3.17.53 \(\int \frac {A+B x}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=414 \[ -\frac {7 b e (a+b x) (5 a B e-9 A b e+4 b B d)}{4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5}-\frac {7 e (a+b x) (5 a B e-9 A b e+4 b B d)}{12 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}+\frac {-5 a B e+9 A b e-4 b B d}{4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac {7 e (a+b x) (5 a B e-9 A b e+4 b B d)}{20 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac {A b-a B}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac {7 b^{3/2} e (a+b x) (5 a B e-9 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}} \]

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Rubi [A]  time = 0.37, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {7 b e (a+b x) (5 a B e-9 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)^5}-\frac {7 e (a+b x) (5 a B e-9 A b e+4 b B d)}{12 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}-\frac {5 a B e-9 A b e+4 b B d}{4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac {7 e (a+b x) (5 a B e-9 A b e+4 b B d)}{20 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac {A b-a B}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac {7 b^{3/2} e (a+b x) (5 a B e-9 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)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]

[Out]

-(4*b*B*d - 9*A*b*e + 5*a*B*e)/(4*b*(b*d - a*e)^2*(d + e*x)^(5/2)*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)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(4*b*B*d - 9*A*b*e + 5*a*B*e
)*(a + b*x))/(20*b*(b*d - a*e)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(4*b*B*d - 9*A*b*e + 5*
a*B*e)*(a + b*x))/(12*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*b*e*(4*b*B*d - 9*A*b*e
+ 5*a*B*e)*(a + b*x))/(4*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*b^(3/2)*e*(4*b*B*d -
9*A*b*e + 5*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(11/2)*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)^{7/2} \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)^{7/2}} \, 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)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((4 b B d-9 A b e+5 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)^{7/2}} \, dx}{4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (d+e x)^{5/2} \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)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 e (4 b B d-9 A b e+5 a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{7/2}} \, dx}{8 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (d+e x)^{5/2} \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)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{20 b (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 e (4 b B d-9 A b e+5 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 d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (d+e x)^{5/2} \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)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{20 b (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{12 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 b e (4 b B d-9 A b e+5 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)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (d+e x)^{5/2} \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)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{20 b (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{12 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 b e (4 b B d-9 A b e+5 a B e) (a+b x)}{4 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 b^2 e (4 b B d-9 A b e+5 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)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (d+e x)^{5/2} \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)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{20 b (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{12 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 b e (4 b B d-9 A b e+5 a B e) (a+b x)}{4 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 b^2 (4 b B d-9 A b e+5 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)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (d+e x)^{5/2} \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)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{20 b (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{12 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 b e (4 b B d-9 A b e+5 a B e) (a+b x)}{4 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 b^{3/2} e (4 b B d-9 A b e+5 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)^{11/2} \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 = 0.27 \begin {gather*} \frac {(a+b x) \left (\frac {e (a+b x)^2 (-5 a B e+9 A b e-4 b B d) \, _2F_1\left (-\frac {5}{2},2;-\frac {3}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+5 a B-5 A b\right )}{10 b \left ((a+b x)^2\right )^{3/2} (d+e x)^{5/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]

[Out]

((a + b*x)*(-5*A*b + 5*a*B + (e*(-4*b*B*d + 9*A*b*e - 5*a*B*e)*(a + b*x)^2*Hypergeometric2F1[-5/2, 2, -3/2, (b
*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^2))/(10*b*(b*d - a*e)*((a + b*x)^2)^(3/2)*(d + e*x)^(5/2))

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IntegrateAlgebraic [A]  time = 70.06, size = 687, normalized size = 1.66 \begin {gather*} \frac {(-a e-b e x) \left (-\frac {e \left (24 a^4 A e^5+40 a^4 B e^4 (d+e x)-24 a^4 B d e^4-72 a^3 A b e^4 (d+e x)-96 a^3 A b d e^4+96 a^3 b B d^2 e^3-88 a^3 b B d e^3 (d+e x)-280 a^3 b B e^3 (d+e x)^2+144 a^2 A b^2 d^2 e^3+216 a^2 A b^2 d e^3 (d+e x)+504 a^2 A b^2 e^3 (d+e x)^2-144 a^2 b^2 B d^3 e^2+24 a^2 b^2 B d^2 e^2 (d+e x)+336 a^2 b^2 B d e^2 (d+e x)^2-875 a^2 b^2 B e^2 (d+e x)^3-96 a A b^3 d^3 e^2-216 a A b^3 d^2 e^2 (d+e x)-1008 a A b^3 d e^2 (d+e x)^2+1575 a A b^3 e^2 (d+e x)^3+96 a b^3 B d^4 e+56 a b^3 B d^3 e (d+e x)+168 a b^3 B d^2 e (d+e x)^2+175 a b^3 B d e (d+e x)^3-525 a b^3 B e (d+e x)^4+24 A b^4 d^4 e+72 A b^4 d^3 e (d+e x)+504 A b^4 d^2 e (d+e x)^2-1575 A b^4 d e (d+e x)^3+945 A b^4 e (d+e x)^4-24 b^4 B d^5-32 b^4 B d^4 (d+e x)-224 b^4 B d^3 (d+e x)^2+700 b^4 B d^2 (d+e x)^3-420 b^4 B d (d+e x)^4\right )}{60 (d+e x)^{5/2} (b d-a e)^5 (-a e-b (d+e x)+b d)^2}-\frac {7 \left (5 a b^{3/2} B e^2-9 A b^{5/2} e^2+4 b^{5/2} 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)^5 \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)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]

[Out]

((-(a*e) - b*e*x)*(-1/60*(e*(-24*b^4*B*d^5 + 24*A*b^4*d^4*e + 96*a*b^3*B*d^4*e - 96*a*A*b^3*d^3*e^2 - 144*a^2*
b^2*B*d^3*e^2 + 144*a^2*A*b^2*d^2*e^3 + 96*a^3*b*B*d^2*e^3 - 96*a^3*A*b*d*e^4 - 24*a^4*B*d*e^4 + 24*a^4*A*e^5
- 32*b^4*B*d^4*(d + e*x) + 72*A*b^4*d^3*e*(d + e*x) + 56*a*b^3*B*d^3*e*(d + e*x) - 216*a*A*b^3*d^2*e^2*(d + e*
x) + 24*a^2*b^2*B*d^2*e^2*(d + e*x) + 216*a^2*A*b^2*d*e^3*(d + e*x) - 88*a^3*b*B*d*e^3*(d + e*x) - 72*a^3*A*b*
e^4*(d + e*x) + 40*a^4*B*e^4*(d + e*x) - 224*b^4*B*d^3*(d + e*x)^2 + 504*A*b^4*d^2*e*(d + e*x)^2 + 168*a*b^3*B
*d^2*e*(d + e*x)^2 - 1008*a*A*b^3*d*e^2*(d + e*x)^2 + 336*a^2*b^2*B*d*e^2*(d + e*x)^2 + 504*a^2*A*b^2*e^3*(d +
e*x)^2 - 280*a^3*b*B*e^3*(d + e*x)^2 + 700*b^4*B*d^2*(d + e*x)^3 - 1575*A*b^4*d*e*(d + e*x)^3 + 175*a*b^3*B*d
*e*(d + e*x)^3 + 1575*a*A*b^3*e^2*(d + e*x)^3 - 875*a^2*b^2*B*e^2*(d + e*x)^3 - 420*b^4*B*d*(d + e*x)^4 + 945*
A*b^4*e*(d + e*x)^4 - 525*a*b^3*B*e*(d + e*x)^4))/((b*d - a*e)^5*(d + e*x)^(5/2)*(b*d - a*e - b*(d + e*x))^2)
- (7*(4*b^(5/2)*B*d*e - 9*A*b^(5/2)*e^2 + 5*a*b^(3/2)*B*e^2)*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])
/(b*d - a*e)])/(4*(b*d - a*e)^5*Sqrt[-(b*d) + a*e])))/(e*Sqrt[(a*e + b*e*x)^2/e^2])

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fricas [B]  time = 0.49, size = 2675, normalized size = 6.46

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

[1/120*(105*(4*B*a^2*b^2*d^4*e + (5*B*a^3*b - 9*A*a^2*b^2)*d^3*e^2 + (4*B*b^4*d*e^4 + (5*B*a*b^3 - 9*A*b^4)*e^
5)*x^5 + (12*B*b^4*d^2*e^3 + (23*B*a*b^3 - 27*A*b^4)*d*e^4 + 2*(5*B*a^2*b^2 - 9*A*a*b^3)*e^5)*x^4 + (12*B*b^4*
d^3*e^2 + 3*(13*B*a*b^3 - 9*A*b^4)*d^2*e^3 + 2*(17*B*a^2*b^2 - 27*A*a*b^3)*d*e^4 + (5*B*a^3*b - 9*A*a^2*b^2)*e
^5)*x^3 + (4*B*b^4*d^4*e + (29*B*a*b^3 - 9*A*b^4)*d^3*e^2 + 6*(7*B*a^2*b^2 - 9*A*a*b^3)*d^2*e^3 + 3*(5*B*a^3*b
- 9*A*a^2*b^2)*d*e^4)*x^2 + (8*B*a*b^3*d^4*e + 2*(11*B*a^2*b^2 - 9*A*a*b^3)*d^3*e^2 + 3*(5*B*a^3*b - 9*A*a^2*
b^2)*d^2*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*(24*A*a^4*e^4 - 30*(B*a*b^3 + A*b^4)*d^4 - (659*B*a^2*b^2 - 255*A*a*b^3)*d^3*e - 16*(17*B*a^
3*b - 54*A*a^2*b^2)*d^2*e^2 + 8*(2*B*a^4 - 21*A*a^3*b)*d*e^3 - 105*(4*B*b^4*d*e^3 + (5*B*a*b^3 - 9*A*b^4)*e^4)
*x^4 - 35*(28*B*b^4*d^2*e^2 + (55*B*a*b^3 - 63*A*b^4)*d*e^3 + 5*(5*B*a^2*b^2 - 9*A*a*b^3)*e^4)*x^3 - 7*(92*B*b
^4*d^3*e + 9*(39*B*a*b^3 - 23*A*b^4)*d^2*e^2 + 3*(109*B*a^2*b^2 - 177*A*a*b^3)*d*e^3 + 8*(5*B*a^3*b - 9*A*a^2*
b^2)*e^4)*x^2 - (60*B*b^4*d^4 + (1183*B*a*b^3 - 135*A*b^4)*d^3*e + 3*(643*B*a^2*b^2 - 831*A*a*b^3)*d^2*e^2 + 7
2*(9*B*a^3*b - 17*A*a^2*b^2)*d*e^3 - 8*(5*B*a^4 - 9*A*a^3*b)*e^4)*x)*sqrt(e*x + d))/(a^2*b^5*d^8 - 5*a^3*b^4*d
^7*e + 10*a^4*b^3*d^6*e^2 - 10*a^5*b^2*d^5*e^3 + 5*a^6*b*d^4*e^4 - a^7*d^3*e^5 + (b^7*d^5*e^3 - 5*a*b^6*d^4*e^
4 + 10*a^2*b^5*d^3*e^5 - 10*a^3*b^4*d^2*e^6 + 5*a^4*b^3*d*e^7 - a^5*b^2*e^8)*x^5 + (3*b^7*d^6*e^2 - 13*a*b^6*d
^5*e^3 + 20*a^2*b^5*d^4*e^4 - 10*a^3*b^4*d^3*e^5 - 5*a^4*b^3*d^2*e^6 + 7*a^5*b^2*d*e^7 - 2*a^6*b*e^8)*x^4 + (3
*b^7*d^7*e - 9*a*b^6*d^6*e^2 + a^2*b^5*d^5*e^3 + 25*a^3*b^4*d^4*e^4 - 35*a^4*b^3*d^3*e^5 + 17*a^5*b^2*d^2*e^6
- a^6*b*d*e^7 - a^7*e^8)*x^3 + (b^7*d^8 + a*b^6*d^7*e - 17*a^2*b^5*d^6*e^2 + 35*a^3*b^4*d^5*e^3 - 25*a^4*b^3*d
^4*e^4 - a^5*b^2*d^3*e^5 + 9*a^6*b*d^2*e^6 - 3*a^7*d*e^7)*x^2 + (2*a*b^6*d^8 - 7*a^2*b^5*d^7*e + 5*a^3*b^4*d^6
*e^2 + 10*a^4*b^3*d^5*e^3 - 20*a^5*b^2*d^4*e^4 + 13*a^6*b*d^3*e^5 - 3*a^7*d^2*e^6)*x), 1/60*(105*(4*B*a^2*b^2*
d^4*e + (5*B*a^3*b - 9*A*a^2*b^2)*d^3*e^2 + (4*B*b^4*d*e^4 + (5*B*a*b^3 - 9*A*b^4)*e^5)*x^5 + (12*B*b^4*d^2*e^
3 + (23*B*a*b^3 - 27*A*b^4)*d*e^4 + 2*(5*B*a^2*b^2 - 9*A*a*b^3)*e^5)*x^4 + (12*B*b^4*d^3*e^2 + 3*(13*B*a*b^3 -
9*A*b^4)*d^2*e^3 + 2*(17*B*a^2*b^2 - 27*A*a*b^3)*d*e^4 + (5*B*a^3*b - 9*A*a^2*b^2)*e^5)*x^3 + (4*B*b^4*d^4*e
+ (29*B*a*b^3 - 9*A*b^4)*d^3*e^2 + 6*(7*B*a^2*b^2 - 9*A*a*b^3)*d^2*e^3 + 3*(5*B*a^3*b - 9*A*a^2*b^2)*d*e^4)*x^
2 + (8*B*a*b^3*d^4*e + 2*(11*B*a^2*b^2 - 9*A*a*b^3)*d^3*e^2 + 3*(5*B*a^3*b - 9*A*a^2*b^2)*d^2*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)) + (24*A*a^4*e^4 - 30*(B*a*b
^3 + A*b^4)*d^4 - (659*B*a^2*b^2 - 255*A*a*b^3)*d^3*e - 16*(17*B*a^3*b - 54*A*a^2*b^2)*d^2*e^2 + 8*(2*B*a^4 -
21*A*a^3*b)*d*e^3 - 105*(4*B*b^4*d*e^3 + (5*B*a*b^3 - 9*A*b^4)*e^4)*x^4 - 35*(28*B*b^4*d^2*e^2 + (55*B*a*b^3 -
63*A*b^4)*d*e^3 + 5*(5*B*a^2*b^2 - 9*A*a*b^3)*e^4)*x^3 - 7*(92*B*b^4*d^3*e + 9*(39*B*a*b^3 - 23*A*b^4)*d^2*e^
2 + 3*(109*B*a^2*b^2 - 177*A*a*b^3)*d*e^3 + 8*(5*B*a^3*b - 9*A*a^2*b^2)*e^4)*x^2 - (60*B*b^4*d^4 + (1183*B*a*b
^3 - 135*A*b^4)*d^3*e + 3*(643*B*a^2*b^2 - 831*A*a*b^3)*d^2*e^2 + 72*(9*B*a^3*b - 17*A*a^2*b^2)*d*e^3 - 8*(5*B
*a^4 - 9*A*a^3*b)*e^4)*x)*sqrt(e*x + d))/(a^2*b^5*d^8 - 5*a^3*b^4*d^7*e + 10*a^4*b^3*d^6*e^2 - 10*a^5*b^2*d^5*
e^3 + 5*a^6*b*d^4*e^4 - a^7*d^3*e^5 + (b^7*d^5*e^3 - 5*a*b^6*d^4*e^4 + 10*a^2*b^5*d^3*e^5 - 10*a^3*b^4*d^2*e^6
+ 5*a^4*b^3*d*e^7 - a^5*b^2*e^8)*x^5 + (3*b^7*d^6*e^2 - 13*a*b^6*d^5*e^3 + 20*a^2*b^5*d^4*e^4 - 10*a^3*b^4*d^
3*e^5 - 5*a^4*b^3*d^2*e^6 + 7*a^5*b^2*d*e^7 - 2*a^6*b*e^8)*x^4 + (3*b^7*d^7*e - 9*a*b^6*d^6*e^2 + a^2*b^5*d^5*
e^3 + 25*a^3*b^4*d^4*e^4 - 35*a^4*b^3*d^3*e^5 + 17*a^5*b^2*d^2*e^6 - a^6*b*d*e^7 - a^7*e^8)*x^3 + (b^7*d^8 + a
*b^6*d^7*e - 17*a^2*b^5*d^6*e^2 + 35*a^3*b^4*d^5*e^3 - 25*a^4*b^3*d^4*e^4 - a^5*b^2*d^3*e^5 + 9*a^6*b*d^2*e^6
- 3*a^7*d*e^7)*x^2 + (2*a*b^6*d^8 - 7*a^2*b^5*d^7*e + 5*a^3*b^4*d^6*e^2 + 10*a^4*b^3*d^5*e^3 - 20*a^5*b^2*d^4*
e^4 + 13*a^6*b*d^3*e^5 - 3*a^7*d^2*e^6)*x)]

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giac [B]  time = 0.57, size = 1011, normalized size = 2.44 \begin {gather*} -\frac {7 \, {\left (4 \, B b^{3} d e^{2} + 5 \, B a b^{2} e^{3} - 9 \, A b^{3} e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{5} d^{5} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 5 \, a b^{4} d^{4} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 10 \, a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 10 \, a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 5 \, a^{4} b d e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a^{5} e^{6} \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 {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d e^{2} - 4 \, \sqrt {x e + d} B b^{4} d^{2} e^{2} + 11 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} e^{3} - 15 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} e^{3} - 9 \, \sqrt {x e + d} B a b^{3} d e^{3} + 17 \, \sqrt {x e + d} A b^{4} d e^{3} + 13 \, \sqrt {x e + d} B a^{2} b^{2} e^{4} - 17 \, \sqrt {x e + d} A a b^{3} e^{4}}{4 \, {\left (b^{5} d^{5} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 5 \, a b^{4} d^{4} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 10 \, a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 10 \, a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 5 \, a^{4} b d e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a^{5} e^{6} \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}} - \frac {2 \, {\left (45 \, {\left (x e + d\right )}^{2} B b^{2} d e^{2} + 10 \, {\left (x e + d\right )} B b^{2} d^{2} e^{2} + 3 \, B b^{2} d^{3} e^{2} + 45 \, {\left (x e + d\right )}^{2} B a b e^{3} - 90 \, {\left (x e + d\right )}^{2} A b^{2} e^{3} - 5 \, {\left (x e + d\right )} B a b d e^{3} - 15 \, {\left (x e + d\right )} A b^{2} d e^{3} - 6 \, B a b d^{2} e^{3} - 3 \, A b^{2} d^{2} e^{3} - 5 \, {\left (x e + d\right )} B a^{2} e^{4} + 15 \, {\left (x e + d\right )} A a b e^{4} + 3 \, B a^{2} d e^{4} + 6 \, A a b d e^{4} - 3 \, A a^{2} e^{5}\right )}}{15 \, {\left (b^{5} d^{5} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 5 \, a b^{4} d^{4} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 10 \, a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 10 \, a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 5 \, a^{4} b d e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a^{5} e^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\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)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

-7/4*(4*B*b^3*d*e^2 + 5*B*a*b^2*e^3 - 9*A*b^3*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^5*e*sg
n((x*e + d)*b*e - b*d*e + a*e^2) - 5*a*b^4*d^4*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 10*a^2*b^3*d^3*e^3*sgn
((x*e + d)*b*e - b*d*e + a*e^2) - 10*a^3*b^2*d^2*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 5*a^4*b*d*e^5*sgn((x
*e + d)*b*e - b*d*e + a*e^2) - a^5*e^6*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(-b^2*d + a*b*e)) - 1/4*(4*(x*e
+ d)^(3/2)*B*b^4*d*e^2 - 4*sqrt(x*e + d)*B*b^4*d^2*e^2 + 11*(x*e + d)^(3/2)*B*a*b^3*e^3 - 15*(x*e + d)^(3/2)*
A*b^4*e^3 - 9*sqrt(x*e + d)*B*a*b^3*d*e^3 + 17*sqrt(x*e + d)*A*b^4*d*e^3 + 13*sqrt(x*e + d)*B*a^2*b^2*e^4 - 17
*sqrt(x*e + d)*A*a*b^3*e^4)/((b^5*d^5*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 5*a*b^4*d^4*e^2*sgn((x*e + d)*b*e
- b*d*e + a*e^2) + 10*a^2*b^3*d^3*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 10*a^3*b^2*d^2*e^4*sgn((x*e + d)*b
*e - b*d*e + a*e^2) + 5*a^4*b*d*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) - a^5*e^6*sgn((x*e + d)*b*e - b*d*e + a
*e^2))*((x*e + d)*b - b*d + a*e)^2) - 2/15*(45*(x*e + d)^2*B*b^2*d*e^2 + 10*(x*e + d)*B*b^2*d^2*e^2 + 3*B*b^2*
d^3*e^2 + 45*(x*e + d)^2*B*a*b*e^3 - 90*(x*e + d)^2*A*b^2*e^3 - 5*(x*e + d)*B*a*b*d*e^3 - 15*(x*e + d)*A*b^2*d
*e^3 - 6*B*a*b*d^2*e^3 - 3*A*b^2*d^2*e^3 - 5*(x*e + d)*B*a^2*e^4 + 15*(x*e + d)*A*a*b*e^4 + 3*B*a^2*d*e^4 + 6*
A*a*b*d*e^4 - 3*A*a^2*e^5)/((b^5*d^5*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 5*a*b^4*d^4*e^2*sgn((x*e + d)*b*e
- b*d*e + a*e^2) + 10*a^2*b^3*d^3*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 10*a^3*b^2*d^2*e^4*sgn((x*e + d)*b*
e - b*d*e + a*e^2) + 5*a^4*b*d*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) - a^5*e^6*sgn((x*e + d)*b*e - b*d*e + a*
e^2))*(x*e + d)^(5/2))

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maple [B]  time = 0.08, size = 1230, normalized size = 2.97

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

-1/60*(945*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*x^2*b^5*e^2-30*A*((a*e-b*d)*b)^(1/2)*b^
4*d^4+16*B*((a*e-b*d)*b)^(1/2)*a^4*d*e^3-840*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*x*a*b
^4*d*e+24*A*((a*e-b*d)*b)^(1/2)*a^4*e^4+945*A*((a*e-b*d)*b)^(1/2)*x^4*b^4*e^4+40*B*((a*e-b*d)*b)^(1/2)*x*a^4*e
^4-60*B*((a*e-b*d)*b)^(1/2)*x*b^4*d^4-30*B*((a*e-b*d)*b)^(1/2)*a*b^3*d^4-525*B*arctan((e*x+d)^(1/2)/((a*e-b*d)
*b)^(1/2)*b)*(e*x+d)^(5/2)*a^3*b^2*e^2+1575*A*((a*e-b*d)*b)^(1/2)*x^3*a*b^3*e^4+2205*A*((a*e-b*d)*b)^(1/2)*x^3
*b^4*d*e^3-875*B*((a*e-b*d)*b)^(1/2)*x^3*a^2*b^2*e^4-980*B*((a*e-b*d)*b)^(1/2)*x^3*b^4*d^2*e^2+504*A*((a*e-b*d
)*b)^(1/2)*x^2*a^2*b^2*e^4+1449*A*((a*e-b*d)*b)^(1/2)*x^2*b^4*d^2*e^2-280*B*((a*e-b*d)*b)^(1/2)*x^2*a^3*b*e^4-
644*B*((a*e-b*d)*b)^(1/2)*x^2*b^4*d^3*e-72*A*((a*e-b*d)*b)^(1/2)*x*a^3*b*e^4+135*A*((a*e-b*d)*b)^(1/2)*x*b^4*d
^3*e+945*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*a^2*b^3*e^2-525*B*((a*e-b*d)*b)^(1/2)*x^4
*a*b^3*e^4-420*B*((a*e-b*d)*b)^(1/2)*x^4*b^4*d*e^3-168*A*((a*e-b*d)*b)^(1/2)*a^3*b*d*e^3+864*A*((a*e-b*d)*b)^(
1/2)*a^2*b^2*d^2*e^2+255*A*((a*e-b*d)*b)^(1/2)*a*b^3*d^3*e-272*B*((a*e-b*d)*b)^(1/2)*a^3*b*d^2*e^2-659*B*((a*e
-b*d)*b)^(1/2)*a^2*b^2*d^3*e-2289*B*((a*e-b*d)*b)^(1/2)*x^2*a^2*b^2*d*e^3-2457*B*((a*e-b*d)*b)^(1/2)*x^2*a*b^3
*d^2*e^2+1224*A*((a*e-b*d)*b)^(1/2)*x*a^2*b^2*d*e^3+2493*A*((a*e-b*d)*b)^(1/2)*x*a*b^3*d^2*e^2-648*B*((a*e-b*d
)*b)^(1/2)*x*a^3*b*d*e^3-1929*B*((a*e-b*d)*b)^(1/2)*x*a^2*b^2*d^2*e^2-1183*B*((a*e-b*d)*b)^(1/2)*x*a*b^3*d^3*e
-525*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*x^2*a*b^4*e^2-420*B*arctan((e*x+d)^(1/2)/((a*
e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*x^2*b^5*d*e+1890*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*
x*a*b^4*e^2-1050*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*x*a^2*b^3*e^2-420*B*arctan((e*x+d
)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*a^2*b^3*d*e-1925*B*((a*e-b*d)*b)^(1/2)*x^3*a*b^3*d*e^3+3717*A*((a
*e-b*d)*b)^(1/2)*x^2*a*b^3*d*e^3)*(b*x+a)/(e*x+d)^(5/2)/((a*e-b*d)*b)^(1/2)/(a*e-b*d)^5/((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 {7}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(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)^(7/2)), 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 )}^{7/2}\,{\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)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2)),x)

[Out]

int((A + B*x)/((d + e*x)^(7/2)*(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)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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