3.1750 \(\int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^2} \, dx\)

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

[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 = 0.23, 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 = {78, 50, 63, 208} \[ \frac {(d+e x)^{7/2} (-9 a B e+7 A b e+2 b B d)}{7 b^2 (b d-a e)}+\frac {(d+e x)^{5/2} (-9 a B e+7 A b e+2 b B d)}{5 b^3}+\frac {(d+e x)^{3/2} (b d-a e) (-9 a B e+7 A b e+2 b B d)}{3 b^4}+\frac {\sqrt {d+e x} (b d-a e)^2 (-9 a B e+7 A b e+2 b B d)}{b^5}-\frac {(b d-a e)^{5/2} (-9 a B e+7 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}}-\frac {(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)} \]

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 50

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 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}}{(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 ) \operatorname {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 = 0.65, size = 193, normalized size = 0.75 \[ \frac {\frac {2 \left (-\frac {9 a B e}{2}+\frac {7 A b e}{2}+b B d\right ) \left (7 (b d-a e) \left (5 (b d-a e) \left (\sqrt {b} \sqrt {d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )+3 b^{5/2} (d+e x)^{5/2}\right )+15 b^{7/2} (d+e x)^{7/2}\right )}{105 b^{9/2}}+\frac {(d+e x)^{9/2} (a B-A b)}{a+b x}}{b (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-(A*b) + a*B)*(d + e*x)^(9/2))/(a + b*x) + (2*(b*B*d + (7*A*b*e)/2 - (9*a*B*e)/2)*(15*b^(7/2)*(d + e*x)^(7/
2) + 7*(b*d - a*e)*(3*b^(5/2)*(d + e*x)^(5/2) + 5*(b*d - a*e)*(Sqrt[b]*Sqrt[d + e*x]*(4*b*d - 3*a*e + b*e*x) -
3*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]))))/(105*b^(9/2)))/(b*(b*d - a*e))

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fricas [B]  time = 0.96, size = 1006, normalized size = 3.93 \[ \left [\frac {105 \, {\left (2 \, B a b^{3} d^{3} - {\left (13 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d^{2} e + 2 \, {\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d e^{2} - {\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} e^{3} + {\left (2 \, B b^{4} d^{3} - {\left (13 \, B a b^{3} - 7 \, A b^{4}\right )} d^{2} e + 2 \, {\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d e^{2} - {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (30 \, B b^{4} e^{3} x^{4} + {\left (457 \, B a b^{3} - 105 \, A b^{4}\right )} d^{3} - 7 \, {\left (277 \, B a^{2} b^{2} - 161 \, A a b^{3}\right )} d^{2} e + 35 \, {\left (69 \, B a^{3} b - 49 \, A a^{2} b^{2}\right )} d e^{2} - 105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} e^{3} + 6 \, {\left (22 \, B b^{4} d e^{2} - {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} e^{3}\right )} x^{3} + 2 \, {\left (122 \, B b^{4} d^{2} e - 2 \, {\left (79 \, B a b^{3} - 56 \, A b^{4}\right )} d e^{2} + 7 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} e^{3}\right )} x^{2} + 2 \, {\left (176 \, B b^{4} d^{3} - 2 \, {\left (345 \, B a b^{3} - 203 \, A b^{4}\right )} d^{2} e + 14 \, {\left (59 \, B a^{2} b^{2} - 42 \, A a b^{3}\right )} d e^{2} - 35 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{210 \, {\left (b^{6} x + a b^{5}\right )}}, -\frac {105 \, {\left (2 \, B a b^{3} d^{3} - {\left (13 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d^{2} e + 2 \, {\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d e^{2} - {\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} e^{3} + {\left (2 \, B b^{4} d^{3} - {\left (13 \, B a b^{3} - 7 \, A b^{4}\right )} d^{2} e + 2 \, {\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d e^{2} - {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (30 \, B b^{4} e^{3} x^{4} + {\left (457 \, B a b^{3} - 105 \, A b^{4}\right )} d^{3} - 7 \, {\left (277 \, B a^{2} b^{2} - 161 \, A a b^{3}\right )} d^{2} e + 35 \, {\left (69 \, B a^{3} b - 49 \, A a^{2} b^{2}\right )} d e^{2} - 105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} e^{3} + 6 \, {\left (22 \, B b^{4} d e^{2} - {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} e^{3}\right )} x^{3} + 2 \, {\left (122 \, B b^{4} d^{2} e - 2 \, {\left (79 \, B a b^{3} - 56 \, A b^{4}\right )} d e^{2} + 7 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} e^{3}\right )} x^{2} + 2 \, {\left (176 \, B b^{4} d^{3} - 2 \, {\left (345 \, B a b^{3} - 203 \, A b^{4}\right )} d^{2} e + 14 \, {\left (59 \, B a^{2} b^{2} - 42 \, A a b^{3}\right )} d e^{2} - 35 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (b^{6} x + a b^{5}\right )}}\right ] \]

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*a*b^3*d^3 - (13*B*a^2*b^2 - 7*A*a*b^3)*d^2*e + 2*(10*B*a^3*b - 7*A*a^2*b^2)*d*e^2 - (9*B*a^4
- 7*A*a^3*b)*e^3 + (2*B*b^4*d^3 - (13*B*a*b^3 - 7*A*b^4)*d^2*e + 2*(10*B*a^2*b^2 - 7*A*a*b^3)*d*e^2 - (9*B*a^3
*b - 7*A*a^2*b^2)*e^3)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b)
)/(b*x + a)) + 2*(30*B*b^4*e^3*x^4 + (457*B*a*b^3 - 105*A*b^4)*d^3 - 7*(277*B*a^2*b^2 - 161*A*a*b^3)*d^2*e + 3
5*(69*B*a^3*b - 49*A*a^2*b^2)*d*e^2 - 105*(9*B*a^4 - 7*A*a^3*b)*e^3 + 6*(22*B*b^4*d*e^2 - (9*B*a*b^3 - 7*A*b^4
)*e^3)*x^3 + 2*(122*B*b^4*d^2*e - 2*(79*B*a*b^3 - 56*A*b^4)*d*e^2 + 7*(9*B*a^2*b^2 - 7*A*a*b^3)*e^3)*x^2 + 2*(
176*B*b^4*d^3 - 2*(345*B*a*b^3 - 203*A*b^4)*d^2*e + 14*(59*B*a^2*b^2 - 42*A*a*b^3)*d*e^2 - 35*(9*B*a^3*b - 7*A
*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^6*x + a*b^5), -1/105*(105*(2*B*a*b^3*d^3 - (13*B*a^2*b^2 - 7*A*a*b^3)*d^2*
e + 2*(10*B*a^3*b - 7*A*a^2*b^2)*d*e^2 - (9*B*a^4 - 7*A*a^3*b)*e^3 + (2*B*b^4*d^3 - (13*B*a*b^3 - 7*A*b^4)*d^2
*e + 2*(10*B*a^2*b^2 - 7*A*a*b^3)*d*e^2 - (9*B*a^3*b - 7*A*a^2*b^2)*e^3)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(
e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (30*B*b^4*e^3*x^4 + (457*B*a*b^3 - 105*A*b^4)*d^3 - 7*(277*B*a^
2*b^2 - 161*A*a*b^3)*d^2*e + 35*(69*B*a^3*b - 49*A*a^2*b^2)*d*e^2 - 105*(9*B*a^4 - 7*A*a^3*b)*e^3 + 6*(22*B*b^
4*d*e^2 - (9*B*a*b^3 - 7*A*b^4)*e^3)*x^3 + 2*(122*B*b^4*d^2*e - 2*(79*B*a*b^3 - 56*A*b^4)*d*e^2 + 7*(9*B*a^2*b
^2 - 7*A*a*b^3)*e^3)*x^2 + 2*(176*B*b^4*d^3 - 2*(345*B*a*b^3 - 203*A*b^4)*d^2*e + 14*(59*B*a^2*b^2 - 42*A*a*b^
3)*d*e^2 - 35*(9*B*a^3*b - 7*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^6*x + a*b^5)]

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giac [B]  time = 1.35, size = 604, normalized size = 2.36 \[ \frac {{\left (2 \, B b^{4} d^{4} - 15 \, B a b^{3} d^{3} e + 7 \, A b^{4} d^{3} e + 33 \, B a^{2} b^{2} d^{2} e^{2} - 21 \, A a b^{3} d^{2} e^{2} - 29 \, B a^{3} b d e^{3} + 21 \, A a^{2} b^{2} d e^{3} + 9 \, B a^{4} e^{4} - 7 \, A a^{3} b e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{5}} + \frac {\sqrt {x e + d} B a b^{3} d^{3} e - \sqrt {x e + d} A b^{4} d^{3} e - 3 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{2} + 3 \, \sqrt {x e + d} A a b^{3} d^{2} e^{2} + 3 \, \sqrt {x e + d} B a^{3} b d e^{3} - 3 \, \sqrt {x e + d} A a^{2} b^{2} d e^{3} - \sqrt {x e + d} B a^{4} e^{4} + \sqrt {x e + d} A a^{3} b e^{4}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{5}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{12} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{12} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{12} d^{2} + 105 \, \sqrt {x e + d} B b^{12} d^{3} - 42 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{11} e + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{12} e - 140 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{11} d e + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{12} d e - 630 \, \sqrt {x e + d} B a b^{11} d^{2} e + 315 \, \sqrt {x e + d} A b^{12} d^{2} e + 105 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{10} e^{2} - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{11} e^{2} + 945 \, \sqrt {x e + d} B a^{2} b^{10} d e^{2} - 630 \, \sqrt {x e + d} A a b^{11} d e^{2} - 420 \, \sqrt {x e + d} B a^{3} b^{9} e^{3} + 315 \, \sqrt {x e + d} A a^{2} b^{10} e^{3}\right )}}{105 \, b^{14}} \]

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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maple [B]  time = 0.02, size = 915, normalized size = 3.57 \[ -\frac {7 A \,a^{3} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {21 A \,a^{2} d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{3}}-\frac {21 A a \,d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{2}}+\frac {7 A \,d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b}+\frac {9 B \,a^{4} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{5}}-\frac {29 B \,a^{3} d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {33 B \,a^{2} d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{3}}-\frac {15 B a \,d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{2}}+\frac {2 B \,d^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b}+\frac {\sqrt {e x +d}\, A \,a^{3} e^{4}}{\left (b x e +a e \right ) b^{4}}-\frac {3 \sqrt {e x +d}\, A \,a^{2} d \,e^{3}}{\left (b x e +a e \right ) b^{3}}+\frac {3 \sqrt {e x +d}\, A a \,d^{2} e^{2}}{\left (b x e +a e \right ) b^{2}}-\frac {\sqrt {e x +d}\, A \,d^{3} e}{\left (b x e +a e \right ) b}-\frac {\sqrt {e x +d}\, B \,a^{4} e^{4}}{\left (b x e +a e \right ) b^{5}}+\frac {3 \sqrt {e x +d}\, B \,a^{3} d \,e^{3}}{\left (b x e +a e \right ) b^{4}}-\frac {3 \sqrt {e x +d}\, B \,a^{2} d^{2} e^{2}}{\left (b x e +a e \right ) b^{3}}+\frac {\sqrt {e x +d}\, B a \,d^{3} e}{\left (b x e +a e \right ) b^{2}}+\frac {6 \sqrt {e x +d}\, A \,a^{2} e^{3}}{b^{4}}-\frac {12 \sqrt {e x +d}\, A a d \,e^{2}}{b^{3}}+\frac {6 \sqrt {e x +d}\, A \,d^{2} e}{b^{2}}-\frac {8 \sqrt {e x +d}\, B \,a^{3} e^{3}}{b^{5}}+\frac {18 \sqrt {e x +d}\, B \,a^{2} d \,e^{2}}{b^{4}}-\frac {12 \sqrt {e x +d}\, B a \,d^{2} e}{b^{3}}+\frac {2 \sqrt {e x +d}\, B \,d^{3}}{b^{2}}-\frac {4 \left (e x +d \right )^{\frac {3}{2}} A a \,e^{2}}{3 b^{3}}+\frac {4 \left (e x +d \right )^{\frac {3}{2}} A d e}{3 b^{2}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} e^{2}}{b^{4}}-\frac {8 \left (e x +d \right )^{\frac {3}{2}} B a d e}{3 b^{3}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} B \,d^{2}}{3 b^{2}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} A e}{5 b^{2}}-\frac {4 \left (e x +d \right )^{\frac {5}{2}} B a e}{5 b^{3}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} B d}{5 b^{2}}+\frac {2 \left (e x +d \right )^{\frac {7}{2}} B}{7 b^{2}} \]

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)

[Out]

2/5/b^2*A*(e*x+d)^(5/2)*e+2/5/b^2*B*(e*x+d)^(5/2)*d+2/3/b^2*B*(e*x+d)^(3/2)*d^2+2/b^2*B*(e*x+d)^(1/2)*d^3+2/b/
((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*d^4-4/5/b^3*B*(e*x+d)^(5/2)*a*e-4/3/b^3*A*(e
*x+d)^(3/2)*a*e^2+4/3/b^2*A*(e*x+d)^(3/2)*d*e+9/b^5/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/
2)*b)*B*a^4*e^4-7/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A*a^3*e^4+2/7/b^2*B*(e*x
+d)^(7/2)+6/b^2*A*(e*x+d)^(1/2)*d^2*e-8/b^5*B*(e*x+d)^(1/2)*a^3*e^3-8/3/b^3*B*(e*x+d)^(3/2)*a*d*e-12/b^3*A*(e*
x+d)^(1/2)*a*d*e^2+18/b^4*B*(e*x+d)^(1/2)*a^2*d*e^2-12/b^3*B*(e*x+d)^(1/2)*a*d^2*e-1/b^5*(e*x+d)^(1/2)/(b*e*x+
a*e)*B*a^4*e^4+7/b/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A*d^3*e+2/b^4*B*(e*x+d)^(3/
2)*a^2*e^2+6/b^4*A*(e*x+d)^(1/2)*a^2*e^3-1/b*(e*x+d)^(1/2)/(b*e*x+a*e)*A*d^3*e+1/b^4*(e*x+d)^(1/2)/(b*e*x+a*e)
*A*a^3*e^4+1/b^2*(e*x+d)^(1/2)/(b*e*x+a*e)*B*a*d^3*e+21/b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d
)*b)^(1/2)*b)*A*a^2*d*e^3-21/b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A*a*d^2*e^2-2
9/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*a^3*d*e^3+33/b^3/((a*e-b*d)*b)^(1/2)*a
rctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*a^2*d^2*e^2-15/b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e
-b*d)*b)^(1/2)*b)*B*a*d^3*e-3/b^3*(e*x+d)^(1/2)/(b*e*x+a*e)*A*a^2*d*e^3-3/b^3*(e*x+d)^(1/2)/(b*e*x+a*e)*B*a^2*
d^2*e^2+3/b^2*(e*x+d)^(1/2)/(b*e*x+a*e)*A*a*d^2*e^2+3/b^4*(e*x+d)^(1/2)/(b*e*x+a*e)*B*a^3*d*e^3

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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((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(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 = 0.16, size = 562, normalized size = 2.20 \[ \left (\frac {2\,A\,e-2\,B\,d}{5\,b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{5\,b^4}\right )\,{\left (d+e\,x\right )}^{5/2}+\left (\frac {\left (\frac {\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )}{b^2}-\frac {2\,B\,{\left (a\,e-b\,d\right )}^2}{b^4}\right )\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^2}-\frac {{\left (a\,e-b\,d\right )}^2\,\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )}{b^2}\right )\,\sqrt {d+e\,x}+\left (\frac {\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )}{3\,b^2}-\frac {2\,B\,{\left (a\,e-b\,d\right )}^2}{3\,b^4}\right )\,{\left (d+e\,x\right )}^{3/2}-\frac {\sqrt {d+e\,x}\,\left (B\,a^4\,e^4-3\,B\,a^3\,b\,d\,e^3-A\,a^3\,b\,e^4+3\,B\,a^2\,b^2\,d^2\,e^2+3\,A\,a^2\,b^2\,d\,e^3-B\,a\,b^3\,d^3\,e-3\,A\,a\,b^3\,d^2\,e^2+A\,b^4\,d^3\,e\right )}{b^6\,\left (d+e\,x\right )-b^6\,d+a\,b^5\,e}+\frac {2\,B\,{\left (d+e\,x\right )}^{7/2}}{7\,b^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}\,\left (7\,A\,b\,e-9\,B\,a\,e+2\,B\,b\,d\right )}{9\,B\,a^4\,e^4-29\,B\,a^3\,b\,d\,e^3-7\,A\,a^3\,b\,e^4+33\,B\,a^2\,b^2\,d^2\,e^2+21\,A\,a^2\,b^2\,d\,e^3-15\,B\,a\,b^3\,d^3\,e-21\,A\,a\,b^3\,d^2\,e^2+2\,B\,b^4\,d^4+7\,A\,b^4\,d^3\,e}\right )\,{\left (a\,e-b\,d\right )}^{5/2}\,\left (7\,A\,b\,e-9\,B\,a\,e+2\,B\,b\,d\right )}{b^{11/2}} \]

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

[Out]

Timed out

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