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

Optimal. Leaf size=348 \[ -\frac {A b-a B}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac {5 e (a+b x) (3 a B e-7 A b e+4 b B d)}{4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}+\frac {-3 a B e+7 A b e-4 b B d}{4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac {5 e (a+b x) (3 a B e-7 A b e+4 b B d)}{12 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac {5 \sqrt {b} e (a+b x) (3 a B e-7 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}} \]

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Rubi [A]  time = 0.32, antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {770, 78, 51, 63, 208} \begin {gather*} -\frac {A b-a B}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac {5 e (a+b x) (3 a B e-7 A b e+4 b B d)}{4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}-\frac {3 a B e-7 A b e+4 b B d}{4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac {5 e (a+b x) (3 a B e-7 A b e+4 b B d)}{12 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac {5 \sqrt {b} e (a+b x) (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 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}} \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)^(3/2)),x]

[Out]

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

Rule 51

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 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]

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

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

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Mathematica [C]  time = 0.07, size = 111, normalized size = 0.32 \begin {gather*} \frac {(a+b x) \left (\frac {e (a+b x)^2 (-3 a B e+7 A b e-4 b B d) \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+3 a B-3 A b\right )}{6 b \left ((a+b x)^2\right )^{3/2} (d+e x)^{3/2} (b d-a e)} \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)^(3/2)),x]

[Out]

((a + b*x)*(-3*A*b + 3*a*B + (e*(-4*b*B*d + 7*A*b*e - 3*a*B*e)*(a + b*x)^2*Hypergeometric2F1[-3/2, 2, -1/2, (b
*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^2))/(6*b*(b*d - a*e)*((a + b*x)^2)^(3/2)*(d + e*x)^(3/2))

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IntegrateAlgebraic [A]  time = 63.32, size = 490, normalized size = 1.41 \begin {gather*} \frac {(-a e-b e x) \left (\frac {e \left (8 a^3 A e^4+24 a^3 B e^3 (d+e x)-8 a^3 B d e^3-56 a^2 A b e^3 (d+e x)-24 a^2 A b d e^3+24 a^2 b B d^2 e^2-16 a^2 b B d e^2 (d+e x)+75 a^2 b B e^2 (d+e x)^2+24 a A b^2 d^2 e^2+112 a A b^2 d e^2 (d+e x)-175 a A b^2 e^2 (d+e x)^2-24 a b^2 B d^3 e-40 a b^2 B d^2 e (d+e x)+25 a b^2 B d e (d+e x)^2+45 a b^2 B e (d+e x)^3-8 A b^3 d^3 e-56 A b^3 d^2 e (d+e x)+175 A b^3 d e (d+e x)^2-105 A b^3 e (d+e x)^3+8 b^3 B d^4+32 b^3 B d^3 (d+e x)-100 b^3 B d^2 (d+e x)^2+60 b^3 B d (d+e x)^3\right )}{12 (d+e x)^{3/2} (b d-a e)^4 (-a e-b (d+e x)+b d)^2}-\frac {5 \left (3 a \sqrt {b} B e^2-7 A b^{3/2} e^2+4 b^{3/2} B d e\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{4 (b d-a e)^4 \sqrt {a e-b d}}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-(a*e) - b*e*x)*((e*(8*b^3*B*d^4 - 8*A*b^3*d^3*e - 24*a*b^2*B*d^3*e + 24*a*A*b^2*d^2*e^2 + 24*a^2*b*B*d^2*e^
2 - 24*a^2*A*b*d*e^3 - 8*a^3*B*d*e^3 + 8*a^3*A*e^4 + 32*b^3*B*d^3*(d + e*x) - 56*A*b^3*d^2*e*(d + e*x) - 40*a*
b^2*B*d^2*e*(d + e*x) + 112*a*A*b^2*d*e^2*(d + e*x) - 16*a^2*b*B*d*e^2*(d + e*x) - 56*a^2*A*b*e^3*(d + e*x) +
24*a^3*B*e^3*(d + e*x) - 100*b^3*B*d^2*(d + e*x)^2 + 175*A*b^3*d*e*(d + e*x)^2 + 25*a*b^2*B*d*e*(d + e*x)^2 -
175*a*A*b^2*e^2*(d + e*x)^2 + 75*a^2*b*B*e^2*(d + e*x)^2 + 60*b^3*B*d*(d + e*x)^3 - 105*A*b^3*e*(d + e*x)^3 +
45*a*b^2*B*e*(d + e*x)^3))/(12*(b*d - a*e)^4*(d + e*x)^(3/2)*(b*d - a*e - b*(d + e*x))^2) - (5*(4*b^(3/2)*B*d*
e - 7*A*b^(3/2)*e^2 + 3*a*Sqrt[b]*B*e^2)*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(4*(b
*d - a*e)^4*Sqrt[-(b*d) + a*e])))/(e*Sqrt[(a*e + b*e*x)^2/e^2])

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fricas [B]  time = 0.55, size = 1776, normalized size = 5.10

result too large to display

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)^(3/2),x, algorithm="fricas")

[Out]

[-1/24*(15*(4*B*a^2*b*d^3*e + (3*B*a^3 - 7*A*a^2*b)*d^2*e^2 + (4*B*b^3*d*e^3 + (3*B*a*b^2 - 7*A*b^3)*e^4)*x^4
+ 2*(4*B*b^3*d^2*e^2 + 7*(B*a*b^2 - A*b^3)*d*e^3 + (3*B*a^2*b - 7*A*a*b^2)*e^4)*x^3 + (4*B*b^3*d^3*e + (19*B*a
*b^2 - 7*A*b^3)*d^2*e^2 + 4*(4*B*a^2*b - 7*A*a*b^2)*d*e^3 + (3*B*a^3 - 7*A*a^2*b)*e^4)*x^2 + 2*(4*B*a*b^2*d^3*
e + 7*(B*a^2*b - A*a*b^2)*d^2*e^2 + (3*B*a^3 - 7*A*a^2*b)*d*e^3)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a
*e - 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 2*(8*A*a^3*e^3 + 6*(B*a*b^2 + A*b^3)*d^3 +
(83*B*a^2*b - 39*A*a*b^2)*d^2*e + 16*(B*a^3 - 5*A*a^2*b)*d*e^2 + 15*(4*B*b^3*d*e^2 + (3*B*a*b^2 - 7*A*b^3)*e^3
)*x^3 + 5*(16*B*b^3*d^2*e + 4*(8*B*a*b^2 - 7*A*b^3)*d*e^2 + 5*(3*B*a^2*b - 7*A*a*b^2)*e^3)*x^2 + (12*B*b^3*d^3
+ (145*B*a*b^2 - 21*A*b^3)*d^2*e + 2*(67*B*a^2*b - 119*A*a*b^2)*d*e^2 + 8*(3*B*a^3 - 7*A*a^2*b)*e^3)*x)*sqrt(
e*x + d))/(a^2*b^4*d^6 - 4*a^3*b^3*d^5*e + 6*a^4*b^2*d^4*e^2 - 4*a^5*b*d^3*e^3 + a^6*d^2*e^4 + (b^6*d^4*e^2 -
4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^4 + 2*(b^6*d^5*e - 3*a*b^5*d^4*e^2 + 2*
a^2*b^4*d^3*e^3 + 2*a^3*b^3*d^2*e^4 - 3*a^4*b^2*d*e^5 + a^5*b*e^6)*x^3 + (b^6*d^6 - 9*a^2*b^4*d^4*e^2 + 16*a^3
*b^3*d^3*e^3 - 9*a^4*b^2*d^2*e^4 + a^6*e^6)*x^2 + 2*(a*b^5*d^6 - 3*a^2*b^4*d^5*e + 2*a^3*b^3*d^4*e^2 + 2*a^4*b
^2*d^3*e^3 - 3*a^5*b*d^2*e^4 + a^6*d*e^5)*x), 1/12*(15*(4*B*a^2*b*d^3*e + (3*B*a^3 - 7*A*a^2*b)*d^2*e^2 + (4*B
*b^3*d*e^3 + (3*B*a*b^2 - 7*A*b^3)*e^4)*x^4 + 2*(4*B*b^3*d^2*e^2 + 7*(B*a*b^2 - A*b^3)*d*e^3 + (3*B*a^2*b - 7*
A*a*b^2)*e^4)*x^3 + (4*B*b^3*d^3*e + (19*B*a*b^2 - 7*A*b^3)*d^2*e^2 + 4*(4*B*a^2*b - 7*A*a*b^2)*d*e^3 + (3*B*a
^3 - 7*A*a^2*b)*e^4)*x^2 + 2*(4*B*a*b^2*d^3*e + 7*(B*a^2*b - A*a*b^2)*d^2*e^2 + (3*B*a^3 - 7*A*a^2*b)*d*e^3)*x
)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (8*A*a^3*e^3 +
6*(B*a*b^2 + A*b^3)*d^3 + (83*B*a^2*b - 39*A*a*b^2)*d^2*e + 16*(B*a^3 - 5*A*a^2*b)*d*e^2 + 15*(4*B*b^3*d*e^2 +
(3*B*a*b^2 - 7*A*b^3)*e^3)*x^3 + 5*(16*B*b^3*d^2*e + 4*(8*B*a*b^2 - 7*A*b^3)*d*e^2 + 5*(3*B*a^2*b - 7*A*a*b^2
)*e^3)*x^2 + (12*B*b^3*d^3 + (145*B*a*b^2 - 21*A*b^3)*d^2*e + 2*(67*B*a^2*b - 119*A*a*b^2)*d*e^2 + 8*(3*B*a^3
- 7*A*a^2*b)*e^3)*x)*sqrt(e*x + d))/(a^2*b^4*d^6 - 4*a^3*b^3*d^5*e + 6*a^4*b^2*d^4*e^2 - 4*a^5*b*d^3*e^3 + a^6
*d^2*e^4 + (b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^4 + 2*(b^6*d^
5*e - 3*a*b^5*d^4*e^2 + 2*a^2*b^4*d^3*e^3 + 2*a^3*b^3*d^2*e^4 - 3*a^4*b^2*d*e^5 + a^5*b*e^6)*x^3 + (b^6*d^6 -
9*a^2*b^4*d^4*e^2 + 16*a^3*b^3*d^3*e^3 - 9*a^4*b^2*d^2*e^4 + a^6*e^6)*x^2 + 2*(a*b^5*d^6 - 3*a^2*b^4*d^5*e + 2
*a^3*b^3*d^4*e^2 + 2*a^4*b^2*d^3*e^3 - 3*a^5*b*d^2*e^4 + a^6*d*e^5)*x)]

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giac [B]  time = 0.49, size = 785, normalized size = 2.26 \begin {gather*} -\frac {5 \, {\left (4 \, B b^{2} d e^{2} + 3 \, B a b e^{3} - 7 \, A b^{2} e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{4} d^{4} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, {\left (6 \, {\left (x e + d\right )} B b d e^{2} + B b d^{2} e^{2} + 3 \, {\left (x e + d\right )} B a e^{3} - 9 \, {\left (x e + d\right )} A b e^{3} - B a d e^{3} - A b d e^{3} + A a e^{4}\right )}}{3 \, {\left (b^{4} d^{4} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} {\left (x e + d\right )}^{\frac {3}{2}}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} d e^{2} - 4 \, \sqrt {x e + d} B b^{3} d^{2} e^{2} + 7 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} e^{3} - 11 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} e^{3} - 5 \, \sqrt {x e + d} B a b^{2} d e^{3} + 13 \, \sqrt {x e + d} A b^{3} d e^{3} + 9 \, \sqrt {x e + d} B a^{2} b e^{4} - 13 \, \sqrt {x e + d} A a b^{2} e^{4}}{4 \, {\left (b^{4} d^{4} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\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)/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.13, size = 928, normalized size = 2.67 \begin {gather*} \frac {\left (105 \left (e x +d \right )^{\frac {3}{2}} A \,b^{4} e^{2} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+105 \sqrt {\left (a e -b d \right ) b}\, A \,b^{3} e^{3} x^{3}-45 \left (e x +d \right )^{\frac {3}{2}} B a \,b^{3} e^{2} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-45 \sqrt {\left (a e -b d \right ) b}\, B a \,b^{2} e^{3} x^{3}-60 \left (e x +d \right )^{\frac {3}{2}} B \,b^{4} d e \,x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-60 \sqrt {\left (a e -b d \right ) b}\, B \,b^{3} d \,e^{2} x^{3}+210 \left (e x +d \right )^{\frac {3}{2}} A a \,b^{3} e^{2} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+175 \sqrt {\left (a e -b d \right ) b}\, A a \,b^{2} e^{3} x^{2}+140 \sqrt {\left (a e -b d \right ) b}\, A \,b^{3} d \,e^{2} x^{2}-90 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} b^{2} e^{2} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-75 \sqrt {\left (a e -b d \right ) b}\, B \,a^{2} b \,e^{3} x^{2}-120 \left (e x +d \right )^{\frac {3}{2}} B a \,b^{3} d e x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-160 \sqrt {\left (a e -b d \right ) b}\, B a \,b^{2} d \,e^{2} x^{2}-80 \sqrt {\left (a e -b d \right ) b}\, B \,b^{3} d^{2} e \,x^{2}+105 \left (e x +d \right )^{\frac {3}{2}} A \,a^{2} b^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+56 \sqrt {\left (a e -b d \right ) b}\, A \,a^{2} b \,e^{3} x +238 \sqrt {\left (a e -b d \right ) b}\, A a \,b^{2} d \,e^{2} x +21 \sqrt {\left (a e -b d \right ) b}\, A \,b^{3} d^{2} e x -45 \left (e x +d \right )^{\frac {3}{2}} B \,a^{3} b \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-24 \sqrt {\left (a e -b d \right ) b}\, B \,a^{3} e^{3} x -60 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} b^{2} d e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-134 \sqrt {\left (a e -b d \right ) b}\, B \,a^{2} b d \,e^{2} x -145 \sqrt {\left (a e -b d \right ) b}\, B a \,b^{2} d^{2} e x -12 \sqrt {\left (a e -b d \right ) b}\, B \,b^{3} d^{3} x -8 \sqrt {\left (a e -b d \right ) b}\, A \,a^{3} e^{3}+80 \sqrt {\left (a e -b d \right ) b}\, A \,a^{2} b d \,e^{2}+39 \sqrt {\left (a e -b d \right ) b}\, A a \,b^{2} d^{2} e -6 \sqrt {\left (a e -b d \right ) b}\, A \,b^{3} d^{3}-16 \sqrt {\left (a e -b d \right ) b}\, B \,a^{3} d \,e^{2}-83 \sqrt {\left (a e -b d \right ) b}\, B \,a^{2} b \,d^{2} e -6 \sqrt {\left (a e -b d \right ) b}\, B a \,b^{2} d^{3}\right ) \left (b x +a \right )}{12 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}} \end {gather*}

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)^(3/2),x)

[Out]

1/12*(105*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(3/2)*a^2*b^2*e^2-120*B*arctan((e*x+d)^(1/2)/(
(a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(3/2)*x*a*b^3*d*e-8*A*((a*e-b*d)*b)^(1/2)*a^3*e^3-6*A*((a*e-b*d)*b)^(1/2)*b^3*d^
3-16*B*((a*e-b*d)*b)^(1/2)*a^3*d*e^2-6*B*((a*e-b*d)*b)^(1/2)*a*b^2*d^3+80*A*((a*e-b*d)*b)^(1/2)*a^2*b*d*e^2+39
*A*((a*e-b*d)*b)^(1/2)*a*b^2*d^2*e-83*B*((a*e-b*d)*b)^(1/2)*a^2*b*d^2*e+105*A*((a*e-b*d)*b)^(1/2)*x^3*b^3*e^3-
24*B*((a*e-b*d)*b)^(1/2)*x*a^3*e^3-12*B*((a*e-b*d)*b)^(1/2)*x*b^3*d^3-45*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^
(1/2)*b)*(e*x+d)^(3/2)*x^2*a*b^3*e^2-60*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(3/2)*x^2*b^4*d*
e+210*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(3/2)*x*a*b^3*e^2-90*B*arctan((e*x+d)^(1/2)/((a*e-
b*d)*b)^(1/2)*b)*(e*x+d)^(3/2)*x*a^2*b^2*e^2-60*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(3/2)*a^
2*b^2*d*e-160*B*((a*e-b*d)*b)^(1/2)*x^2*a*b^2*d*e^2+238*A*((a*e-b*d)*b)^(1/2)*x*a*b^2*d*e^2-134*B*((a*e-b*d)*b
)^(1/2)*x*a^2*b*d*e^2-145*B*((a*e-b*d)*b)^(1/2)*x*a*b^2*d^2*e+21*A*((a*e-b*d)*b)^(1/2)*x*b^3*d^2*e+105*A*arcta
n((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(3/2)*x^2*b^4*e^2-75*B*((a*e-b*d)*b)^(1/2)*x^2*a^2*b*e^3-80*B*(
(a*e-b*d)*b)^(1/2)*x^2*b^3*d^2*e+56*A*((a*e-b*d)*b)^(1/2)*x*a^2*b*e^3-45*B*((a*e-b*d)*b)^(1/2)*x^3*a*b^2*e^3-6
0*B*((a*e-b*d)*b)^(1/2)*x^3*b^3*d*e^2-45*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(3/2)*a^3*b*e^2
+175*A*((a*e-b*d)*b)^(1/2)*x^2*a*b^2*e^3+140*A*((a*e-b*d)*b)^(1/2)*x^2*b^3*d*e^2)*(b*x+a)/(e*x+d)^(3/2)/((a*e-
b*d)*b)^(1/2)/(a*e-b*d)^4/((b*x+a)^2)^(3/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {B x + A}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \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)^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{{\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \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)^(3/2)),x)

[Out]

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

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sympy [F(-1)]  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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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