3.20.20 \(\int \frac {a+b x}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=382 \[ -\frac {231 b^2 e^3 (a+b x)}{8 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^6}-\frac {77 b e^3 (a+b x)}{8 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}-\frac {231 e^3 (a+b x)}{40 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}-\frac {33 e^2}{8 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}+\frac {11 e}{12 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac {1}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac {231 b^{5/2} e^3 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{13/2}} \]

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Rubi [A]  time = 0.25, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {770, 21, 51, 63, 208} \begin {gather*} -\frac {231 b^2 e^3 (a+b x)}{8 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^6}-\frac {77 b e^3 (a+b x)}{8 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}-\frac {231 e^3 (a+b x)}{40 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}-\frac {33 e^2}{8 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}+\frac {231 b^{5/2} e^3 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{13/2}}+\frac {11 e}{12 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac {1}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)} \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)^(5/2)),x]

[Out]

(-33*e^2)/(8*(b*d - a*e)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(3*(b*d - a*e)*(a + b*x)^2*(d +
e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (11*e)/(12*(b*d - a*e)^2*(a + b*x)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2]) - (231*e^3*(a + b*x))/(40*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (77*b
*e^3*(a + b*x))/(8*(b*d - a*e)^5*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (231*b^2*e^3*(a + b*x))/(8*(
b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (231*b^(5/2)*e^3*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d
+ e*x])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 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 )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {a+b x}{\left (a b+b^2 x\right )^5 (d+e x)^{7/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{(a+b x)^4 (d+e x)^{7/2}} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (11 e \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{7/2}} \, dx}{6 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (33 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x)^2 (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 {33 e^2}{8 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (231 e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx}{16 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {33 e^2}{8 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (a+b x)}{40 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (231 e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{16 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {33 e^2}{8 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (a+b x)}{40 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {77 b e^3 (a+b x)}{8 (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (231 b e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {33 e^2}{8 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (a+b x)}{40 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {77 b e^3 (a+b x)}{8 (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 b^2 e^3 (a+b x)}{8 (b d-a e)^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (231 b^2 e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {33 e^2}{8 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (a+b x)}{40 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {77 b e^3 (a+b x)}{8 (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 b^2 e^3 (a+b x)}{8 (b d-a e)^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (231 b^2 e^2 \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {33 e^2}{8 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 e}{12 (b d-a e)^2 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (a+b x)}{40 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {77 b e^3 (a+b x)}{8 (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 b^2 e^3 (a+b x)}{8 (b d-a e)^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {231 b^{5/2} e^3 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 68, normalized size = 0.18 \begin {gather*} -\frac {2 e^3 (a+b x) \, _2F_1\left (-\frac {5}{2},4;-\frac {3}{2};-\frac {b (d+e x)}{a e-b d}\right )}{5 \sqrt {(a+b x)^2} (d+e x)^{5/2} (a e-b d)^4} \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)^(5/2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 65.86, size = 448, normalized size = 1.17 \begin {gather*} \frac {(-a e-b e x) \left (-\frac {e^3 \left (48 a^5 e^5-176 a^4 b e^4 (d+e x)-240 a^4 b d e^4+480 a^3 b^2 d^2 e^3+1584 a^3 b^2 e^3 (d+e x)^2+704 a^3 b^2 d e^3 (d+e x)-480 a^2 b^3 d^3 e^2-1056 a^2 b^3 d^2 e^2 (d+e x)+7623 a^2 b^3 e^2 (d+e x)^3-4752 a^2 b^3 d e^2 (d+e x)^2+240 a b^4 d^4 e+704 a b^4 d^3 e (d+e x)+4752 a b^4 d^2 e (d+e x)^2+9240 a b^4 e (d+e x)^4-15246 a b^4 d e (d+e x)^3-48 b^5 d^5-176 b^5 d^4 (d+e x)-1584 b^5 d^3 (d+e x)^2+7623 b^5 d^2 (d+e x)^3+3465 b^5 (d+e x)^5-9240 b^5 d (d+e x)^4\right )}{120 (d+e x)^{5/2} (b d-a e)^6 (-a e-b (d+e x)+b d)^3}-\frac {231 b^{5/2} e^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{8 (b d-a e)^6 \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)^(5/2)),x]

[Out]

((-(a*e) - b*e*x)*(-1/120*(e^3*(-48*b^5*d^5 + 240*a*b^4*d^4*e - 480*a^2*b^3*d^3*e^2 + 480*a^3*b^2*d^2*e^3 - 24
0*a^4*b*d*e^4 + 48*a^5*e^5 - 176*b^5*d^4*(d + e*x) + 704*a*b^4*d^3*e*(d + e*x) - 1056*a^2*b^3*d^2*e^2*(d + e*x
) + 704*a^3*b^2*d*e^3*(d + e*x) - 176*a^4*b*e^4*(d + e*x) - 1584*b^5*d^3*(d + e*x)^2 + 4752*a*b^4*d^2*e*(d + e
*x)^2 - 4752*a^2*b^3*d*e^2*(d + e*x)^2 + 1584*a^3*b^2*e^3*(d + e*x)^2 + 7623*b^5*d^2*(d + e*x)^3 - 15246*a*b^4
*d*e*(d + e*x)^3 + 7623*a^2*b^3*e^2*(d + e*x)^3 - 9240*b^5*d*(d + e*x)^4 + 9240*a*b^4*e*(d + e*x)^4 + 3465*b^5
*(d + e*x)^5))/((b*d - a*e)^6*(d + e*x)^(5/2)*(b*d - a*e - b*(d + e*x))^3) - (231*b^(5/2)*e^3*ArcTan[(Sqrt[b]*
Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(8*(b*d - a*e)^6*Sqrt[-(b*d) + a*e])))/(e*Sqrt[(a*e + b*e*x)^2
/e^2])

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fricas [B]  time = 0.50, size = 2550, normalized size = 6.68

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)^(5/2),x, algorithm="fricas")

[Out]

[1/240*(3465*(b^5*e^6*x^6 + a^3*b^2*d^3*e^3 + 3*(b^5*d*e^5 + a*b^4*e^6)*x^5 + 3*(b^5*d^2*e^4 + 3*a*b^4*d*e^5 +
a^2*b^3*e^6)*x^4 + (b^5*d^3*e^3 + 9*a*b^4*d^2*e^4 + 9*a^2*b^3*d*e^5 + a^3*b^2*e^6)*x^3 + 3*(a*b^4*d^3*e^3 + 3
*a^2*b^3*d^2*e^4 + a^3*b^2*d*e^5)*x^2 + 3*(a^2*b^3*d^3*e^3 + a^3*b^2*d^2*e^4)*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*(3465*b^5*e^5*x^5 + 40*b^5*d
^5 - 310*a*b^4*d^4*e + 1335*a^2*b^3*d^3*e^2 + 2768*a^3*b^2*d^2*e^3 - 416*a^4*b*d*e^4 + 48*a^5*e^5 + 1155*(7*b^
5*d*e^4 + 8*a*b^4*e^5)*x^4 + 231*(23*b^5*d^2*e^3 + 94*a*b^4*d*e^4 + 33*a^2*b^3*e^5)*x^3 + 99*(5*b^5*d^3*e^2 +
146*a*b^4*d^2*e^3 + 183*a^2*b^3*d*e^4 + 16*a^3*b^2*e^5)*x^2 - 11*(10*b^5*d^4*e - 130*a*b^4*d^3*e^2 - 1119*a^2*
b^3*d^2*e^3 - 352*a^3*b^2*d*e^4 + 16*a^4*b*e^5)*x)*sqrt(e*x + d))/(a^3*b^6*d^9 - 6*a^4*b^5*d^8*e + 15*a^5*b^4*
d^7*e^2 - 20*a^6*b^3*d^6*e^3 + 15*a^7*b^2*d^5*e^4 - 6*a^8*b*d^4*e^5 + a^9*d^3*e^6 + (b^9*d^6*e^3 - 6*a*b^8*d^5
*e^4 + 15*a^2*b^7*d^4*e^5 - 20*a^3*b^6*d^3*e^6 + 15*a^4*b^5*d^2*e^7 - 6*a^5*b^4*d*e^8 + a^6*b^3*e^9)*x^6 + 3*(
b^9*d^7*e^2 - 5*a*b^8*d^6*e^3 + 9*a^2*b^7*d^5*e^4 - 5*a^3*b^6*d^4*e^5 - 5*a^4*b^5*d^3*e^6 + 9*a^5*b^4*d^2*e^7
- 5*a^6*b^3*d*e^8 + a^7*b^2*e^9)*x^5 + 3*(b^9*d^8*e - 3*a*b^8*d^7*e^2 - 2*a^2*b^7*d^6*e^3 + 19*a^3*b^6*d^5*e^4
- 30*a^4*b^5*d^4*e^5 + 19*a^5*b^4*d^3*e^6 - 2*a^6*b^3*d^2*e^7 - 3*a^7*b^2*d*e^8 + a^8*b*e^9)*x^4 + (b^9*d^9 +
3*a*b^8*d^8*e - 30*a^2*b^7*d^7*e^2 + 62*a^3*b^6*d^6*e^3 - 36*a^4*b^5*d^5*e^4 - 36*a^5*b^4*d^4*e^5 + 62*a^6*b^
3*d^3*e^6 - 30*a^7*b^2*d^2*e^7 + 3*a^8*b*d*e^8 + a^9*e^9)*x^3 + 3*(a*b^8*d^9 - 3*a^2*b^7*d^8*e - 2*a^3*b^6*d^7
*e^2 + 19*a^4*b^5*d^6*e^3 - 30*a^5*b^4*d^5*e^4 + 19*a^6*b^3*d^4*e^5 - 2*a^7*b^2*d^3*e^6 - 3*a^8*b*d^2*e^7 + a^
9*d*e^8)*x^2 + 3*(a^2*b^7*d^9 - 5*a^3*b^6*d^8*e + 9*a^4*b^5*d^7*e^2 - 5*a^5*b^4*d^6*e^3 - 5*a^6*b^3*d^5*e^4 +
9*a^7*b^2*d^4*e^5 - 5*a^8*b*d^3*e^6 + a^9*d^2*e^7)*x), 1/120*(3465*(b^5*e^6*x^6 + a^3*b^2*d^3*e^3 + 3*(b^5*d*e
^5 + a*b^4*e^6)*x^5 + 3*(b^5*d^2*e^4 + 3*a*b^4*d*e^5 + a^2*b^3*e^6)*x^4 + (b^5*d^3*e^3 + 9*a*b^4*d^2*e^4 + 9*a
^2*b^3*d*e^5 + a^3*b^2*e^6)*x^3 + 3*(a*b^4*d^3*e^3 + 3*a^2*b^3*d^2*e^4 + a^3*b^2*d*e^5)*x^2 + 3*(a^2*b^3*d^3*e
^3 + a^3*b^2*d^2*e^4)*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)) - (3465*b^5*e^5*x^5 + 40*b^5*d^5 - 310*a*b^4*d^4*e + 1335*a^2*b^3*d^3*e^2 + 2768*a^3*b^2*d^2*e^3 - 416*a
^4*b*d*e^4 + 48*a^5*e^5 + 1155*(7*b^5*d*e^4 + 8*a*b^4*e^5)*x^4 + 231*(23*b^5*d^2*e^3 + 94*a*b^4*d*e^4 + 33*a^2
*b^3*e^5)*x^3 + 99*(5*b^5*d^3*e^2 + 146*a*b^4*d^2*e^3 + 183*a^2*b^3*d*e^4 + 16*a^3*b^2*e^5)*x^2 - 11*(10*b^5*d
^4*e - 130*a*b^4*d^3*e^2 - 1119*a^2*b^3*d^2*e^3 - 352*a^3*b^2*d*e^4 + 16*a^4*b*e^5)*x)*sqrt(e*x + d))/(a^3*b^6
*d^9 - 6*a^4*b^5*d^8*e + 15*a^5*b^4*d^7*e^2 - 20*a^6*b^3*d^6*e^3 + 15*a^7*b^2*d^5*e^4 - 6*a^8*b*d^4*e^5 + a^9*
d^3*e^6 + (b^9*d^6*e^3 - 6*a*b^8*d^5*e^4 + 15*a^2*b^7*d^4*e^5 - 20*a^3*b^6*d^3*e^6 + 15*a^4*b^5*d^2*e^7 - 6*a^
5*b^4*d*e^8 + a^6*b^3*e^9)*x^6 + 3*(b^9*d^7*e^2 - 5*a*b^8*d^6*e^3 + 9*a^2*b^7*d^5*e^4 - 5*a^3*b^6*d^4*e^5 - 5*
a^4*b^5*d^3*e^6 + 9*a^5*b^4*d^2*e^7 - 5*a^6*b^3*d*e^8 + a^7*b^2*e^9)*x^5 + 3*(b^9*d^8*e - 3*a*b^8*d^7*e^2 - 2*
a^2*b^7*d^6*e^3 + 19*a^3*b^6*d^5*e^4 - 30*a^4*b^5*d^4*e^5 + 19*a^5*b^4*d^3*e^6 - 2*a^6*b^3*d^2*e^7 - 3*a^7*b^2
*d*e^8 + a^8*b*e^9)*x^4 + (b^9*d^9 + 3*a*b^8*d^8*e - 30*a^2*b^7*d^7*e^2 + 62*a^3*b^6*d^6*e^3 - 36*a^4*b^5*d^5*
e^4 - 36*a^5*b^4*d^4*e^5 + 62*a^6*b^3*d^3*e^6 - 30*a^7*b^2*d^2*e^7 + 3*a^8*b*d*e^8 + a^9*e^9)*x^3 + 3*(a*b^8*d
^9 - 3*a^2*b^7*d^8*e - 2*a^3*b^6*d^7*e^2 + 19*a^4*b^5*d^6*e^3 - 30*a^5*b^4*d^5*e^4 + 19*a^6*b^3*d^4*e^5 - 2*a^
7*b^2*d^3*e^6 - 3*a^8*b*d^2*e^7 + a^9*d*e^8)*x^2 + 3*(a^2*b^7*d^9 - 5*a^3*b^6*d^8*e + 9*a^4*b^5*d^7*e^2 - 5*a^
5*b^4*d^6*e^3 - 5*a^6*b^3*d^5*e^4 + 9*a^7*b^2*d^4*e^5 - 5*a^8*b*d^3*e^6 + a^9*d^2*e^7)*x)]

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giac [B]  time = 0.55, size = 932, normalized size = 2.44 \begin {gather*} -\frac {231 \, b^{3} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{3}}{8 \, {\left (b^{6} d^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{6} 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 {2 \, {\left (150 \, {\left (x e + d\right )}^{2} b^{2} e^{3} + 20 \, {\left (x e + d\right )} b^{2} d e^{3} + 3 \, b^{2} d^{2} e^{3} - 20 \, {\left (x e + d\right )} a b e^{4} - 6 \, a b d e^{4} + 3 \, a^{2} e^{5}\right )}}{15 \, {\left (b^{6} d^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{6} 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}}} - \frac {213 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} e^{3} - 472 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d e^{3} + 267 \, \sqrt {x e + d} b^{5} d^{2} e^{3} + 472 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} e^{4} - 534 \, \sqrt {x e + d} a b^{4} d e^{4} + 267 \, \sqrt {x e + d} a^{2} b^{3} e^{5}}{24 \, {\left (b^{6} d^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{6} 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 )}^{3}} \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)^(5/2),x, algorithm="giac")

[Out]

-231/8*b^3*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((b^6*d^6*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a
*b^5*d^5*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^2*b^4*d^4*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 20*a^3
*b^3*d^3*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^4*b^2*d^2*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a^
5*b*d*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^6*e^6*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(-b^2*d + a*b*e
)) - 2/15*(150*(x*e + d)^2*b^2*e^3 + 20*(x*e + d)*b^2*d*e^3 + 3*b^2*d^2*e^3 - 20*(x*e + d)*a*b*e^4 - 6*a*b*d*e
^4 + 3*a^2*e^5)/((b^6*d^6*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a*b^5*d^5*e*sgn((x*e + d)*b*e - b*d*e + a*e^2
) + 15*a^2*b^4*d^4*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 20*a^3*b^3*d^3*e^3*sgn((x*e + d)*b*e - b*d*e + a*e
^2) + 15*a^4*b^2*d^2*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a^5*b*d*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2)
+ a^6*e^6*sgn((x*e + d)*b*e - b*d*e + a*e^2))*(x*e + d)^(5/2)) - 1/24*(213*(x*e + d)^(5/2)*b^5*e^3 - 472*(x*e
+ d)^(3/2)*b^5*d*e^3 + 267*sqrt(x*e + d)*b^5*d^2*e^3 + 472*(x*e + d)^(3/2)*a*b^4*e^4 - 534*sqrt(x*e + d)*a*b^
4*d*e^4 + 267*sqrt(x*e + d)*a^2*b^3*e^5)/((b^6*d^6*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a*b^5*d^5*e*sgn((x*e
+ d)*b*e - b*d*e + a*e^2) + 15*a^2*b^4*d^4*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 20*a^3*b^3*d^3*e^3*sgn((x
*e + d)*b*e - b*d*e + a*e^2) + 15*a^4*b^2*d^2*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a^5*b*d*e^5*sgn((x*e
+ d)*b*e - b*d*e + a*e^2) + a^6*e^6*sgn((x*e + d)*b*e - b*d*e + a*e^2))*((x*e + d)*b - b*d + a*e)^3)

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maple [B]  time = 0.08, size = 722, normalized size = 1.89 \begin {gather*} -\frac {\left (3465 \sqrt {\left (a e -b d \right ) b}\, b^{5} e^{5} x^{5}+9240 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} e^{5} x^{4}+8085 \sqrt {\left (a e -b d \right ) b}\, b^{5} d \,e^{4} x^{4}+7623 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{3} e^{5} x^{3}+21714 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} d \,e^{4} x^{3}+3465 \left (e x +d \right )^{\frac {5}{2}} b^{6} e^{3} x^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+5313 \sqrt {\left (a e -b d \right ) b}\, b^{5} d^{2} e^{3} x^{3}+1584 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{2} e^{5} x^{2}+18117 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{3} d \,e^{4} x^{2}+10395 \left (e x +d \right )^{\frac {5}{2}} a \,b^{5} e^{3} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+14454 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} d^{2} e^{3} x^{2}+495 \sqrt {\left (a e -b d \right ) b}\, b^{5} d^{3} e^{2} x^{2}-176 \sqrt {\left (a e -b d \right ) b}\, a^{4} b \,e^{5} x +3872 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{2} d \,e^{4} x +10395 \left (e x +d \right )^{\frac {5}{2}} a^{2} b^{4} e^{3} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+12309 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{3} d^{2} e^{3} x +1430 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} d^{3} e^{2} x -110 \sqrt {\left (a e -b d \right ) b}\, b^{5} d^{4} e x +48 \sqrt {\left (a e -b d \right ) b}\, a^{5} e^{5}-416 \sqrt {\left (a e -b d \right ) b}\, a^{4} b d \,e^{4}+3465 \left (e x +d \right )^{\frac {5}{2}} a^{3} b^{3} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+2768 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{2} d^{2} e^{3}+1335 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{3} d^{3} e^{2}-310 \sqrt {\left (a e -b d \right ) b}\, a \,b^{4} d^{4} e +40 \sqrt {\left (a e -b d \right ) b}\, b^{5} d^{5}\right ) \left (b x +a \right )^{2}}{120 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}} \end {gather*}

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)^(5/2),x)

[Out]

-1/120*(12309*((a*e-b*d)*b)^(1/2)*x*a^2*b^3*d^2*e^3-416*((a*e-b*d)*b)^(1/2)*a^4*b*d*e^4+2768*((a*e-b*d)*b)^(1/
2)*a^3*b^2*d^2*e^3+40*((a*e-b*d)*b)^(1/2)*b^5*d^5+9240*((a*e-b*d)*b)^(1/2)*x^4*a*b^4*e^5+7623*((a*e-b*d)*b)^(1
/2)*x^3*a^2*b^3*e^5-176*((a*e-b*d)*b)^(1/2)*x*a^4*b*e^5-110*((a*e-b*d)*b)^(1/2)*x*b^5*d^4*e+5313*((a*e-b*d)*b)
^(1/2)*x^3*b^5*d^2*e^3+1584*((a*e-b*d)*b)^(1/2)*x^2*a^3*b^2*e^5+3465*(e*x+d)^(5/2)*arctan((e*x+d)^(1/2)/((a*e-
b*d)*b)^(1/2)*b)*x^3*b^6*e^3+3465*(e*x+d)^(5/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^3*b^3*e^3+495*((
a*e-b*d)*b)^(1/2)*x^2*b^5*d^3*e^2+8085*((a*e-b*d)*b)^(1/2)*x^4*b^5*d*e^4+10395*(e*x+d)^(5/2)*arctan((e*x+d)^(1
/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a*b^5*e^3+10395*(e*x+d)^(5/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^2
*b^4*e^3+1430*((a*e-b*d)*b)^(1/2)*x*a*b^4*d^3*e^2+48*((a*e-b*d)*b)^(1/2)*a^5*e^5+3465*((a*e-b*d)*b)^(1/2)*x^5*
b^5*e^5+18117*((a*e-b*d)*b)^(1/2)*x^2*a^2*b^3*d*e^4+14454*((a*e-b*d)*b)^(1/2)*x^2*a*b^4*d^2*e^3+3872*((a*e-b*d
)*b)^(1/2)*x*a^3*b^2*d*e^4+21714*((a*e-b*d)*b)^(1/2)*x^3*a*b^4*d*e^4+1335*((a*e-b*d)*b)^(1/2)*a^2*b^3*d^3*e^2-
310*((a*e-b*d)*b)^(1/2)*a*b^4*d^4*e)*(b*x+a)^2/(e*x+d)^(5/2)/((a*e-b*d)*b)^(1/2)/(a*e-b*d)^6/((b*x+a)^2)^(5/2)

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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 {5}{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)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(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 )}^{5/2}} \,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)^(5/2)),x)

[Out]

int((a + b*x)/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), 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)**(5/2),x)

[Out]

Timed out

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