3.18.50 \(\int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^2} \, dx\) [1750]

Optimal. Leaf size=256 \[ \frac {(b d-a e)^2 (2 b B d+7 A b e-9 a B e) \sqrt {d+e x}}{b^5}+\frac {(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}-\frac {(b d-a e)^{5/2} (2 b B d+7 A b e-9 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 52, 65, 214} \begin {gather*} -\frac {(b d-a e)^{5/2} (-9 a B e+7 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}}+\frac {\sqrt {d+e x} (b d-a e)^2 (-9 a B e+7 A b e+2 b B d)}{b^5}+\frac {(d+e x)^{3/2} (b d-a e) (-9 a B e+7 A b e+2 b B d)}{3 b^4}+\frac {(d+e x)^{5/2} (-9 a B e+7 A b e+2 b B d)}{5 b^3}+\frac {(d+e x)^{7/2} (-9 a B e+7 A b e+2 b B d)}{7 b^2 (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.71, size = 305, normalized size = 1.19 \begin {gather*} \frac {\sqrt {d+e x} \left (-7 A b \left (-105 a^3 e^3+35 a^2 b e^2 (7 d-2 e x)+7 a b^2 e \left (-23 d^2+24 d e x+2 e^2 x^2\right )+b^3 \left (15 d^3-116 d^2 e x-32 d e^2 x^2-6 e^3 x^3\right )\right )+B \left (-945 a^4 e^3+105 a^3 b e^2 (23 d-6 e x)+7 a^2 b^2 e \left (-277 d^2+236 d e x+18 e^2 x^2\right )+a b^3 \left (457 d^3-1380 d^2 e x-316 d e^2 x^2-54 e^3 x^3\right )+2 b^4 x \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )\right )}{105 b^5 (a+b x)}-\frac {(-b d+a e)^{5/2} (2 b B d+7 A b e-9 a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(-7*A*b*(-105*a^3*e^3 + 35*a^2*b*e^2*(7*d - 2*e*x) + 7*a*b^2*e*(-23*d^2 + 24*d*e*x + 2*e^2*x^2)
+ b^3*(15*d^3 - 116*d^2*e*x - 32*d*e^2*x^2 - 6*e^3*x^3)) + B*(-945*a^4*e^3 + 105*a^3*b*e^2*(23*d - 6*e*x) + 7
*a^2*b^2*e*(-277*d^2 + 236*d*e*x + 18*e^2*x^2) + a*b^3*(457*d^3 - 1380*d^2*e*x - 316*d*e^2*x^2 - 54*e^3*x^3) +
2*b^4*x*(176*d^3 + 122*d^2*e*x + 66*d*e^2*x^2 + 15*e^3*x^3))))/(105*b^5*(a + b*x)) - ((-(b*d) + a*e)^(5/2)*(2
*b*B*d + 7*A*b*e - 9*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/b^(11/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(524\) vs. \(2(230)=460\).
time = 0.13, size = 525, normalized size = 2.05 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)^2,x,method=_RETURNVERBOSE)

[Out]

2/b^5*(1/7*b^3*B*(e*x+d)^(7/2)+1/5*A*b^3*e*(e*x+d)^(5/2)-2/5*B*a*b^2*e*(e*x+d)^(5/2)+1/5*B*b^3*d*(e*x+d)^(5/2)
-2/3*A*a*b^2*e^2*(e*x+d)^(3/2)+2/3*A*b^3*d*e*(e*x+d)^(3/2)+B*a^2*b*e^2*(e*x+d)^(3/2)-4/3*B*a*b^2*d*e*(e*x+d)^(
3/2)+1/3*B*b^3*d^2*(e*x+d)^(3/2)+3*A*a^2*b*e^3*(e*x+d)^(1/2)-6*A*a*b^2*d*e^2*(e*x+d)^(1/2)+3*A*b^3*d^2*e*(e*x+
d)^(1/2)-4*B*a^3*e^3*(e*x+d)^(1/2)+9*B*a^2*b*d*e^2*(e*x+d)^(1/2)-6*B*a*b^2*d^2*e*(e*x+d)^(1/2)+B*b^3*d^3*(e*x+
d)^(1/2))-2/b^5*((-1/2*A*a^3*b*e^4+3/2*A*a^2*b^2*d*e^3-3/2*A*a*b^3*d^2*e^2+1/2*A*b^4*d^3*e+1/2*B*a^4*e^4-3/2*B
*a^3*b*d*e^3+3/2*B*a^2*b^2*d^2*e^2-1/2*B*a*b^3*d^3*e)*(e*x+d)^(1/2)/(b*(e*x+d)+a*e-b*d)+1/2*(7*A*a^3*b*e^4-21*
A*a^2*b^2*d*e^3+21*A*a*b^3*d^2*e^2-7*A*b^4*d^3*e-9*B*a^4*e^4+29*B*a^3*b*d*e^3-33*B*a^2*b^2*d^2*e^2+15*B*a*b^3*
d^3*e-2*B*b^4*d^4)/((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)^2,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 = 0.86, size = 969, normalized size = 3.79 \begin {gather*} \left [\frac {105 \, {\left (2 \, B b^{4} d^{3} x + 2 \, B a b^{3} d^{3} - {\left (9 \, B a^{4} - 7 \, A a^{3} b + {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} e^{3} + 2 \, {\left ({\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d x + {\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d\right )} e^{2} - {\left ({\left (13 \, B a b^{3} - 7 \, A b^{4}\right )} d^{2} x + {\left (13 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} 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 (352 \, B b^{4} d^{3} x + {\left (457 \, B a b^{3} - 105 \, A b^{4}\right )} d^{3} + {\left (30 \, B b^{4} x^{4} - 945 \, B a^{4} + 735 \, A a^{3} b - 6 \, {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} e^{3} + {\left (132 \, B b^{4} d x^{3} - 4 \, {\left (79 \, B a b^{3} - 56 \, A b^{4}\right )} d x^{2} + 28 \, {\left (59 \, B a^{2} b^{2} - 42 \, A a b^{3}\right )} d x + 35 \, {\left (69 \, B a^{3} b - 49 \, A a^{2} b^{2}\right )} d\right )} e^{2} + {\left (244 \, B b^{4} d^{2} x^{2} - 4 \, {\left (345 \, B a b^{3} - 203 \, A b^{4}\right )} d^{2} x - 7 \, {\left (277 \, B a^{2} b^{2} - 161 \, A a b^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{210 \, {\left (b^{6} x + a b^{5}\right )}}, -\frac {105 \, {\left (2 \, B b^{4} d^{3} x + 2 \, B a b^{3} d^{3} - {\left (9 \, B a^{4} - 7 \, A a^{3} b + {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} e^{3} + 2 \, {\left ({\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d x + {\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d\right )} e^{2} - {\left ({\left (13 \, B a b^{3} - 7 \, A b^{4}\right )} d^{2} x + {\left (13 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} 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 (352 \, B b^{4} d^{3} x + {\left (457 \, B a b^{3} - 105 \, A b^{4}\right )} d^{3} + {\left (30 \, B b^{4} x^{4} - 945 \, B a^{4} + 735 \, A a^{3} b - 6 \, {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} e^{3} + {\left (132 \, B b^{4} d x^{3} - 4 \, {\left (79 \, B a b^{3} - 56 \, A b^{4}\right )} d x^{2} + 28 \, {\left (59 \, B a^{2} b^{2} - 42 \, A a b^{3}\right )} d x + 35 \, {\left (69 \, B a^{3} b - 49 \, A a^{2} b^{2}\right )} d\right )} e^{2} + {\left (244 \, B b^{4} d^{2} x^{2} - 4 \, {\left (345 \, B a b^{3} - 203 \, A b^{4}\right )} d^{2} x - 7 \, {\left (277 \, B a^{2} b^{2} - 161 \, A a b^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{105 \, {\left (b^{6} x + a 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)^2,x, algorithm="fricas")

[Out]

[1/210*(105*(2*B*b^4*d^3*x + 2*B*a*b^3*d^3 - (9*B*a^4 - 7*A*a^3*b + (9*B*a^3*b - 7*A*a^2*b^2)*x)*e^3 + 2*((10*
B*a^2*b^2 - 7*A*a*b^3)*d*x + (10*B*a^3*b - 7*A*a^2*b^2)*d)*e^2 - ((13*B*a*b^3 - 7*A*b^4)*d^2*x + (13*B*a^2*b^2
- 7*A*a*b^3)*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*(352*B*b^4*d^3*x + (457*B*a*b^3 - 105*A*b^4)*d^3 + (30*B*b^4*x^4 - 945*B*a^4 + 735*A*a^3*b - 6*(9
*B*a*b^3 - 7*A*b^4)*x^3 + 14*(9*B*a^2*b^2 - 7*A*a*b^3)*x^2 - 70*(9*B*a^3*b - 7*A*a^2*b^2)*x)*e^3 + (132*B*b^4*
d*x^3 - 4*(79*B*a*b^3 - 56*A*b^4)*d*x^2 + 28*(59*B*a^2*b^2 - 42*A*a*b^3)*d*x + 35*(69*B*a^3*b - 49*A*a^2*b^2)*
d)*e^2 + (244*B*b^4*d^2*x^2 - 4*(345*B*a*b^3 - 203*A*b^4)*d^2*x - 7*(277*B*a^2*b^2 - 161*A*a*b^3)*d^2)*e)*sqrt
(x*e + d))/(b^6*x + a*b^5), -1/105*(105*(2*B*b^4*d^3*x + 2*B*a*b^3*d^3 - (9*B*a^4 - 7*A*a^3*b + (9*B*a^3*b - 7
*A*a^2*b^2)*x)*e^3 + 2*((10*B*a^2*b^2 - 7*A*a*b^3)*d*x + (10*B*a^3*b - 7*A*a^2*b^2)*d)*e^2 - ((13*B*a*b^3 - 7*
A*b^4)*d^2*x + (13*B*a^2*b^2 - 7*A*a*b^3)*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)) - (352*B*b^4*d^3*x + (457*B*a*b^3 - 105*A*b^4)*d^3 + (30*B*b^4*x^4 - 945*B*a^4 + 735*A*a^3*
b - 6*(9*B*a*b^3 - 7*A*b^4)*x^3 + 14*(9*B*a^2*b^2 - 7*A*a*b^3)*x^2 - 70*(9*B*a^3*b - 7*A*a^2*b^2)*x)*e^3 + (13
2*B*b^4*d*x^3 - 4*(79*B*a*b^3 - 56*A*b^4)*d*x^2 + 28*(59*B*a^2*b^2 - 42*A*a*b^3)*d*x + 35*(69*B*a^3*b - 49*A*a
^2*b^2)*d)*e^2 + (244*B*b^4*d^2*x^2 - 4*(345*B*a*b^3 - 203*A*b^4)*d^2*x - 7*(277*B*a^2*b^2 - 161*A*a*b^3)*d^2)
*e)*sqrt(x*e + d))/(b^6*x + a*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)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (251) = 502\).
time = 1.33, size = 604, normalized size = 2.36 \begin {gather*} \frac {{\left (2 \, B b^{4} d^{4} - 15 \, B a b^{3} d^{3} e + 7 \, A b^{4} d^{3} e + 33 \, B a^{2} b^{2} d^{2} e^{2} - 21 \, A a b^{3} d^{2} e^{2} - 29 \, B a^{3} b d e^{3} + 21 \, A a^{2} b^{2} d e^{3} + 9 \, B a^{4} e^{4} - 7 \, A a^{3} b e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{5}} + \frac {\sqrt {x e + d} B a b^{3} d^{3} e - \sqrt {x e + d} A b^{4} d^{3} e - 3 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{2} + 3 \, \sqrt {x e + d} A a b^{3} d^{2} e^{2} + 3 \, \sqrt {x e + d} B a^{3} b d e^{3} - 3 \, \sqrt {x e + d} A a^{2} b^{2} d e^{3} - \sqrt {x e + d} B a^{4} e^{4} + \sqrt {x e + d} A a^{3} b e^{4}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{5}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{12} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{12} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{12} d^{2} + 105 \, \sqrt {x e + d} B b^{12} d^{3} - 42 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{11} e + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{12} e - 140 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{11} d e + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{12} d e - 630 \, \sqrt {x e + d} B a b^{11} d^{2} e + 315 \, \sqrt {x e + d} A b^{12} d^{2} e + 105 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{10} e^{2} - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{11} e^{2} + 945 \, \sqrt {x e + d} B a^{2} b^{10} d e^{2} - 630 \, \sqrt {x e + d} A a b^{11} d e^{2} - 420 \, \sqrt {x e + d} B a^{3} b^{9} e^{3} + 315 \, \sqrt {x e + d} A a^{2} b^{10} e^{3}\right )}}{105 \, b^{14}} \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)^2,x, algorithm="giac")

[Out]

(2*B*b^4*d^4 - 15*B*a*b^3*d^3*e + 7*A*b^4*d^3*e + 33*B*a^2*b^2*d^2*e^2 - 21*A*a*b^3*d^2*e^2 - 29*B*a^3*b*d*e^3
+ 21*A*a^2*b^2*d*e^3 + 9*B*a^4*e^4 - 7*A*a^3*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) + (sqrt(x*e + d)*B*a*b^3*d^3*e - sqrt(x*e + d)*A*b^4*d^3*e - 3*sqrt(x*e + d)*B*a^2*b^2*d^2*e^2
+ 3*sqrt(x*e + d)*A*a*b^3*d^2*e^2 + 3*sqrt(x*e + d)*B*a^3*b*d*e^3 - 3*sqrt(x*e + d)*A*a^2*b^2*d*e^3 - sqrt(x*e
+ d)*B*a^4*e^4 + sqrt(x*e + d)*A*a^3*b*e^4)/(((x*e + d)*b - b*d + a*e)*b^5) + 2/105*(15*(x*e + d)^(7/2)*B*b^1
2 + 21*(x*e + d)^(5/2)*B*b^12*d + 35*(x*e + d)^(3/2)*B*b^12*d^2 + 105*sqrt(x*e + d)*B*b^12*d^3 - 42*(x*e + d)^
(5/2)*B*a*b^11*e + 21*(x*e + d)^(5/2)*A*b^12*e - 140*(x*e + d)^(3/2)*B*a*b^11*d*e + 70*(x*e + d)^(3/2)*A*b^12*
d*e - 630*sqrt(x*e + d)*B*a*b^11*d^2*e + 315*sqrt(x*e + d)*A*b^12*d^2*e + 105*(x*e + d)^(3/2)*B*a^2*b^10*e^2 -
70*(x*e + d)^(3/2)*A*a*b^11*e^2 + 945*sqrt(x*e + d)*B*a^2*b^10*d*e^2 - 630*sqrt(x*e + d)*A*a*b^11*d*e^2 - 420
*sqrt(x*e + d)*B*a^3*b^9*e^3 + 315*sqrt(x*e + d)*A*a^2*b^10*e^3)/b^14

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Mupad [B]
time = 0.16, size = 562, normalized size = 2.20 \begin {gather*} \left (\frac {2\,A\,e-2\,B\,d}{5\,b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{5\,b^4}\right )\,{\left (d+e\,x\right )}^{5/2}+\left (\frac {\left (\frac {\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )}{b^2}-\frac {2\,B\,{\left (a\,e-b\,d\right )}^2}{b^4}\right )\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^2}-\frac {{\left (a\,e-b\,d\right )}^2\,\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )}{b^2}\right )\,\sqrt {d+e\,x}+\left (\frac {\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )}{3\,b^2}-\frac {2\,B\,{\left (a\,e-b\,d\right )}^2}{3\,b^4}\right )\,{\left (d+e\,x\right )}^{3/2}-\frac {\sqrt {d+e\,x}\,\left (B\,a^4\,e^4-3\,B\,a^3\,b\,d\,e^3-A\,a^3\,b\,e^4+3\,B\,a^2\,b^2\,d^2\,e^2+3\,A\,a^2\,b^2\,d\,e^3-B\,a\,b^3\,d^3\,e-3\,A\,a\,b^3\,d^2\,e^2+A\,b^4\,d^3\,e\right )}{b^6\,\left (d+e\,x\right )-b^6\,d+a\,b^5\,e}+\frac {2\,B\,{\left (d+e\,x\right )}^{7/2}}{7\,b^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}\,\left (7\,A\,b\,e-9\,B\,a\,e+2\,B\,b\,d\right )}{9\,B\,a^4\,e^4-29\,B\,a^3\,b\,d\,e^3-7\,A\,a^3\,b\,e^4+33\,B\,a^2\,b^2\,d^2\,e^2+21\,A\,a^2\,b^2\,d\,e^3-15\,B\,a\,b^3\,d^3\,e-21\,A\,a\,b^3\,d^2\,e^2+2\,B\,b^4\,d^4+7\,A\,b^4\,d^3\,e}\right )\,{\left (a\,e-b\,d\right )}^{5/2}\,\left (7\,A\,b\,e-9\,B\,a\,e+2\,B\,b\,d\right )}{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)^2,x)

[Out]

((2*A*e - 2*B*d)/(5*b^2) + (2*B*(2*b^2*d - 2*a*b*e))/(5*b^4))*(d + e*x)^(5/2) + (((((2*b^2*d - 2*a*b*e)*((2*A*
e - 2*B*d)/b^2 + (2*B*(2*b^2*d - 2*a*b*e))/b^4))/b^2 - (2*B*(a*e - b*d)^2)/b^4)*(2*b^2*d - 2*a*b*e))/b^2 - ((a
*e - b*d)^2*((2*A*e - 2*B*d)/b^2 + (2*B*(2*b^2*d - 2*a*b*e))/b^4))/b^2)*(d + e*x)^(1/2) + (((2*b^2*d - 2*a*b*e
)*((2*A*e - 2*B*d)/b^2 + (2*B*(2*b^2*d - 2*a*b*e))/b^4))/(3*b^2) - (2*B*(a*e - b*d)^2)/(3*b^4))*(d + e*x)^(3/2
) - ((d + e*x)^(1/2)*(B*a^4*e^4 - A*a^3*b*e^4 + A*b^4*d^3*e - 3*A*a*b^3*d^2*e^2 + 3*A*a^2*b^2*d*e^3 + 3*B*a^2*
b^2*d^2*e^2 - B*a*b^3*d^3*e - 3*B*a^3*b*d*e^3))/(b^6*(d + e*x) - b^6*d + a*b^5*e) + (2*B*(d + e*x)^(7/2))/(7*b
^2) + (atan((b^(1/2)*(a*e - b*d)^(5/2)*(d + e*x)^(1/2)*(7*A*b*e - 9*B*a*e + 2*B*b*d))/(9*B*a^4*e^4 + 2*B*b^4*d
^4 - 7*A*a^3*b*e^4 + 7*A*b^4*d^3*e - 21*A*a*b^3*d^2*e^2 + 21*A*a^2*b^2*d*e^3 + 33*B*a^2*b^2*d^2*e^2 - 15*B*a*b
^3*d^3*e - 29*B*a^3*b*d*e^3))*(a*e - b*d)^(5/2)*(7*A*b*e - 9*B*a*e + 2*B*b*d))/b^(11/2)

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