3.19.14 \(\int \frac {A+B x}{(d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)} \, dx\) [1814]

Optimal. Leaf size=181 \[ \frac {2 b B d-5 A b e+3 a B e}{3 b (b d-a e)^2 (d+e x)^{3/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{3/2}}+\frac {2 b B d-5 A b e+3 a B e}{(b d-a e)^3 \sqrt {d+e x}}-\frac {\sqrt {b} (2 b B d-5 A b e+3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 79, 53, 65, 214} \begin {gather*} -\frac {A b-a B}{b (a+b x) (d+e x)^{3/2} (b d-a e)}+\frac {3 a B e-5 A b e+2 b B d}{\sqrt {d+e x} (b d-a e)^3}+\frac {3 a B e-5 A b e+2 b B d}{3 b (d+e x)^{3/2} (b d-a e)^2}-\frac {\sqrt {b} (3 a B e-5 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)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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

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Mathematica [A]
time = 0.65, size = 197, normalized size = 1.09 \begin {gather*} \frac {B \left (2 a^2 e (2 d+3 e x)+2 b^2 d x (4 d+3 e x)+a b \left (11 d^2+16 d e x+9 e^2 x^2\right )\right )-A \left (-2 a^2 e^2+2 a b e (7 d+5 e x)+b^2 \left (3 d^2+20 d e x+15 e^2 x^2\right )\right )}{3 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}-\frac {\sqrt {b} (2 b B d-5 A b e+3 a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(2*a^2*e*(2*d + 3*e*x) + 2*b^2*d*x*(4*d + 3*e*x) + a*b*(11*d^2 + 16*d*e*x + 9*e^2*x^2)) - A*(-2*a^2*e^2 + 2
*a*b*e*(7*d + 5*e*x) + b^2*(3*d^2 + 20*d*e*x + 15*e^2*x^2)))/(3*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)) - (Sq
rt[b]*(2*b*B*d - 5*A*b*e + 3*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(7/2)

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Maple [A]
time = 0.94, size = 164, normalized size = 0.91

method	result	size
derivativedivides	\(\frac {2 b \left (\frac {\left (\frac {1}{2} A b e -\frac {1}{2} B a e \right ) \sqrt {e x +d}}{\left (e x +d \right ) b +a e -b d}+\frac {\left (5 A b e -3 B a e -2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{3}}-\frac {2 \left (A e -B d \right )}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-2 A b e +B a e +B b d \right )}{\left (a e -b d \right )^{3} \sqrt {e x +d}}\)	\(164\)
default	\(\frac {2 b \left (\frac {\left (\frac {1}{2} A b e -\frac {1}{2} B a e \right ) \sqrt {e x +d}}{\left (e x +d \right ) b +a e -b d}+\frac {\left (5 A b e -3 B a e -2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{3}}-\frac {2 \left (A e -B d \right )}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-2 A b e +B a e +B b d \right )}{\left (a e -b d \right )^{3} \sqrt {e x +d}}\)	\(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(a*e-b*d)^3*b*((1/2*A*b*e-1/2*B*a*e)*(e*x+d)^(1/2)/((e*x+d)*b+a*e-b*d)+1/2*(5*A*b*e-3*B*a*e-2*B*b*d)/(b*(a*e
-b*d))^(1/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2)))-2/3*(A*e-B*d)/(a*e-b*d)^2/(e*x+d)^(3/2)-2/(a*e-b*d)^
3*(-2*A*b*e+B*a*e+B*b*d)/(e*x+d)^(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)^(5/2)/(b^2*x^2+2*a*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 [B] Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (177) = 354\).
time = 3.44, size = 1099, normalized size = 6.07 \begin {gather*} \left [\frac {3 \, {\left (2 \, B b^{2} d^{3} x + 2 \, B a b d^{3} + {\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{3} + {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} e^{3} + 2 \, {\left (B b^{2} d x^{3} + {\left (4 \, B a b - 5 \, A b^{2}\right )} d x^{2} + {\left (3 \, B a^{2} - 5 \, A a b\right )} d x\right )} e^{2} + {\left (4 \, B b^{2} d^{2} x^{2} + {\left (7 \, B a b - 5 \, A b^{2}\right )} d^{2} x + {\left (3 \, B a^{2} - 5 \, A a b\right )} d^{2}\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 (8 \, B b^{2} d^{2} x + {\left (11 \, B a b - 3 \, A b^{2}\right )} d^{2} + {\left (2 \, A a^{2} + 3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} e^{2} + 2 \, {\left (3 \, B b^{2} d x^{2} + 2 \, {\left (4 \, B a b - 5 \, A b^{2}\right )} d x + {\left (2 \, B a^{2} - 7 \, A a b\right )} d\right )} e\right )} \sqrt {x e + d}}{6 \, {\left (b^{4} d^{5} x + a b^{3} d^{5} - {\left (a^{3} b x^{3} + a^{4} x^{2}\right )} e^{5} + {\left (3 \, a^{2} b^{2} d x^{3} + a^{3} b d x^{2} - 2 \, a^{4} d x\right )} e^{4} - {\left (3 \, a b^{3} d^{2} x^{3} - 3 \, a^{2} b^{2} d^{2} x^{2} - 5 \, a^{3} b d^{2} x + a^{4} d^{2}\right )} e^{3} + {\left (b^{4} d^{3} x^{3} - 5 \, a b^{3} d^{3} x^{2} - 3 \, a^{2} b^{2} d^{3} x + 3 \, a^{3} b d^{3}\right )} e^{2} + {\left (2 \, b^{4} d^{4} x^{2} - a b^{3} d^{4} x - 3 \, a^{2} b^{2} d^{4}\right )} e\right )}}, -\frac {3 \, {\left (2 \, B b^{2} d^{3} x + 2 \, B a b d^{3} + {\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{3} + {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} e^{3} + 2 \, {\left (B b^{2} d x^{3} + {\left (4 \, B a b - 5 \, A b^{2}\right )} d x^{2} + {\left (3 \, B a^{2} - 5 \, A a b\right )} d x\right )} e^{2} + {\left (4 \, B b^{2} d^{2} x^{2} + {\left (7 \, B a b - 5 \, A b^{2}\right )} d^{2} x + {\left (3 \, B a^{2} - 5 \, A a b\right )} d^{2}\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 (8 \, B b^{2} d^{2} x + {\left (11 \, B a b - 3 \, A b^{2}\right )} d^{2} + {\left (2 \, A a^{2} + 3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} e^{2} + 2 \, {\left (3 \, B b^{2} d x^{2} + 2 \, {\left (4 \, B a b - 5 \, A b^{2}\right )} d x + {\left (2 \, B a^{2} - 7 \, A a b\right )} d\right )} e\right )} \sqrt {x e + d}}{3 \, {\left (b^{4} d^{5} x + a b^{3} d^{5} - {\left (a^{3} b x^{3} + a^{4} x^{2}\right )} e^{5} + {\left (3 \, a^{2} b^{2} d x^{3} + a^{3} b d x^{2} - 2 \, a^{4} d x\right )} e^{4} - {\left (3 \, a b^{3} d^{2} x^{3} - 3 \, a^{2} b^{2} d^{2} x^{2} - 5 \, a^{3} b d^{2} x + a^{4} d^{2}\right )} e^{3} + {\left (b^{4} d^{3} x^{3} - 5 \, a b^{3} d^{3} x^{2} - 3 \, a^{2} b^{2} d^{3} x + 3 \, a^{3} b d^{3}\right )} e^{2} + {\left (2 \, b^{4} d^{4} x^{2} - a b^{3} d^{4} x - 3 \, a^{2} b^{2} d^{4}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*(2*B*b^2*d^3*x + 2*B*a*b*d^3 + ((3*B*a*b - 5*A*b^2)*x^3 + (3*B*a^2 - 5*A*a*b)*x^2)*e^3 + 2*(B*b^2*d*x^
3 + (4*B*a*b - 5*A*b^2)*d*x^2 + (3*B*a^2 - 5*A*a*b)*d*x)*e^2 + (4*B*b^2*d^2*x^2 + (7*B*a*b - 5*A*b^2)*d^2*x +
(3*B*a^2 - 5*A*a*b)*d^2)*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*(8*B*b^2*d^2*x + (11*B*a*b - 3*A*b^2)*d^2 + (2*A*a^2 + 3*(3*B*a*b - 5*A*b^2)*x^2
+ 2*(3*B*a^2 - 5*A*a*b)*x)*e^2 + 2*(3*B*b^2*d*x^2 + 2*(4*B*a*b - 5*A*b^2)*d*x + (2*B*a^2 - 7*A*a*b)*d)*e)*sqrt
(x*e + d))/(b^4*d^5*x + a*b^3*d^5 - (a^3*b*x^3 + a^4*x^2)*e^5 + (3*a^2*b^2*d*x^3 + a^3*b*d*x^2 - 2*a^4*d*x)*e^
4 - (3*a*b^3*d^2*x^3 - 3*a^2*b^2*d^2*x^2 - 5*a^3*b*d^2*x + a^4*d^2)*e^3 + (b^4*d^3*x^3 - 5*a*b^3*d^3*x^2 - 3*a
^2*b^2*d^3*x + 3*a^3*b*d^3)*e^2 + (2*b^4*d^4*x^2 - a*b^3*d^4*x - 3*a^2*b^2*d^4)*e), -1/3*(3*(2*B*b^2*d^3*x + 2
*B*a*b*d^3 + ((3*B*a*b - 5*A*b^2)*x^3 + (3*B*a^2 - 5*A*a*b)*x^2)*e^3 + 2*(B*b^2*d*x^3 + (4*B*a*b - 5*A*b^2)*d*
x^2 + (3*B*a^2 - 5*A*a*b)*d*x)*e^2 + (4*B*b^2*d^2*x^2 + (7*B*a*b - 5*A*b^2)*d^2*x + (3*B*a^2 - 5*A*a*b)*d^2)*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)) - (8*B*b^2*d^2*x
+ (11*B*a*b - 3*A*b^2)*d^2 + (2*A*a^2 + 3*(3*B*a*b - 5*A*b^2)*x^2 + 2*(3*B*a^2 - 5*A*a*b)*x)*e^2 + 2*(3*B*b^2*
d*x^2 + 2*(4*B*a*b - 5*A*b^2)*d*x + (2*B*a^2 - 7*A*a*b)*d)*e)*sqrt(x*e + d))/(b^4*d^5*x + a*b^3*d^5 - (a^3*b*x
^3 + a^4*x^2)*e^5 + (3*a^2*b^2*d*x^3 + a^3*b*d*x^2 - 2*a^4*d*x)*e^4 - (3*a*b^3*d^2*x^3 - 3*a^2*b^2*d^2*x^2 - 5
*a^3*b*d^2*x + a^4*d^2)*e^3 + (b^4*d^3*x^3 - 5*a*b^3*d^3*x^2 - 3*a^2*b^2*d^3*x + 3*a^3*b*d^3)*e^2 + (2*b^4*d^4
*x^2 - a*b^3*d^4*x - 3*a^2*b^2*d^4)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\left (a + b x\right )^{2} \left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Integral((A + B*x)/((a + b*x)**2*(d + e*x)**(5/2)), x)

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Giac [A]
time = 0.85, size = 297, normalized size = 1.64 \begin {gather*} \frac {{\left (2 \, B b^{2} d + 3 \, B a b e - 5 \, A b^{2} e\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e}} + \frac {\sqrt {x e + d} B a b e - \sqrt {x e + d} A b^{2} e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )} B b d + B b d^{2} + 3 \, {\left (x e + d\right )} B a e - 6 \, {\left (x e + d\right )} A b e - B a d e - A b d e + A a e^{2}\right )}}{3 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*B*b^2*d + 3*B*a*b*e - 5*A*b^2*e)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^3*d^3 - 3*a*b^2*d^2*e + 3
*a^2*b*d*e^2 - a^3*e^3)*sqrt(-b^2*d + a*b*e)) + (sqrt(x*e + d)*B*a*b*e - sqrt(x*e + d)*A*b^2*e)/((b^3*d^3 - 3*
a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*((x*e + d)*b - b*d + a*e)) + 2/3*(3*(x*e + d)*B*b*d + B*b*d^2 + 3*(x*e
+ d)*B*a*e - 6*(x*e + d)*A*b*e - B*a*d*e - A*b*d*e + A*a*e^2)/((b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*
e^3)*(x*e + d)^(3/2))

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Mupad [B]
time = 2.10, size = 210, normalized size = 1.16 \begin {gather*} -\frac {\frac {2\,\left (A\,e-B\,d\right )}{3\,\left (a\,e-b\,d\right )}+\frac {2\,\left (d+e\,x\right )\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )}{3\,{\left (a\,e-b\,d\right )}^2}+\frac {b\,{\left (d+e\,x\right )}^2\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )}{{\left (a\,e-b\,d\right )}^3}}{b\,{\left (d+e\,x\right )}^{5/2}+\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{3/2}}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^{7/2}}\right )\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )}{{\left (a\,e-b\,d\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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