\left ({\left (9 \, B a^{4} - 5 \, A a^{3} b + {\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 2 \, {\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} e^{3} - {\left ({\left (13 \, B a b^{3} - 5 \, A b^{4}\right )} d x^{2} + 2 \, {\left (13 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} d x + {\left (13 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} d\right )} e^{2} + 4 \, {\left (B b^{4} d^{2} x^{2} + 2 \, B a b^{3} d^{2} x + B a^{2} b^{2} d^{2}\right )} e\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) + {\left (60 \, B b^{4} d^{3} x + 30 \, {\left (B a b^{3} + A b^{4}\right )} d^{3} - {\left (24 \, B b^{4} x^{4} + 945 \, B a^{4} - 525 \, A a^{3} b - 8 \, {\left (9 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 56 \, {\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 175 \, {\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} e^{3} - 2 \, {\left (64 \, B b^{4} d x^{3} - 8 \, {\left (59 \, B a b^{3} - 25 \, A b^{4}\right )} d x^{2} - 7 \, {\left (203 \, B a^{2} b^{2} - 85 \, A a b^{3}\right )} d x - 70 \, {\left (12 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} d\right )} e^{2} - {\left (464 \, B b^{4} d^{2} x^{2} + {\left (1303 \, B a b^{3} - 195 \, A b^{4}\right )} d^{2} x + 7 \, {\left (107 \, B a^{2} b^{2} - 15 \, A a b^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{60 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/120*(105*((9*B*a^4 - 5*A*a^3*b + (9*B*a^2*b^2 - 5*A*a*b^3)*x^2 + 2*(9*B*a^3*b - 5*A*a^2*b^2)*x)*e^3 - ((13*
B*a*b^3 - 5*A*b^4)*d*x^2 + 2*(13*B*a^2*b^2 - 5*A*a*b^3)*d*x + (13*B*a^3*b - 5*A*a^2*b^2)*d)*e^2 + 4*(B*b^4*d^2
*x^2 + 2*B*a*b^3*d^2*x + B*a^2*b^2*d^2)*e)*sqrt((b*d - a*e)/b)*log((2*b*d - 2*sqrt(x*e + d)*b*sqrt((b*d - a*e)
/b) + (b*x - a)*e)/(b*x + a)) - 2*(60*B*b^4*d^3*x + 30*(B*a*b^3 + A*b^4)*d^3 - (24*B*b^4*x^4 + 945*B*a^4 - 525
*A*a^3*b - 8*(9*B*a*b^3 - 5*A*b^4)*x^3 + 56*(9*B*a^2*b^2 - 5*A*a*b^3)*x^2 + 175*(9*B*a^3*b - 5*A*a^2*b^2)*x)*e
^3 - 2*(64*B*b^4*d*x^3 - 8*(59*B*a*b^3 - 25*A*b^4)*d*x^2 - 7*(203*B*a^2*b^2 - 85*A*a*b^3)*d*x - 70*(12*B*a^3*b
- 5*A*a^2*b^2)*d)*e^2 - (464*B*b^4*d^2*x^2 + (1303*B*a*b^3 - 195*A*b^4)*d^2*x + 7*(107*B*a^2*b^2 - 15*A*a*b^3
)*d^2)*e)*sqrt(x*e + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5), -1/60*(105*((9*B*a^4 - 5*A*a^3*b + (9*B*a^2*b^2 - 5*
A*a*b^3)*x^2 + 2*(9*B*a^3*b - 5*A*a^2*b^2)*x)*e^3 - ((13*B*a*b^3 - 5*A*b^4)*d*x^2 + 2*(13*B*a^2*b^2 - 5*A*a*b^
3)*d*x + (13*B*a^3*b - 5*A*a^2*b^2)*d)*e^2 + 4*(B*b^4*d^2*x^2 + 2*B*a*b^3*d^2*x + B*a^2*b^2*d^2)*e)*sqrt(-(b*d
- a*e)/b)*arctan(-sqrt(x*e + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) + (60*B*b^4*d^3*x + 30*(B*a*b^3 + A*b^4)*
d^3 - (24*B*b^4*x^4 + 945*B*a^4 - 525*A*a^3*b - 8*(9*B*a*b^3 - 5*A*b^4)*x^3 + 56*(9*B*a^2*b^2 - 5*A*a*b^3)*x^2
+ 175*(9*B*a^3*b - 5*A*a^2*b^2)*x)*e^3 - 2*(64*B*b^4*d*x^3 - 8*(59*B*a*b^3 - 25*A*b^4)*d*x^2 - 7*(203*B*a^2*b
^2 - 85*A*a*b^3)*d*x - 70*(12*B*a^3*b - 5*A*a^2*b^2)*d)*e^2 - (464*B*b^4*d^2*x^2 + (1303*B*a*b^3 - 195*A*b^4)*
d^2*x + 7*(107*B*a^2*b^2 - 15*A*a*b^3)*d^2)*e)*sqrt(x*e + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)]

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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)**(7/2)/(b*x+a)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 607 vs. \(2 (268) = 536\).
time = 1.69, size = 607, normalized size = 2.22 \begin {gather*} \frac {7 \, {\left (4 \, B b^{3} d^{3} e - 17 \, B a b^{2} d^{2} e^{2} + 5 \, A b^{3} d^{2} e^{2} + 22 \, B a^{2} b d e^{3} - 10 \, A a b^{2} d e^{3} - 9 \, B a^{3} e^{4} + 5 \, A a^{2} b e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, \sqrt {-b^{2} d + a b e} b^{5}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e - 4 \, \sqrt {x e + d} B b^{4} d^{4} e - 25 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{2} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{2} + 27 \, \sqrt {x e + d} B a b^{3} d^{3} e^{2} - 11 \, \sqrt {x e + d} A b^{4} d^{3} e^{2} + 38 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{3} - 26 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{3} - 57 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{3} + 33 \, \sqrt {x e + d} A a b^{3} d^{2} e^{3} - 17 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{4} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{4} + 49 \, \sqrt {x e + d} B a^{3} b d e^{4} - 33 \, \sqrt {x e + d} A a^{2} b^{2} d e^{4} - 15 \, \sqrt {x e + d} B a^{4} e^{5} + 11 \, \sqrt {x e + d} A a^{3} b e^{5}}{4 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{5}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{12} e + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{12} d e + 45 \, \sqrt {x e + d} B b^{12} d^{2} e - 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{11} e^{2} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{12} e^{2} - 135 \, \sqrt {x e + d} B a b^{11} d e^{2} + 45 \, \sqrt {x e + d} A b^{12} d e^{2} + 90 \, \sqrt {x e + d} B a^{2} b^{10} e^{3} - 45 \, \sqrt {x e + d} A a b^{11} e^{3}\right )}}{15 \, b^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/4*(4*B*b^3*d^3*e - 17*B*a*b^2*d^2*e^2 + 5*A*b^3*d^2*e^2 + 22*B*a^2*b*d*e^3 - 10*A*a*b^2*d*e^3 - 9*B*a^3*e^4
+ 5*A*a^2*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5) - 1/4*(4*(x*e + d)^(3
/2)*B*b^4*d^3*e - 4*sqrt(x*e + d)*B*b^4*d^4*e - 25*(x*e + d)^(3/2)*B*a*b^3*d^2*e^2 + 13*(x*e + d)^(3/2)*A*b^4*
d^2*e^2 + 27*sqrt(x*e + d)*B*a*b^3*d^3*e^2 - 11*sqrt(x*e + d)*A*b^4*d^3*e^2 + 38*(x*e + d)^(3/2)*B*a^2*b^2*d*e
^3 - 26*(x*e + d)^(3/2)*A*a*b^3*d*e^3 - 57*sqrt(x*e + d)*B*a^2*b^2*d^2*e^3 + 33*sqrt(x*e + d)*A*a*b^3*d^2*e^3
- 17*(x*e + d)^(3/2)*B*a^3*b*e^4 + 13*(x*e + d)^(3/2)*A*a^2*b^2*e^4 + 49*sqrt(x*e + d)*B*a^3*b*d*e^4 - 33*sqrt
(x*e + d)*A*a^2*b^2*d*e^4 - 15*sqrt(x*e + d)*B*a^4*e^5 + 11*sqrt(x*e + d)*A*a^3*b*e^5)/(((x*e + d)*b - b*d + a
*e)^2*b^5) + 2/15*(3*(x*e + d)^(5/2)*B*b^12*e + 10*(x*e + d)^(3/2)*B*b^12*d*e + 45*sqrt(x*e + d)*B*b^12*d^2*e
- 15*(x*e + d)^(3/2)*B*a*b^11*e^2 + 5*(x*e + d)^(3/2)*A*b^12*e^2 - 135*sqrt(x*e + d)*B*a*b^11*d*e^2 + 45*sqrt(
x*e + d)*A*b^12*d*e^2 + 90*sqrt(x*e + d)*B*a^2*b^10*e^3 - 45*sqrt(x*e + d)*A*a*b^11*e^3)/b^15

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Mupad [B]
time = 1.33, size = 561, normalized size = 2.05 \begin {gather*} \left (\frac {2\,A\,e^2-2\,B\,d\,e}{3\,b^3}+\frac {2\,B\,e\,\left (3\,b^3\,d-3\,a\,b^2\,e\right )}{3\,b^6}\right )\,{\left (d+e\,x\right )}^{3/2}+\left (\frac {\left (\frac {2\,A\,e^2-2\,B\,d\,e}{b^3}+\frac {2\,B\,e\,\left (3\,b^3\,d-3\,a\,b^2\,e\right )}{b^6}\right )\,\left (3\,b^3\,d-3\,a\,b^2\,e\right )}{b^3}-\frac {6\,B\,e\,{\left (a\,e-b\,d\right )}^2}{b^5}\right )\,\sqrt {d+e\,x}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (-\frac {17\,B\,a^3\,b\,e^4}{4}+\frac {19\,B\,a^2\,b^2\,d\,e^3}{2}+\frac {13\,A\,a^2\,b^2\,e^4}{4}-\frac {25\,B\,a\,b^3\,d^2\,e^2}{4}-\frac {13\,A\,a\,b^3\,d\,e^3}{2}+B\,b^4\,d^3\,e+\frac {13\,A\,b^4\,d^2\,e^2}{4}\right )-\sqrt {d+e\,x}\,\left (\frac {15\,B\,a^4\,e^5}{4}-\frac {49\,B\,a^3\,b\,d\,e^4}{4}-\frac {11\,A\,a^3\,b\,e^5}{4}+\frac {57\,B\,a^2\,b^2\,d^2\,e^3}{4}+\frac {33\,A\,a^2\,b^2\,d\,e^4}{4}-\frac {27\,B\,a\,b^3\,d^3\,e^2}{4}-\frac {33\,A\,a\,b^3\,d^2\,e^3}{4}+B\,b^4\,d^4\,e+\frac {11\,A\,b^4\,d^3\,e^2}{4}\right )}{b^7\,{\left (d+e\,x\right )}^2-\left (2\,b^7\,d-2\,a\,b^6\,e\right )\,\left (d+e\,x\right )+b^7\,d^2+a^2\,b^5\,e^2-2\,a\,b^6\,d\,e}+\frac {2\,B\,e\,{\left (d+e\,x\right )}^{5/2}}{5\,b^3}+\frac {7\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,{\left (a\,e-b\,d\right )}^{3/2}\,\sqrt {d+e\,x}\,\left (5\,A\,b\,e-9\,B\,a\,e+4\,B\,b\,d\right )}{-9\,B\,a^3\,e^4+22\,B\,a^2\,b\,d\,e^3+5\,A\,a^2\,b\,e^4-17\,B\,a\,b^2\,d^2\,e^2-10\,A\,a\,b^2\,d\,e^3+4\,B\,b^3\,d^3\,e+5\,A\,b^3\,d^2\,e^2}\right )\,{\left (a\,e-b\,d\right )}^{3/2}\,\left (5\,A\,b\,e-9\,B\,a\,e+4\,B\,b\,d\right )}{4\,b^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*A*e^2 - 2*B*d*e)/(3*b^3) + (2*B*e*(3*b^3*d - 3*a*b^2*e))/(3*b^6))*(d + e*x)^(3/2) + ((((2*A*e^2 - 2*B*d*e)
/b^3 + (2*B*e*(3*b^3*d - 3*a*b^2*e))/b^6)*(3*b^3*d - 3*a*b^2*e))/b^3 - (6*B*e*(a*e - b*d)^2)/b^5)*(d + e*x)^(1
/2) - ((d + e*x)^(3/2)*(B*b^4*d^3*e - (17*B*a^3*b*e^4)/4 + (13*A*a^2*b^2*e^4)/4 + (13*A*b^4*d^2*e^2)/4 - (25*B
*a*b^3*d^2*e^2)/4 + (19*B*a^2*b^2*d*e^3)/2 - (13*A*a*b^3*d*e^3)/2) - (d + e*x)^(1/2)*((15*B*a^4*e^5)/4 - (11*A
*a^3*b*e^5)/4 + B*b^4*d^4*e + (11*A*b^4*d^3*e^2)/4 - (33*A*a*b^3*d^2*e^3)/4 + (33*A*a^2*b^2*d*e^4)/4 - (27*B*a
*b^3*d^3*e^2)/4 + (57*B*a^2*b^2*d^2*e^3)/4 - (49*B*a^3*b*d*e^4)/4))/(b^7*(d + e*x)^2 - (2*b^7*d - 2*a*b^6*e)*(
d + e*x) + b^7*d^2 + a^2*b^5*e^2 - 2*a*b^6*d*e) + (2*B*e*(d + e*x)^(5/2))/(5*b^3) + (7*e*atan((b^(1/2)*e*(a*e
- b*d)^(3/2)*(d + e*x)^(1/2)*(5*A*b*e - 9*B*a*e + 4*B*b*d))/(5*A*a^2*b*e^4 - 9*B*a^3*e^4 + 4*B*b^3*d^3*e + 5*A
*b^3*d^2*e^2 - 17*B*a*b^2*d^2*e^2 - 10*A*a*b^2*d*e^3 + 22*B*a^2*b*d*e^3))*(a*e - b*d)^(3/2)*(5*A*b*e - 9*B*a*e
+ 4*B*b*d))/(4*b^(11/2))

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