3.18.64 \(\int \frac {A+B x}{(a+b x)^3 (d+e x)^{5/2}} \, dx\) [1764]

Optimal. Leaf size=240 \[ -\frac {5 e (4 b B d-7 A b e+3 a B e)}{12 b (b d-a e)^3 (d+e x)^{3/2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac {4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{3/2}}-\frac {5 e (4 b B d-7 A b e+3 a B e)}{4 (b d-a e)^4 \sqrt {d+e x}}+\frac {5 \sqrt {b} e (4 b B d-7 A b e+3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{9/2}} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 44, 53, 65, 214} \begin {gather*} -\frac {A b-a B}{2 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)}-\frac {5 e (3 a B e-7 A b e+4 b B d)}{4 \sqrt {d+e x} (b d-a e)^4}-\frac {5 e (3 a B e-7 A b e+4 b B d)}{12 b (d+e x)^{3/2} (b d-a e)^3}-\frac {3 a B e-7 A b e+4 b B d}{4 b (a+b x) (d+e x)^{3/2} (b d-a e)^2}+\frac {5 \sqrt {b} e (3 a B e-7 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 d-a e)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x)^3*(d + e*x)^(5/2)),x]

[Out]

(-5*e*(4*b*B*d - 7*A*b*e + 3*a*B*e))/(12*b*(b*d - a*e)^3*(d + e*x)^(3/2)) - (A*b - a*B)/(2*b*(b*d - a*e)*(a +
b*x)^2*(d + e*x)^(3/2)) - (4*b*B*d - 7*A*b*e + 3*a*B*e)/(4*b*(b*d - a*e)^2*(a + b*x)*(d + e*x)^(3/2)) - (5*e*(
4*b*B*d - 7*A*b*e + 3*a*B*e))/(4*(b*d - a*e)^4*Sqrt[d + e*x]) + (5*Sqrt[b]*e*(4*b*B*d - 7*A*b*e + 3*a*B*e)*Arc
Tanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(9/2))

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

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)^3 (d+e x)^{5/2}} \, dx &=-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac {(4 b B d-7 A b e+3 a B e) \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{4 b (b d-a e)}\\ &=-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac {4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{3/2}}-\frac {(5 e (4 b B d-7 A b e+3 a B e)) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{8 b (b d-a e)^2}\\ &=-\frac {5 e (4 b B d-7 A b e+3 a B e)}{12 b (b d-a e)^3 (d+e x)^{3/2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac {4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{3/2}}-\frac {(5 e (4 b B d-7 A b e+3 a B e)) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{8 (b d-a e)^3}\\ &=-\frac {5 e (4 b B d-7 A b e+3 a B e)}{12 b (b d-a e)^3 (d+e x)^{3/2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac {4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{3/2}}-\frac {5 e (4 b B d-7 A b e+3 a B e)}{4 (b d-a e)^4 \sqrt {d+e x}}-\frac {(5 b e (4 b B d-7 A b e+3 a B e)) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{8 (b d-a e)^4}\\ &=-\frac {5 e (4 b B d-7 A b e+3 a B e)}{12 b (b d-a e)^3 (d+e x)^{3/2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac {4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{3/2}}-\frac {5 e (4 b B d-7 A b e+3 a B e)}{4 (b d-a e)^4 \sqrt {d+e x}}-\frac {(5 b (4 b B d-7 A b e+3 a B e)) \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 d-a e)^4}\\ &=-\frac {5 e (4 b B d-7 A b e+3 a B e)}{12 b (b d-a e)^3 (d+e x)^{3/2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac {4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{3/2}}-\frac {5 e (4 b B d-7 A b e+3 a B e)}{4 (b d-a e)^4 \sqrt {d+e x}}+\frac {5 \sqrt {b} e (4 b B d-7 A b e+3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 1.20, size = 291, normalized size = 1.21 \begin {gather*} \frac {-B \left (8 a^3 e^2 (2 d+3 e x)+4 b^3 d x \left (3 d^2+20 d e x+15 e^2 x^2\right )+a^2 b e \left (83 d^2+134 d e x+75 e^2 x^2\right )+a b^2 \left (6 d^3+145 d^2 e x+160 d e^2 x^2+45 e^3 x^3\right )\right )+A \left (-8 a^3 e^3+8 a^2 b e^2 (10 d+7 e x)+a b^2 e \left (39 d^2+238 d e x+175 e^2 x^2\right )+b^3 \left (-6 d^3+21 d^2 e x+140 d e^2 x^2+105 e^3 x^3\right )\right )}{12 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {5 \sqrt {b} e (4 b B d-7 A b e+3 a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{4 (-b d+a e)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x)^3*(d + e*x)^(5/2)),x]

[Out]

(-(B*(8*a^3*e^2*(2*d + 3*e*x) + 4*b^3*d*x*(3*d^2 + 20*d*e*x + 15*e^2*x^2) + a^2*b*e*(83*d^2 + 134*d*e*x + 75*e
^2*x^2) + a*b^2*(6*d^3 + 145*d^2*e*x + 160*d*e^2*x^2 + 45*e^3*x^3))) + A*(-8*a^3*e^3 + 8*a^2*b*e^2*(10*d + 7*e
*x) + a*b^2*e*(39*d^2 + 238*d*e*x + 175*e^2*x^2) + b^3*(-6*d^3 + 21*d^2*e*x + 140*d*e^2*x^2 + 105*e^3*x^3)))/(
12*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(3/2)) - (5*Sqrt[b]*e*(4*b*B*d - 7*A*b*e + 3*a*B*e)*ArcTan[(Sqrt[b]*Sqr
t[d + e*x])/Sqrt[-(b*d) + a*e]])/(4*(-(b*d) + a*e)^(9/2))

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Maple [A]
time = 0.08, size = 229, normalized size = 0.95

method	result	size
derivativedivides	\(2 e \left (-\frac {A e -B d}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {-3 A b e +B a e +2 B b d}{\left (a e -b d \right )^{4} \sqrt {e x +d}}+\frac {b \left (\frac {\left (\frac {11}{8} A \,b^{2} e -\frac {7}{8} B a b e -\frac {1}{2} b^{2} B d \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {13}{8} A a b \,e^{2}-\frac {13}{8} A \,b^{2} d e -\frac {9}{8} B \,a^{2} e^{2}+\frac {5}{8} B a b d e +\frac {1}{2} b^{2} B \,d^{2}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {5 \left (7 A b e -3 B a e -4 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{4}}\right )\)	\(229\)
default	\(2 e \left (-\frac {A e -B d}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {-3 A b e +B a e +2 B b d}{\left (a e -b d \right )^{4} \sqrt {e x +d}}+\frac {b \left (\frac {\left (\frac {11}{8} A \,b^{2} e -\frac {7}{8} B a b e -\frac {1}{2} b^{2} B d \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {13}{8} A a b \,e^{2}-\frac {13}{8} A \,b^{2} d e -\frac {9}{8} B \,a^{2} e^{2}+\frac {5}{8} B a b d e +\frac {1}{2} b^{2} B \,d^{2}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {5 \left (7 A b e -3 B a e -4 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{4}}\right )\)	\(229\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)^3/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2*e*(-1/3*(A*e-B*d)/(a*e-b*d)^3/(e*x+d)^(3/2)-1/(a*e-b*d)^4*(-3*A*b*e+B*a*e+2*B*b*d)/(e*x+d)^(1/2)+1/(a*e-b*d)
^4*b*(((11/8*A*b^2*e-7/8*B*a*b*e-1/2*b^2*B*d)*(e*x+d)^(3/2)+(13/8*A*a*b*e^2-13/8*A*b^2*d*e-9/8*B*a^2*e^2+5/8*B
*a*b*d*e+1/2*b^2*B*d^2)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^2+5/8*(7*A*b*e-3*B*a*e-4*B*b*d)/((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)/(b*x+a)^3/(e*x+d)^(5/2),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 [B] Leaf count of result is larger than twice the leaf count of optimal. 866 vs. \(2 (234) = 468\).
time = 0.98, size = 1743, normalized size = 7.26 \begin {gather*} \left [-\frac {15 \, {\left ({\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} e^{4} + 2 \, {\left (2 \, B b^{3} d x^{4} + 7 \, {\left (B a b^{2} - A b^{3}\right )} d x^{3} + 2 \, {\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} d x^{2} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} d x\right )} e^{3} + {\left (8 \, B b^{3} d^{2} x^{3} + {\left (19 \, B a b^{2} - 7 \, A b^{3}\right )} d^{2} x^{2} + 14 \, {\left (B a^{2} b - A a b^{2}\right )} d^{2} x + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} d^{2}\right )} e^{2} + 4 \, {\left (B b^{3} d^{3} x^{2} + 2 \, B a b^{2} d^{3} x + B a^{2} b 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 (12 \, B b^{3} d^{3} x + 6 \, {\left (B a b^{2} + A b^{3}\right )} d^{3} + {\left (8 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} e^{3} + 2 \, {\left (30 \, B b^{3} d x^{3} + 10 \, {\left (8 \, B a b^{2} - 7 \, A b^{3}\right )} d x^{2} + {\left (67 \, B a^{2} b - 119 \, A a b^{2}\right )} d x + 8 \, {\left (B a^{3} - 5 \, A a^{2} b\right )} d\right )} e^{2} + {\left (80 \, B b^{3} d^{2} x^{2} + {\left (145 \, B a b^{2} - 21 \, A b^{3}\right )} d^{2} x + {\left (83 \, B a^{2} b - 39 \, A a b^{2}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{24 \, {\left (b^{6} d^{6} x^{2} + 2 \, a b^{5} d^{6} x + a^{2} b^{4} d^{6} + {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )} e^{6} - 2 \, {\left (2 \, a^{3} b^{3} d x^{4} + 3 \, a^{4} b^{2} d x^{3} - a^{6} d x\right )} e^{5} + {\left (6 \, a^{2} b^{4} d^{2} x^{4} + 4 \, a^{3} b^{3} d^{2} x^{3} - 9 \, a^{4} b^{2} d^{2} x^{2} - 6 \, a^{5} b d^{2} x + a^{6} d^{2}\right )} e^{4} - 4 \, {\left (a b^{5} d^{3} x^{4} - a^{2} b^{4} d^{3} x^{3} - 4 \, a^{3} b^{3} d^{3} x^{2} - a^{4} b^{2} d^{3} x + a^{5} b d^{3}\right )} e^{3} + {\left (b^{6} d^{4} x^{4} - 6 \, a b^{5} d^{4} x^{3} - 9 \, a^{2} b^{4} d^{4} x^{2} + 4 \, a^{3} b^{3} d^{4} x + 6 \, a^{4} b^{2} d^{4}\right )} e^{2} + 2 \, {\left (b^{6} d^{5} x^{3} - 3 \, a^{2} b^{4} d^{5} x - 2 \, a^{3} b^{3} d^{5}\right )} e\right )}}, \frac {15 \, {\left ({\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} e^{4} + 2 \, {\left (2 \, B b^{3} d x^{4} + 7 \, {\left (B a b^{2} - A b^{3}\right )} d x^{3} + 2 \, {\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} d x^{2} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} d x\right )} e^{3} + {\left (8 \, B b^{3} d^{2} x^{3} + {\left (19 \, B a b^{2} - 7 \, A b^{3}\right )} d^{2} x^{2} + 14 \, {\left (B a^{2} b - A a b^{2}\right )} d^{2} x + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} d^{2}\right )} e^{2} + 4 \, {\left (B b^{3} d^{3} x^{2} + 2 \, B a b^{2} d^{3} x + B a^{2} b 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 (12 \, B b^{3} d^{3} x + 6 \, {\left (B a b^{2} + A b^{3}\right )} d^{3} + {\left (8 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} e^{3} + 2 \, {\left (30 \, B b^{3} d x^{3} + 10 \, {\left (8 \, B a b^{2} - 7 \, A b^{3}\right )} d x^{2} + {\left (67 \, B a^{2} b - 119 \, A a b^{2}\right )} d x + 8 \, {\left (B a^{3} - 5 \, A a^{2} b\right )} d\right )} e^{2} + {\left (80 \, B b^{3} d^{2} x^{2} + {\left (145 \, B a b^{2} - 21 \, A b^{3}\right )} d^{2} x + {\left (83 \, B a^{2} b - 39 \, A a b^{2}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{12 \, {\left (b^{6} d^{6} x^{2} + 2 \, a b^{5} d^{6} x + a^{2} b^{4} d^{6} + {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )} e^{6} - 2 \, {\left (2 \, a^{3} b^{3} d x^{4} + 3 \, a^{4} b^{2} d x^{3} - a^{6} d x\right )} e^{5} + {\left (6 \, a^{2} b^{4} d^{2} x^{4} + 4 \, a^{3} b^{3} d^{2} x^{3} - 9 \, a^{4} b^{2} d^{2} x^{2} - 6 \, a^{5} b d^{2} x + a^{6} d^{2}\right )} e^{4} - 4 \, {\left (a b^{5} d^{3} x^{4} - a^{2} b^{4} d^{3} x^{3} - 4 \, a^{3} b^{3} d^{3} x^{2} - a^{4} b^{2} d^{3} x + a^{5} b d^{3}\right )} e^{3} + {\left (b^{6} d^{4} x^{4} - 6 \, a b^{5} d^{4} x^{3} - 9 \, a^{2} b^{4} d^{4} x^{2} + 4 \, a^{3} b^{3} d^{4} x + 6 \, a^{4} b^{2} d^{4}\right )} e^{2} + 2 \, {\left (b^{6} d^{5} x^{3} - 3 \, a^{2} b^{4} d^{5} x - 2 \, a^{3} b^{3} d^{5}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^3/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

[-1/24*(15*(((3*B*a*b^2 - 7*A*b^3)*x^4 + 2*(3*B*a^2*b - 7*A*a*b^2)*x^3 + (3*B*a^3 - 7*A*a^2*b)*x^2)*e^4 + 2*(2
*B*b^3*d*x^4 + 7*(B*a*b^2 - A*b^3)*d*x^3 + 2*(4*B*a^2*b - 7*A*a*b^2)*d*x^2 + (3*B*a^3 - 7*A*a^2*b)*d*x)*e^3 +
(8*B*b^3*d^2*x^3 + (19*B*a*b^2 - 7*A*b^3)*d^2*x^2 + 14*(B*a^2*b - A*a*b^2)*d^2*x + (3*B*a^3 - 7*A*a^2*b)*d^2)*
e^2 + 4*(B*b^3*d^3*x^2 + 2*B*a*b^2*d^3*x + B*a^2*b*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*(12*B*b^3*d^3*x + 6*(B*a*b^2 + A*b^3)*d^3 + (8*A*a
^3 + 15*(3*B*a*b^2 - 7*A*b^3)*x^3 + 25*(3*B*a^2*b - 7*A*a*b^2)*x^2 + 8*(3*B*a^3 - 7*A*a^2*b)*x)*e^3 + 2*(30*B*
b^3*d*x^3 + 10*(8*B*a*b^2 - 7*A*b^3)*d*x^2 + (67*B*a^2*b - 119*A*a*b^2)*d*x + 8*(B*a^3 - 5*A*a^2*b)*d)*e^2 + (
80*B*b^3*d^2*x^2 + (145*B*a*b^2 - 21*A*b^3)*d^2*x + (83*B*a^2*b - 39*A*a*b^2)*d^2)*e)*sqrt(x*e + d))/(b^6*d^6*
x^2 + 2*a*b^5*d^6*x + a^2*b^4*d^6 + (a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2)*e^6 - 2*(2*a^3*b^3*d*x^4 + 3*a^4*b^2
*d*x^3 - a^6*d*x)*e^5 + (6*a^2*b^4*d^2*x^4 + 4*a^3*b^3*d^2*x^3 - 9*a^4*b^2*d^2*x^2 - 6*a^5*b*d^2*x + a^6*d^2)*
e^4 - 4*(a*b^5*d^3*x^4 - a^2*b^4*d^3*x^3 - 4*a^3*b^3*d^3*x^2 - a^4*b^2*d^3*x + a^5*b*d^3)*e^3 + (b^6*d^4*x^4 -
6*a*b^5*d^4*x^3 - 9*a^2*b^4*d^4*x^2 + 4*a^3*b^3*d^4*x + 6*a^4*b^2*d^4)*e^2 + 2*(b^6*d^5*x^3 - 3*a^2*b^4*d^5*x
- 2*a^3*b^3*d^5)*e), 1/12*(15*(((3*B*a*b^2 - 7*A*b^3)*x^4 + 2*(3*B*a^2*b - 7*A*a*b^2)*x^3 + (3*B*a^3 - 7*A*a^
2*b)*x^2)*e^4 + 2*(2*B*b^3*d*x^4 + 7*(B*a*b^2 - A*b^3)*d*x^3 + 2*(4*B*a^2*b - 7*A*a*b^2)*d*x^2 + (3*B*a^3 - 7*
A*a^2*b)*d*x)*e^3 + (8*B*b^3*d^2*x^3 + (19*B*a*b^2 - 7*A*b^3)*d^2*x^2 + 14*(B*a^2*b - A*a*b^2)*d^2*x + (3*B*a^
3 - 7*A*a^2*b)*d^2)*e^2 + 4*(B*b^3*d^3*x^2 + 2*B*a*b^2*d^3*x + B*a^2*b*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)) - (12*B*b^3*d^3*x + 6*(B*a*b^2 + A*b^3)*d^3 + (8*A
*a^3 + 15*(3*B*a*b^2 - 7*A*b^3)*x^3 + 25*(3*B*a^2*b - 7*A*a*b^2)*x^2 + 8*(3*B*a^3 - 7*A*a^2*b)*x)*e^3 + 2*(30*
B*b^3*d*x^3 + 10*(8*B*a*b^2 - 7*A*b^3)*d*x^2 + (67*B*a^2*b - 119*A*a*b^2)*d*x + 8*(B*a^3 - 5*A*a^2*b)*d)*e^2 +
(80*B*b^3*d^2*x^2 + (145*B*a*b^2 - 21*A*b^3)*d^2*x + (83*B*a^2*b - 39*A*a*b^2)*d^2)*e)*sqrt(x*e + d))/(b^6*d^
6*x^2 + 2*a*b^5*d^6*x + a^2*b^4*d^6 + (a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2)*e^6 - 2*(2*a^3*b^3*d*x^4 + 3*a^4*b
^2*d*x^3 - a^6*d*x)*e^5 + (6*a^2*b^4*d^2*x^4 + 4*a^3*b^3*d^2*x^3 - 9*a^4*b^2*d^2*x^2 - 6*a^5*b*d^2*x + a^6*d^2
)*e^4 - 4*(a*b^5*d^3*x^4 - a^2*b^4*d^3*x^3 - 4*a^3*b^3*d^3*x^2 - a^4*b^2*d^3*x + a^5*b*d^3)*e^3 + (b^6*d^4*x^4
- 6*a*b^5*d^4*x^3 - 9*a^2*b^4*d^4*x^2 + 4*a^3*b^3*d^4*x + 6*a^4*b^2*d^4)*e^2 + 2*(b^6*d^5*x^3 - 3*a^2*b^4*d^5
*x - 2*a^3*b^3*d^5)*e)]

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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)/(b*x+a)**3/(e*x+d)**(5/2),x)

[Out]

Timed out

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Giac [A]
time = 1.38, size = 449, normalized size = 1.87 \begin {gather*} -\frac {5 \, {\left (4 \, B b^{2} d e + 3 \, B a b e^{2} - 7 \, A b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\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 {2 \, {\left (6 \, {\left (x e + d\right )} B b d e + B b d^{2} e + 3 \, {\left (x e + d\right )} B a e^{2} - 9 \, {\left (x e + d\right )} A b e^{2} - B a d e^{2} - A b d e^{2} + A a e^{3}\right )}}{3 \, {\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 {3}{2}}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} d e - 4 \, \sqrt {x e + d} B b^{3} d^{2} e + 7 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} e^{2} - 11 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} e^{2} - 5 \, \sqrt {x e + d} B a b^{2} d e^{2} + 13 \, \sqrt {x e + d} A b^{3} d e^{2} + 9 \, \sqrt {x e + d} B a^{2} b e^{3} - 13 \, \sqrt {x e + d} A a b^{2} e^{3}}{4 \, {\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 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^3/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

-5/4*(4*B*b^2*d*e + 3*B*a*b*e^2 - 7*A*b^2*e^2)*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)) - 2/3*(6*(x*e + d)*B*b*d*e + B*b*
d^2*e + 3*(x*e + d)*B*a*e^2 - 9*(x*e + d)*A*b*e^2 - B*a*d*e^2 - A*b*d*e^2 + A*a*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)^(3/2)) - 1/4*(4*(x*e + d)^(3/2)*B*b^3*d*e - 4*sqrt(x
*e + d)*B*b^3*d^2*e + 7*(x*e + d)^(3/2)*B*a*b^2*e^2 - 11*(x*e + d)^(3/2)*A*b^3*e^2 - 5*sqrt(x*e + d)*B*a*b^2*d
*e^2 + 13*sqrt(x*e + d)*A*b^3*d*e^2 + 9*sqrt(x*e + d)*B*a^2*b*e^3 - 13*sqrt(x*e + d)*A*a*b^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)*b - b*d + a*e)^2)

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Mupad [B]
time = 1.52, size = 361, normalized size = 1.50 \begin {gather*} -\frac {\frac {2\,\left (A\,e^2-B\,d\,e\right )}{3\,\left (a\,e-b\,d\right )}+\frac {2\,\left (d+e\,x\right )\,\left (3\,B\,a\,e^2-7\,A\,b\,e^2+4\,B\,b\,d\,e\right )}{3\,{\left (a\,e-b\,d\right )}^2}+\frac {25\,{\left (d+e\,x\right )}^2\,\left (-7\,A\,b^2\,e^2+4\,B\,d\,b^2\,e+3\,B\,a\,b\,e^2\right )}{12\,{\left (a\,e-b\,d\right )}^3}+\frac {5\,b^2\,{\left (d+e\,x\right )}^3\,\left (3\,B\,a\,e^2-7\,A\,b\,e^2+4\,B\,b\,d\,e\right )}{4\,{\left (a\,e-b\,d\right )}^4}}{b^2\,{\left (d+e\,x\right )}^{7/2}-\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (d+e\,x\right )}^{5/2}+{\left (d+e\,x\right )}^{3/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}-\frac {5\,\sqrt {b}\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,\sqrt {d+e\,x}\,\left (3\,B\,a\,e-7\,A\,b\,e+4\,B\,b\,d\right )\,\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}\,\left (3\,B\,a\,e^2-7\,A\,b\,e^2+4\,B\,b\,d\,e\right )}\right )\,\left (3\,B\,a\,e-7\,A\,b\,e+4\,B\,b\,d\right )}{4\,{\left (a\,e-b\,d\right )}^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((a + b*x)^3*(d + e*x)^(5/2)),x)

[Out]

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

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