3.19.65 \(\int \frac {a+b x}{(d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=206 \[ \frac {315 e^4}{64 \sqrt {d+e x} (b d-a e)^5}-\frac {315 \sqrt {b} e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{11/2}}+\frac {105 e^3}{64 (a+b x) \sqrt {d+e x} (b d-a e)^4}-\frac {21 e^2}{32 (a+b x)^2 \sqrt {d+e x} (b d-a e)^3}+\frac {3 e}{8 (a+b x)^3 \sqrt {d+e x} (b d-a e)^2}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {27, 51, 63, 208} \begin {gather*} \frac {315 e^4}{64 \sqrt {d+e x} (b d-a e)^5}+\frac {105 e^3}{64 (a+b x) \sqrt {d+e x} (b d-a e)^4}-\frac {21 e^2}{32 (a+b x)^2 \sqrt {d+e x} (b d-a e)^3}-\frac {315 \sqrt {b} e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{11/2}}+\frac {3 e}{8 (a+b x)^3 \sqrt {d+e x} (b d-a e)^2}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(315*e^4)/(64*(b*d - a*e)^5*Sqrt[d + e*x]) - 1/(4*(b*d - a*e)*(a + b*x)^4*Sqrt[d + e*x]) + (3*e)/(8*(b*d - a*e
)^2*(a + b*x)^3*Sqrt[d + e*x]) - (21*e^2)/(32*(b*d - a*e)^3*(a + b*x)^2*Sqrt[d + e*x]) + (105*e^3)/(64*(b*d -
a*e)^4*(a + b*x)*Sqrt[d + e*x]) - (315*Sqrt[b]*e^4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*(b*d
- a*e)^(11/2))

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 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]

Rubi steps

\begin {align*} \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^5 (d+e x)^{3/2}} \, dx\\ &=-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}-\frac {(9 e) \int \frac {1}{(a+b x)^4 (d+e x)^{3/2}} \, dx}{8 (b d-a e)}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt {d+e x}}+\frac {\left (21 e^2\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{16 (b d-a e)^2}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt {d+e x}}-\frac {\left (105 e^3\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{64 (b d-a e)^3}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt {d+e x}}+\frac {105 e^3}{64 (b d-a e)^4 (a+b x) \sqrt {d+e x}}+\frac {\left (315 e^4\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{128 (b d-a e)^4}\\ &=\frac {315 e^4}{64 (b d-a e)^5 \sqrt {d+e x}}-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt {d+e x}}+\frac {105 e^3}{64 (b d-a e)^4 (a+b x) \sqrt {d+e x}}+\frac {\left (315 b e^4\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{128 (b d-a e)^5}\\ &=\frac {315 e^4}{64 (b d-a e)^5 \sqrt {d+e x}}-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt {d+e x}}+\frac {105 e^3}{64 (b d-a e)^4 (a+b x) \sqrt {d+e x}}+\frac {\left (315 b e^3\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 (b d-a e)^5}\\ &=\frac {315 e^4}{64 (b d-a e)^5 \sqrt {d+e x}}-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt {d+e x}}+\frac {105 e^3}{64 (b d-a e)^4 (a+b x) \sqrt {d+e x}}-\frac {315 \sqrt {b} e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{11/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 50, normalized size = 0.24 \begin {gather*} -\frac {2 e^4 \, _2F_1\left (-\frac {1}{2},5;\frac {1}{2};-\frac {b (d+e x)}{a e-b d}\right )}{\sqrt {d+e x} (a e-b d)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e^4*Hypergeometric2F1[-1/2, 5, 1/2, -((b*(d + e*x))/(-(b*d) + a*e))])/((-(b*d) + a*e)^5*Sqrt[d + e*x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.41, size = 304, normalized size = 1.48 \begin {gather*} \frac {e^4 \left (128 a^4 e^4+837 a^3 b e^3 (d+e x)-512 a^3 b d e^3+768 a^2 b^2 d^2 e^2+1533 a^2 b^2 e^2 (d+e x)^2-2511 a^2 b^2 d e^2 (d+e x)-512 a b^3 d^3 e+2511 a b^3 d^2 e (d+e x)+1155 a b^3 e (d+e x)^3-3066 a b^3 d e (d+e x)^2+128 b^4 d^4-837 b^4 d^3 (d+e x)+1533 b^4 d^2 (d+e x)^2+315 b^4 (d+e x)^4-1155 b^4 d (d+e x)^3\right )}{64 \sqrt {d+e x} (b d-a e)^5 (-a e-b (d+e x)+b d)^4}+\frac {315 \sqrt {b} e^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 (a e-b d)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*(128*b^4*d^4 - 512*a*b^3*d^3*e + 768*a^2*b^2*d^2*e^2 - 512*a^3*b*d*e^3 + 128*a^4*e^4 - 837*b^4*d^3*(d + e
*x) + 2511*a*b^3*d^2*e*(d + e*x) - 2511*a^2*b^2*d*e^2*(d + e*x) + 837*a^3*b*e^3*(d + e*x) + 1533*b^4*d^2*(d +
e*x)^2 - 3066*a*b^3*d*e*(d + e*x)^2 + 1533*a^2*b^2*e^2*(d + e*x)^2 - 1155*b^4*d*(d + e*x)^3 + 1155*a*b^3*e*(d
+ e*x)^3 + 315*b^4*(d + e*x)^4))/(64*(b*d - a*e)^5*Sqrt[d + e*x]*(b*d - a*e - b*(d + e*x))^4) + (315*Sqrt[b]*e
^4*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(64*(-(b*d) + a*e)^(11/2))

________________________________________________________________________________________

fricas [B]  time = 0.48, size = 1734, normalized size = 8.42

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/128*(315*(b^4*e^5*x^5 + a^4*d*e^4 + (b^4*d*e^4 + 4*a*b^3*e^5)*x^4 + 2*(2*a*b^3*d*e^4 + 3*a^2*b^2*e^5)*x^3
+ 2*(3*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*x^2 + (4*a^3*b*d*e^4 + a^4*e^5)*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*(315*b^4*e^4*x^4 - 16*b^4*d^4 + 88*a*b
^3*d^3*e - 210*a^2*b^2*d^2*e^2 + 325*a^3*b*d*e^3 + 128*a^4*e^4 + 105*(b^4*d*e^3 + 11*a*b^3*e^4)*x^3 - 21*(2*b^
4*d^2*e^2 - 19*a*b^3*d*e^3 - 73*a^2*b^2*e^4)*x^2 + 3*(8*b^4*d^3*e - 52*a*b^3*d^2*e^2 + 185*a^2*b^2*d*e^3 + 279
*a^3*b*e^4)*x)*sqrt(e*x + d))/(a^4*b^5*d^6 - 5*a^5*b^4*d^5*e + 10*a^6*b^3*d^4*e^2 - 10*a^7*b^2*d^3*e^3 + 5*a^8
*b*d^2*e^4 - a^9*d*e^5 + (b^9*d^5*e - 5*a*b^8*d^4*e^2 + 10*a^2*b^7*d^3*e^3 - 10*a^3*b^6*d^2*e^4 + 5*a^4*b^5*d*
e^5 - a^5*b^4*e^6)*x^5 + (b^9*d^6 - a*b^8*d^5*e - 10*a^2*b^7*d^4*e^2 + 30*a^3*b^6*d^3*e^3 - 35*a^4*b^5*d^2*e^4
+ 19*a^5*b^4*d*e^5 - 4*a^6*b^3*e^6)*x^4 + 2*(2*a*b^8*d^6 - 7*a^2*b^7*d^5*e + 5*a^3*b^6*d^4*e^2 + 10*a^4*b^5*d
^3*e^3 - 20*a^5*b^4*d^2*e^4 + 13*a^6*b^3*d*e^5 - 3*a^7*b^2*e^6)*x^3 + 2*(3*a^2*b^7*d^6 - 13*a^3*b^6*d^5*e + 20
*a^4*b^5*d^4*e^2 - 10*a^5*b^4*d^3*e^3 - 5*a^6*b^3*d^2*e^4 + 7*a^7*b^2*d*e^5 - 2*a^8*b*e^6)*x^2 + (4*a^3*b^6*d^
6 - 19*a^4*b^5*d^5*e + 35*a^5*b^4*d^4*e^2 - 30*a^6*b^3*d^3*e^3 + 10*a^7*b^2*d^2*e^4 + a^8*b*d*e^5 - a^9*e^6)*x
), -1/64*(315*(b^4*e^5*x^5 + a^4*d*e^4 + (b^4*d*e^4 + 4*a*b^3*e^5)*x^4 + 2*(2*a*b^3*d*e^4 + 3*a^2*b^2*e^5)*x^3
+ 2*(3*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*x^2 + (4*a^3*b*d*e^4 + a^4*e^5)*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)) - (315*b^4*e^4*x^4 - 16*b^4*d^4 + 88*a*b^3*d^3*e - 210*a^
2*b^2*d^2*e^2 + 325*a^3*b*d*e^3 + 128*a^4*e^4 + 105*(b^4*d*e^3 + 11*a*b^3*e^4)*x^3 - 21*(2*b^4*d^2*e^2 - 19*a*
b^3*d*e^3 - 73*a^2*b^2*e^4)*x^2 + 3*(8*b^4*d^3*e - 52*a*b^3*d^2*e^2 + 185*a^2*b^2*d*e^3 + 279*a^3*b*e^4)*x)*sq
rt(e*x + d))/(a^4*b^5*d^6 - 5*a^5*b^4*d^5*e + 10*a^6*b^3*d^4*e^2 - 10*a^7*b^2*d^3*e^3 + 5*a^8*b*d^2*e^4 - a^9*
d*e^5 + (b^9*d^5*e - 5*a*b^8*d^4*e^2 + 10*a^2*b^7*d^3*e^3 - 10*a^3*b^6*d^2*e^4 + 5*a^4*b^5*d*e^5 - a^5*b^4*e^6
)*x^5 + (b^9*d^6 - a*b^8*d^5*e - 10*a^2*b^7*d^4*e^2 + 30*a^3*b^6*d^3*e^3 - 35*a^4*b^5*d^2*e^4 + 19*a^5*b^4*d*e
^5 - 4*a^6*b^3*e^6)*x^4 + 2*(2*a*b^8*d^6 - 7*a^2*b^7*d^5*e + 5*a^3*b^6*d^4*e^2 + 10*a^4*b^5*d^3*e^3 - 20*a^5*b
^4*d^2*e^4 + 13*a^6*b^3*d*e^5 - 3*a^7*b^2*e^6)*x^3 + 2*(3*a^2*b^7*d^6 - 13*a^3*b^6*d^5*e + 20*a^4*b^5*d^4*e^2
- 10*a^5*b^4*d^3*e^3 - 5*a^6*b^3*d^2*e^4 + 7*a^7*b^2*d*e^5 - 2*a^8*b*e^6)*x^2 + (4*a^3*b^6*d^6 - 19*a^4*b^5*d^
5*e + 35*a^5*b^4*d^4*e^2 - 30*a^6*b^3*d^3*e^3 + 10*a^7*b^2*d^2*e^4 + a^8*b*d*e^5 - a^9*e^6)*x)]

________________________________________________________________________________________

giac [B]  time = 0.25, size = 440, normalized size = 2.14 \begin {gather*} \frac {315 \, b \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{4}}{64 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, e^{4}}{{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {x e + d}} + \frac {187 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} e^{4} - 643 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d e^{4} + 765 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{2} e^{4} - 325 \, \sqrt {x e + d} b^{4} d^{3} e^{4} + 643 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} e^{5} - 1530 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d e^{5} + 975 \, \sqrt {x e + d} a b^{3} d^{2} e^{5} + 765 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} e^{6} - 975 \, \sqrt {x e + d} a^{2} b^{2} d e^{6} + 325 \, \sqrt {x e + d} a^{3} b e^{7}}{64 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

315/64*b*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*
a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(-b^2*d + a*b*e)) + 2*e^4/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^
3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(x*e + d)) + 1/64*(187*(x*e + d)^(7/2)*b^4*e^4 -
643*(x*e + d)^(5/2)*b^4*d*e^4 + 765*(x*e + d)^(3/2)*b^4*d^2*e^4 - 325*sqrt(x*e + d)*b^4*d^3*e^4 + 643*(x*e +
d)^(5/2)*a*b^3*e^5 - 1530*(x*e + d)^(3/2)*a*b^3*d*e^5 + 975*sqrt(x*e + d)*a*b^3*d^2*e^5 + 765*(x*e + d)^(3/2)*
a^2*b^2*e^6 - 975*sqrt(x*e + d)*a^2*b^2*d*e^6 + 325*sqrt(x*e + d)*a^3*b*e^7)/((b^5*d^5 - 5*a*b^4*d^4*e + 10*a^
2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*((x*e + d)*b - b*d + a*e)^4)

________________________________________________________________________________________

maple [B]  time = 0.07, size = 446, normalized size = 2.17 \begin {gather*} -\frac {325 \sqrt {e x +d}\, a^{3} b \,e^{7}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}+\frac {975 \sqrt {e x +d}\, a^{2} b^{2} d \,e^{6}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}-\frac {975 \sqrt {e x +d}\, a \,b^{3} d^{2} e^{5}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}+\frac {325 \sqrt {e x +d}\, b^{4} d^{3} e^{4}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}-\frac {765 \left (e x +d \right )^{\frac {3}{2}} a^{2} b^{2} e^{6}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}+\frac {765 \left (e x +d \right )^{\frac {3}{2}} a \,b^{3} d \,e^{5}}{32 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}-\frac {765 \left (e x +d \right )^{\frac {3}{2}} b^{4} d^{2} e^{4}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}-\frac {643 \left (e x +d \right )^{\frac {5}{2}} a \,b^{3} e^{5}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}+\frac {643 \left (e x +d \right )^{\frac {5}{2}} b^{4} d \,e^{4}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}-\frac {187 \left (e x +d \right )^{\frac {7}{2}} b^{4} e^{4}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}-\frac {315 b \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{64 \left (a e -b d \right )^{5} \sqrt {\left (a e -b d \right ) b}}-\frac {2 e^{4}}{\left (a e -b d \right )^{5} \sqrt {e x +d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-187/64*e^4/(a*e-b*d)^5*b^4/(b*e*x+a*e)^4*(e*x+d)^(7/2)-643/64*e^5/(a*e-b*d)^5*b^3/(b*e*x+a*e)^4*(e*x+d)^(5/2)
*a+643/64*e^4/(a*e-b*d)^5*b^4/(b*e*x+a*e)^4*(e*x+d)^(5/2)*d-765/64*e^6/(a*e-b*d)^5*b^2/(b*e*x+a*e)^4*(e*x+d)^(
3/2)*a^2+765/32*e^5/(a*e-b*d)^5*b^3/(b*e*x+a*e)^4*(e*x+d)^(3/2)*a*d-765/64*e^4/(a*e-b*d)^5*b^4/(b*e*x+a*e)^4*(
e*x+d)^(3/2)*d^2-325/64*e^7/(a*e-b*d)^5*b/(b*e*x+a*e)^4*(e*x+d)^(1/2)*a^3+975/64*e^6/(a*e-b*d)^5*b^2/(b*e*x+a*
e)^4*(e*x+d)^(1/2)*a^2*d-975/64*e^5/(a*e-b*d)^5*b^3/(b*e*x+a*e)^4*(e*x+d)^(1/2)*a*d^2+325/64*e^4/(a*e-b*d)^5*b
^4/(b*e*x+a*e)^4*(e*x+d)^(1/2)*d^3-315/64*e^4/(a*e-b*d)^5*b/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d
)*b)^(1/2)*b)-2*e^4/(a*e-b*d)^5/(e*x+d)^(1/2)

________________________________________________________________________________________

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)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^3,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(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d positive or negative?

________________________________________________________________________________________

mupad [B]  time = 2.39, size = 398, normalized size = 1.93 \begin {gather*} -\frac {\frac {2\,e^4}{a\,e-b\,d}+\frac {1533\,b^2\,e^4\,{\left (d+e\,x\right )}^2}{64\,{\left (a\,e-b\,d\right )}^3}+\frac {1155\,b^3\,e^4\,{\left (d+e\,x\right )}^3}{64\,{\left (a\,e-b\,d\right )}^4}+\frac {315\,b^4\,e^4\,{\left (d+e\,x\right )}^4}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {837\,b\,e^4\,\left (d+e\,x\right )}{64\,{\left (a\,e-b\,d\right )}^2}}{b^4\,{\left (d+e\,x\right )}^{9/2}-\left (4\,b^4\,d-4\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{7/2}+\sqrt {d+e\,x}\,\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 (d+e\,x\right )}^{5/2}\,\left (6\,a^2\,b^2\,e^2-12\,a\,b^3\,d\,e+6\,b^4\,d^2\right )-{\left (d+e\,x\right )}^{3/2}\,\left (-4\,a^3\,b\,e^3+12\,a^2\,b^2\,d\,e^2-12\,a\,b^3\,d^2\,e+4\,b^4\,d^3\right )}-\frac {315\,\sqrt {b}\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^{11/2}}\right )}{64\,{\left (a\,e-b\,d\right )}^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*e^4)/(a*e - b*d) + (1533*b^2*e^4*(d + e*x)^2)/(64*(a*e - b*d)^3) + (1155*b^3*e^4*(d + e*x)^3)/(64*(a*e -
b*d)^4) + (315*b^4*e^4*(d + e*x)^4)/(64*(a*e - b*d)^5) + (837*b*e^4*(d + e*x))/(64*(a*e - b*d)^2))/(b^4*(d +
e*x)^(9/2) - (4*b^4*d - 4*a*b^3*e)*(d + e*x)^(7/2) + (d + e*x)^(1/2)*(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) + (d + e*x)^(5/2)*(6*b^4*d^2 + 6*a^2*b^2*e^2 - 12*a*b^3*d*e) - (d + e*x)^(3/2)*
(4*b^4*d^3 - 4*a^3*b*e^3 + 12*a^2*b^2*d*e^2 - 12*a*b^3*d^2*e)) - (315*b^(1/2)*e^4*atan((b^(1/2)*(d + e*x)^(1/2
)*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4))/(a*e - b*d)^(
11/2)))/(64*(a*e - b*d)^(11/2))

________________________________________________________________________________________

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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

________________________________________________________________________________________