3.18.57 \(\int \frac {A+B x}{(a+b x)^2 (d+e x)^{7/2}} \, dx\) [1757]

Optimal. Leaf size=221 \[ \frac {2 b B d-7 A b e+5 a B e}{5 b (b d-a e)^2 (d+e x)^{5/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{5/2}}+\frac {2 b B d-7 A b e+5 a B e}{3 (b d-a e)^3 (d+e x)^{3/2}}+\frac {b (2 b B d-7 A b e+5 a B e)}{(b d-a e)^4 \sqrt {d+e x}}-\frac {b^{3/2} (2 b B d-7 A b e+5 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{9/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 53, 65, 214} \begin {gather*} -\frac {b^{3/2} (5 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 d-a e)^{9/2}}+\frac {b (5 a B e-7 A b e+2 b B d)}{\sqrt {d+e x} (b d-a e)^4}+\frac {5 a B e-7 A b e+2 b B d}{3 (d+e x)^{3/2} (b d-a e)^3}+\frac {5 a B e-7 A b e+2 b B d}{5 b (d+e x)^{5/2} (b d-a e)^2}-\frac {A b-a B}{b (a+b x) (d+e x)^{5/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 53

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] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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}{(a+b x)^2 (d+e x)^{7/2}} \, dx &=-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{5/2}}+\frac {(2 b B d-7 A b e+5 a B e) \int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx}{2 b (b d-a e)}\\ &=\frac {2 b B d-7 A b e+5 a B e}{5 b (b d-a e)^2 (d+e x)^{5/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{5/2}}+\frac {(2 b B d-7 A b e+5 a B e) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{2 (b d-a e)^2}\\ &=\frac {2 b B d-7 A b e+5 a B e}{5 b (b d-a e)^2 (d+e x)^{5/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{5/2}}+\frac {2 b B d-7 A b e+5 a B e}{3 (b d-a e)^3 (d+e x)^{3/2}}+\frac {(b (2 b B d-7 A b e+5 a B e)) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 (b d-a e)^3}\\ &=\frac {2 b B d-7 A b e+5 a B e}{5 b (b d-a e)^2 (d+e x)^{5/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{5/2}}+\frac {2 b B d-7 A b e+5 a B e}{3 (b d-a e)^3 (d+e x)^{3/2}}+\frac {b (2 b B d-7 A b e+5 a B e)}{(b d-a e)^4 \sqrt {d+e x}}+\frac {\left (b^2 (2 b B d-7 A b e+5 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 (b d-a e)^4}\\ &=\frac {2 b B d-7 A b e+5 a B e}{5 b (b d-a e)^2 (d+e x)^{5/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{5/2}}+\frac {2 b B d-7 A b e+5 a B e}{3 (b d-a e)^3 (d+e x)^{3/2}}+\frac {b (2 b B d-7 A b e+5 a B e)}{(b d-a e)^4 \sqrt {d+e x}}+\frac {\left (b^2 (2 b B d-7 A b e+5 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 )}{e (b d-a e)^4}\\ &=\frac {2 b B d-7 A b e+5 a B e}{5 b (b d-a e)^2 (d+e x)^{5/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{5/2}}+\frac {2 b B d-7 A b e+5 a B e}{3 (b d-a e)^3 (d+e x)^{3/2}}+\frac {b (2 b B d-7 A b e+5 a B e)}{(b d-a e)^4 \sqrt {d+e x}}-\frac {b^{3/2} (2 b B d-7 A b e+5 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{9/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.88, size = 289, normalized size = 1.31 \begin {gather*} \frac {B \left (-2 a^3 e^2 (2 d+5 e x)+2 b^3 d x \left (23 d^2+35 d e x+15 e^2 x^2\right )+2 a^2 b e \left (24 d^2+58 d e x+25 e^2 x^2\right )+a b^2 \left (61 d^3+163 d^2 e x+195 d e^2 x^2+75 e^3 x^3\right )\right )-A \left (6 a^3 e^3-2 a^2 b e^2 (16 d+7 e x)+2 a b^2 e \left (58 d^2+84 d e x+35 e^2 x^2\right )+b^3 \left (15 d^3+161 d^2 e x+245 d e^2 x^2+105 e^3 x^3\right )\right )}{15 (b d-a e)^4 (a+b x) (d+e x)^{5/2}}+\frac {b^{3/2} (2 b B d-7 A b e+5 a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(-2*a^3*e^2*(2*d + 5*e*x) + 2*b^3*d*x*(23*d^2 + 35*d*e*x + 15*e^2*x^2) + 2*a^2*b*e*(24*d^2 + 58*d*e*x + 25*
e^2*x^2) + a*b^2*(61*d^3 + 163*d^2*e*x + 195*d*e^2*x^2 + 75*e^3*x^3)) - A*(6*a^3*e^3 - 2*a^2*b*e^2*(16*d + 7*e
*x) + 2*a*b^2*e*(58*d^2 + 84*d*e*x + 35*e^2*x^2) + b^3*(15*d^3 + 161*d^2*e*x + 245*d*e^2*x^2 + 105*e^3*x^3)))/
(15*(b*d - a*e)^4*(a + b*x)*(d + e*x)^(5/2)) + (b^(3/2)*(2*b*B*d - 7*A*b*e + 5*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d +
e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(9/2)

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 202, normalized size = 0.91

method	result	size
derivativedivides	\(-\frac {2 \left (A e -B d \right )}{5 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (-2 A b e +B a e +B b d \right )}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 b \left (3 A b e -2 B a e -B b d \right )}{\left (a e -b d \right )^{4} \sqrt {e x +d}}-\frac {2 b^{2} \left (\frac {\left (\frac {1}{2} A b e -\frac {1}{2} B a e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (7 A b e -5 B a e -2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{4}}\)	\(202\)
default	\(-\frac {2 \left (A e -B d \right )}{5 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 \left (-2 A b e +B a e +B b d \right )}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 b \left (3 A b e -2 B a e -B b d \right )}{\left (a e -b d \right )^{4} \sqrt {e x +d}}-\frac {2 b^{2} \left (\frac {\left (\frac {1}{2} A b e -\frac {1}{2} B a e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (7 A b e -5 B a e -2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{4}}\)	\(202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 862 vs. \(2 (217) = 434\).
time = 1.50, size = 1735, normalized size = 7.85 \begin {gather*} \left [-\frac {15 \, {\left (2 \, B b^{3} d^{4} x + 2 \, B a b^{2} d^{4} + {\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} e^{4} + {\left (2 \, B b^{3} d x^{4} + {\left (17 \, B a b^{2} - 21 \, A b^{3}\right )} d x^{3} + 3 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} d x^{2}\right )} e^{3} + 3 \, {\left (2 \, B b^{3} d^{2} x^{3} + 7 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} x^{2} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} d^{2} x\right )} e^{2} + {\left (6 \, B b^{3} d^{3} x^{2} + {\left (11 \, B a b^{2} - 7 \, A b^{3}\right )} d^{3} x + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} d^{3}\right )} e\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {2 \, b d + 2 \, {\left (b d - a e\right )} \sqrt {x e + d} \sqrt {\frac {b}{b d - a e}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (46 \, B b^{3} d^{3} x + {\left (61 \, B a b^{2} - 15 \, A b^{3}\right )} d^{3} - {\left (6 \, A a^{3} - 15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} e^{3} + {\left (30 \, B b^{3} d x^{3} + 5 \, {\left (39 \, B a b^{2} - 49 \, A b^{3}\right )} d x^{2} + 4 \, {\left (29 \, B a^{2} b - 42 \, A a b^{2}\right )} d x - 4 \, {\left (B a^{3} - 8 \, A a^{2} b\right )} d\right )} e^{2} + {\left (70 \, B b^{3} d^{2} x^{2} + {\left (163 \, B a b^{2} - 161 \, A b^{3}\right )} d^{2} x + 4 \, {\left (12 \, B a^{2} b - 29 \, A a b^{2}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{30 \, {\left (b^{5} d^{7} x + a b^{4} d^{7} + {\left (a^{4} b x^{4} + a^{5} x^{3}\right )} e^{7} - {\left (4 \, a^{3} b^{2} d x^{4} + a^{4} b d x^{3} - 3 \, a^{5} d x^{2}\right )} e^{6} + 3 \, {\left (2 \, a^{2} b^{3} d^{2} x^{4} - 2 \, a^{3} b^{2} d^{2} x^{3} - 3 \, a^{4} b d^{2} x^{2} + a^{5} d^{2} x\right )} e^{5} - {\left (4 \, a b^{4} d^{3} x^{4} - 14 \, a^{2} b^{3} d^{3} x^{3} - 6 \, a^{3} b^{2} d^{3} x^{2} + 11 \, a^{4} b d^{3} x - a^{5} d^{3}\right )} e^{4} + {\left (b^{5} d^{4} x^{4} - 11 \, a b^{4} d^{4} x^{3} + 6 \, a^{2} b^{3} d^{4} x^{2} + 14 \, a^{3} b^{2} d^{4} x - 4 \, a^{4} b d^{4}\right )} e^{3} + 3 \, {\left (b^{5} d^{5} x^{3} - 3 \, a b^{4} d^{5} x^{2} - 2 \, a^{2} b^{3} d^{5} x + 2 \, a^{3} b^{2} d^{5}\right )} e^{2} + {\left (3 \, b^{5} d^{6} x^{2} - a b^{4} d^{6} x - 4 \, a^{2} b^{3} d^{6}\right )} e\right )}}, -\frac {15 \, {\left (2 \, B b^{3} d^{4} x + 2 \, B a b^{2} d^{4} + {\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} e^{4} + {\left (2 \, B b^{3} d x^{4} + {\left (17 \, B a b^{2} - 21 \, A b^{3}\right )} d x^{3} + 3 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} d x^{2}\right )} e^{3} + 3 \, {\left (2 \, B b^{3} d^{2} x^{3} + 7 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} x^{2} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} d^{2} x\right )} e^{2} + {\left (6 \, B b^{3} d^{3} x^{2} + {\left (11 \, B a b^{2} - 7 \, A b^{3}\right )} d^{3} x + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} d^{3}\right )} e\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {x e + d} \sqrt {-\frac {b}{b d - a e}}}{b x e + b d}\right ) - {\left (46 \, B b^{3} d^{3} x + {\left (61 \, B a b^{2} - 15 \, A b^{3}\right )} d^{3} - {\left (6 \, A a^{3} - 15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} e^{3} + {\left (30 \, B b^{3} d x^{3} + 5 \, {\left (39 \, B a b^{2} - 49 \, A b^{3}\right )} d x^{2} + 4 \, {\left (29 \, B a^{2} b - 42 \, A a b^{2}\right )} d x - 4 \, {\left (B a^{3} - 8 \, A a^{2} b\right )} d\right )} e^{2} + {\left (70 \, B b^{3} d^{2} x^{2} + {\left (163 \, B a b^{2} - 161 \, A b^{3}\right )} d^{2} x + 4 \, {\left (12 \, B a^{2} b - 29 \, A a b^{2}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (b^{5} d^{7} x + a b^{4} d^{7} + {\left (a^{4} b x^{4} + a^{5} x^{3}\right )} e^{7} - {\left (4 \, a^{3} b^{2} d x^{4} + a^{4} b d x^{3} - 3 \, a^{5} d x^{2}\right )} e^{6} + 3 \, {\left (2 \, a^{2} b^{3} d^{2} x^{4} - 2 \, a^{3} b^{2} d^{2} x^{3} - 3 \, a^{4} b d^{2} x^{2} + a^{5} d^{2} x\right )} e^{5} - {\left (4 \, a b^{4} d^{3} x^{4} - 14 \, a^{2} b^{3} d^{3} x^{3} - 6 \, a^{3} b^{2} d^{3} x^{2} + 11 \, a^{4} b d^{3} x - a^{5} d^{3}\right )} e^{4} + {\left (b^{5} d^{4} x^{4} - 11 \, a b^{4} d^{4} x^{3} + 6 \, a^{2} b^{3} d^{4} x^{2} + 14 \, a^{3} b^{2} d^{4} x - 4 \, a^{4} b d^{4}\right )} e^{3} + 3 \, {\left (b^{5} d^{5} x^{3} - 3 \, a b^{4} d^{5} x^{2} - 2 \, a^{2} b^{3} d^{5} x + 2 \, a^{3} b^{2} d^{5}\right )} e^{2} + {\left (3 \, b^{5} d^{6} x^{2} - a b^{4} d^{6} x - 4 \, a^{2} b^{3} d^{6}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/30*(15*(2*B*b^3*d^4*x + 2*B*a*b^2*d^4 + ((5*B*a*b^2 - 7*A*b^3)*x^4 + (5*B*a^2*b - 7*A*a*b^2)*x^3)*e^4 + (2
*B*b^3*d*x^4 + (17*B*a*b^2 - 21*A*b^3)*d*x^3 + 3*(5*B*a^2*b - 7*A*a*b^2)*d*x^2)*e^3 + 3*(2*B*b^3*d^2*x^3 + 7*(
B*a*b^2 - A*b^3)*d^2*x^2 + (5*B*a^2*b - 7*A*a*b^2)*d^2*x)*e^2 + (6*B*b^3*d^3*x^2 + (11*B*a*b^2 - 7*A*b^3)*d^3*
x + (5*B*a^2*b - 7*A*a*b^2)*d^3)*e)*sqrt(b/(b*d - a*e))*log((2*b*d + 2*(b*d - a*e)*sqrt(x*e + d)*sqrt(b/(b*d -
a*e)) + (b*x - a)*e)/(b*x + a)) - 2*(46*B*b^3*d^3*x + (61*B*a*b^2 - 15*A*b^3)*d^3 - (6*A*a^3 - 15*(5*B*a*b^2
- 7*A*b^3)*x^3 - 10*(5*B*a^2*b - 7*A*a*b^2)*x^2 + 2*(5*B*a^3 - 7*A*a^2*b)*x)*e^3 + (30*B*b^3*d*x^3 + 5*(39*B*a
*b^2 - 49*A*b^3)*d*x^2 + 4*(29*B*a^2*b - 42*A*a*b^2)*d*x - 4*(B*a^3 - 8*A*a^2*b)*d)*e^2 + (70*B*b^3*d^2*x^2 +
(163*B*a*b^2 - 161*A*b^3)*d^2*x + 4*(12*B*a^2*b - 29*A*a*b^2)*d^2)*e)*sqrt(x*e + d))/(b^5*d^7*x + a*b^4*d^7 +
(a^4*b*x^4 + a^5*x^3)*e^7 - (4*a^3*b^2*d*x^4 + a^4*b*d*x^3 - 3*a^5*d*x^2)*e^6 + 3*(2*a^2*b^3*d^2*x^4 - 2*a^3*b
^2*d^2*x^3 - 3*a^4*b*d^2*x^2 + a^5*d^2*x)*e^5 - (4*a*b^4*d^3*x^4 - 14*a^2*b^3*d^3*x^3 - 6*a^3*b^2*d^3*x^2 + 11
*a^4*b*d^3*x - a^5*d^3)*e^4 + (b^5*d^4*x^4 - 11*a*b^4*d^4*x^3 + 6*a^2*b^3*d^4*x^2 + 14*a^3*b^2*d^4*x - 4*a^4*b
*d^4)*e^3 + 3*(b^5*d^5*x^3 - 3*a*b^4*d^5*x^2 - 2*a^2*b^3*d^5*x + 2*a^3*b^2*d^5)*e^2 + (3*b^5*d^6*x^2 - a*b^4*d
^6*x - 4*a^2*b^3*d^6)*e), -1/15*(15*(2*B*b^3*d^4*x + 2*B*a*b^2*d^4 + ((5*B*a*b^2 - 7*A*b^3)*x^4 + (5*B*a^2*b -
7*A*a*b^2)*x^3)*e^4 + (2*B*b^3*d*x^4 + (17*B*a*b^2 - 21*A*b^3)*d*x^3 + 3*(5*B*a^2*b - 7*A*a*b^2)*d*x^2)*e^3 +
3*(2*B*b^3*d^2*x^3 + 7*(B*a*b^2 - A*b^3)*d^2*x^2 + (5*B*a^2*b - 7*A*a*b^2)*d^2*x)*e^2 + (6*B*b^3*d^3*x^2 + (1
1*B*a*b^2 - 7*A*b^3)*d^3*x + (5*B*a^2*b - 7*A*a*b^2)*d^3)*e)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(x*e
+ d)*sqrt(-b/(b*d - a*e))/(b*x*e + b*d)) - (46*B*b^3*d^3*x + (61*B*a*b^2 - 15*A*b^3)*d^3 - (6*A*a^3 - 15*(5*B
*a*b^2 - 7*A*b^3)*x^3 - 10*(5*B*a^2*b - 7*A*a*b^2)*x^2 + 2*(5*B*a^3 - 7*A*a^2*b)*x)*e^3 + (30*B*b^3*d*x^3 + 5*
(39*B*a*b^2 - 49*A*b^3)*d*x^2 + 4*(29*B*a^2*b - 42*A*a*b^2)*d*x - 4*(B*a^3 - 8*A*a^2*b)*d)*e^2 + (70*B*b^3*d^2
*x^2 + (163*B*a*b^2 - 161*A*b^3)*d^2*x + 4*(12*B*a^2*b - 29*A*a*b^2)*d^2)*e)*sqrt(x*e + d))/(b^5*d^7*x + a*b^4
*d^7 + (a^4*b*x^4 + a^5*x^3)*e^7 - (4*a^3*b^2*d*x^4 + a^4*b*d*x^3 - 3*a^5*d*x^2)*e^6 + 3*(2*a^2*b^3*d^2*x^4 -
2*a^3*b^2*d^2*x^3 - 3*a^4*b*d^2*x^2 + a^5*d^2*x)*e^5 - (4*a*b^4*d^3*x^4 - 14*a^2*b^3*d^3*x^3 - 6*a^3*b^2*d^3*x
^2 + 11*a^4*b*d^3*x - a^5*d^3)*e^4 + (b^5*d^4*x^4 - 11*a*b^4*d^4*x^3 + 6*a^2*b^3*d^4*x^2 + 14*a^3*b^2*d^4*x -
4*a^4*b*d^4)*e^3 + 3*(b^5*d^5*x^3 - 3*a*b^4*d^5*x^2 - 2*a^2*b^3*d^5*x + 2*a^3*b^2*d^5)*e^2 + (3*b^5*d^6*x^2 -
a*b^4*d^6*x - 4*a^2*b^3*d^6)*e)]

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (217) = 434\).
time = 1.61, size = 435, normalized size = 1.97 \begin {gather*} \frac {{\left (2 \, B b^{3} d + 5 \, B a b^{2} e - 7 \, A b^{3} e\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e}} + \frac {\sqrt {x e + d} B a b^{2} e - \sqrt {x e + d} A b^{3} e}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} B b^{2} d + 5 \, {\left (x e + d\right )} B b^{2} d^{2} + 3 \, B b^{2} d^{3} + 30 \, {\left (x e + d\right )}^{2} B a b e - 45 \, {\left (x e + d\right )}^{2} A b^{2} e - 10 \, {\left (x e + d\right )} A b^{2} d e - 6 \, B a b d^{2} e - 3 \, A b^{2} d^{2} e - 5 \, {\left (x e + d\right )} B a^{2} e^{2} + 10 \, {\left (x e + d\right )} A a b e^{2} + 3 \, B a^{2} d e^{2} + 6 \, A a b d e^{2} - 3 \, A a^{2} e^{3}\right )}}{15 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*B*b^3*d + 5*B*a*b^2*e - 7*A*b^3*e)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^4*d^4 - 4*a*b^3*d^3*e +
6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*sqrt(-b^2*d + a*b*e)) + (sqrt(x*e + d)*B*a*b^2*e - sqrt(x*e + d)
*A*b^3*e)/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*((x*e + d)*b - b*d + a*e))
+ 2/15*(15*(x*e + d)^2*B*b^2*d + 5*(x*e + d)*B*b^2*d^2 + 3*B*b^2*d^3 + 30*(x*e + d)^2*B*a*b*e - 45*(x*e + d)^2
*A*b^2*e - 10*(x*e + d)*A*b^2*d*e - 6*B*a*b*d^2*e - 3*A*b^2*d^2*e - 5*(x*e + d)*B*a^2*e^2 + 10*(x*e + d)*A*a*b
*e^2 + 3*B*a^2*d*e^2 + 6*A*a*b*d*e^2 - 3*A*a^2*e^3)/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*
e^3 + a^4*e^4)*(x*e + d)^(5/2))

________________________________________________________________________________________

Mupad [B]
time = 1.40, size = 261, normalized size = 1.18 \begin {gather*} \frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^{9/2}}\right )\,\left (5\,B\,a\,e-7\,A\,b\,e+2\,B\,b\,d\right )}{{\left (a\,e-b\,d\right )}^{9/2}}-\frac {\frac {2\,\left (A\,e-B\,d\right )}{5\,\left (a\,e-b\,d\right )}+\frac {2\,\left (d+e\,x\right )\,\left (5\,B\,a\,e-7\,A\,b\,e+2\,B\,b\,d\right )}{15\,{\left (a\,e-b\,d\right )}^2}-\frac {b^2\,{\left (d+e\,x\right )}^3\,\left (5\,B\,a\,e-7\,A\,b\,e+2\,B\,b\,d\right )}{{\left (a\,e-b\,d\right )}^4}-\frac {2\,b\,{\left (d+e\,x\right )}^2\,\left (5\,B\,a\,e-7\,A\,b\,e+2\,B\,b\,d\right )}{3\,{\left (a\,e-b\,d\right )}^3}}{b\,{\left (d+e\,x\right )}^{7/2}+\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________