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

Optimal. Leaf size=276 \[ -\frac {35 e^3 (a+b x)}{8 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}+\frac {35 \sqrt {b} e^3 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}-\frac {35 e^2}{24 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}+\frac {7 e}{12 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2}-\frac {1}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)} \]

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Rubi [A]  time = 0.18, antiderivative size = 276, 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, 21, 51, 63, 208} \begin {gather*} -\frac {35 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)^4}-\frac {35 e^2}{24 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}+\frac {35 \sqrt {b} 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)^{9/2}}+\frac {7 e}{12 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2}-\frac {1}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \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)^(5/2)),x]

[Out]

(-35*e^2)/(24*(b*d - a*e)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(3*(b*d - a*e)*(a + b*x)^2*Sqrt[d
+ e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e)/(12*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) - (35*e^3*(a + b*x))/(8*(b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*Sqrt[b]*e^3
*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(9/2)*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)^{3/2} \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)^{3/2}} \, 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)^{3/2}} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{3 (b d-a e) (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 e \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}} \, 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 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e}{12 (b d-a e)^2 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{24 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2}{24 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e}{12 (b d-a e)^2 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2}{24 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e}{12 (b d-a e)^2 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (a+b x)}{8 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 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)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2}{24 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e}{12 (b d-a e)^2 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (a+b x)}{8 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 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)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2}{24 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e}{12 (b d-a e)^2 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (a+b x)}{8 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 \sqrt {b} 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)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 66, normalized size = 0.24 \begin {gather*} -\frac {2 e^3 (a+b x) \, _2F_1\left (-\frac {1}{2},4;\frac {1}{2};-\frac {b (d+e x)}{a e-b d}\right )}{\sqrt {(a+b x)^2} \sqrt {d+e x} (a e-b d)^4} \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)^(5/2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 50.23, size = 255, normalized size = 0.92 \begin {gather*} \frac {(-a e-b e x) \left (-\frac {e^3 \left (48 a^3 e^3+231 a^2 b e^2 (d+e x)-144 a^2 b d e^2+144 a b^2 d^2 e+280 a b^2 e (d+e x)^2-462 a b^2 d e (d+e x)-48 b^3 d^3+231 b^3 d^2 (d+e x)+105 b^3 (d+e x)^3-280 b^3 d (d+e x)^2\right )}{24 \sqrt {d+e x} (b d-a e)^4 (-a e-b (d+e x)+b d)^3}-\frac {35 \sqrt {b} e^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{8 (a e-b d)^{9/2}}\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)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

((-(a*e) - b*e*x)*(-1/24*(e^3*(-48*b^3*d^3 + 144*a*b^2*d^2*e - 144*a^2*b*d*e^2 + 48*a^3*e^3 + 231*b^3*d^2*(d +
e*x) - 462*a*b^2*d*e*(d + e*x) + 231*a^2*b*e^2*(d + e*x) - 280*b^3*d*(d + e*x)^2 + 280*a*b^2*e*(d + e*x)^2 +
105*b^3*(d + e*x)^3))/((b*d - a*e)^4*Sqrt[d + e*x]*(b*d - a*e - b*(d + e*x))^3) - (35*Sqrt[b]*e^3*ArcTan[(Sqrt
[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(8*(-(b*d) + a*e)^(9/2))))/(e*Sqrt[(a*e + b*e*x)^2/e^2])

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fricas [B]  time = 0.47, size = 1204, normalized size = 4.36 \begin {gather*} \left [\frac {105 \, {\left (b^{3} e^{4} x^{4} + a^{3} d e^{3} + {\left (b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{3} + 3 \, {\left (a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{2} + {\left (3 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) - 2 \, {\left (105 \, b^{3} e^{3} x^{3} + 8 \, b^{3} d^{3} - 38 \, a b^{2} d^{2} e + 87 \, a^{2} b d e^{2} + 48 \, a^{3} e^{3} + 35 \, {\left (b^{3} d e^{2} + 8 \, a b^{2} e^{3}\right )} x^{2} - 7 \, {\left (2 \, b^{3} d^{2} e - 14 \, a b^{2} d e^{2} - 33 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (a^{3} b^{4} d^{5} - 4 \, a^{4} b^{3} d^{4} e + 6 \, a^{5} b^{2} d^{3} e^{2} - 4 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4} + {\left (b^{7} d^{4} e - 4 \, a b^{6} d^{3} e^{2} + 6 \, a^{2} b^{5} d^{2} e^{3} - 4 \, a^{3} b^{4} d e^{4} + a^{4} b^{3} e^{5}\right )} x^{4} + {\left (b^{7} d^{5} - a b^{6} d^{4} e - 6 \, a^{2} b^{5} d^{3} e^{2} + 14 \, a^{3} b^{4} d^{2} e^{3} - 11 \, a^{4} b^{3} d e^{4} + 3 \, a^{5} b^{2} e^{5}\right )} x^{3} + 3 \, {\left (a b^{6} d^{5} - 3 \, a^{2} b^{5} d^{4} e + 2 \, a^{3} b^{4} d^{3} e^{2} + 2 \, a^{4} b^{3} d^{2} e^{3} - 3 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} d^{5} - 11 \, a^{3} b^{4} d^{4} e + 14 \, a^{4} b^{3} d^{3} e^{2} - 6 \, a^{5} b^{2} d^{2} e^{3} - a^{6} b d e^{4} + a^{7} e^{5}\right )} x\right )}}, \frac {105 \, {\left (b^{3} e^{4} x^{4} + a^{3} d e^{3} + {\left (b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{3} + 3 \, {\left (a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{2} + {\left (3 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (105 \, b^{3} e^{3} x^{3} + 8 \, b^{3} d^{3} - 38 \, a b^{2} d^{2} e + 87 \, a^{2} b d e^{2} + 48 \, a^{3} e^{3} + 35 \, {\left (b^{3} d e^{2} + 8 \, a b^{2} e^{3}\right )} x^{2} - 7 \, {\left (2 \, b^{3} d^{2} e - 14 \, a b^{2} d e^{2} - 33 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (a^{3} b^{4} d^{5} - 4 \, a^{4} b^{3} d^{4} e + 6 \, a^{5} b^{2} d^{3} e^{2} - 4 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4} + {\left (b^{7} d^{4} e - 4 \, a b^{6} d^{3} e^{2} + 6 \, a^{2} b^{5} d^{2} e^{3} - 4 \, a^{3} b^{4} d e^{4} + a^{4} b^{3} e^{5}\right )} x^{4} + {\left (b^{7} d^{5} - a b^{6} d^{4} e - 6 \, a^{2} b^{5} d^{3} e^{2} + 14 \, a^{3} b^{4} d^{2} e^{3} - 11 \, a^{4} b^{3} d e^{4} + 3 \, a^{5} b^{2} e^{5}\right )} x^{3} + 3 \, {\left (a b^{6} d^{5} - 3 \, a^{2} b^{5} d^{4} e + 2 \, a^{3} b^{4} d^{3} e^{2} + 2 \, a^{4} b^{3} d^{2} e^{3} - 3 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} d^{5} - 11 \, a^{3} b^{4} d^{4} e + 14 \, a^{4} b^{3} d^{3} e^{2} - 6 \, a^{5} b^{2} d^{2} e^{3} - a^{6} b d e^{4} + a^{7} e^{5}\right )} x\right )}}\right ] \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)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(105*(b^3*e^4*x^4 + a^3*d*e^3 + (b^3*d*e^3 + 3*a*b^2*e^4)*x^3 + 3*(a*b^2*d*e^3 + a^2*b*e^4)*x^2 + (3*a^2
*b*d*e^3 + a^3*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*(105*b^3*e^3*x^3 + 8*b^3*d^3 - 38*a*b^2*d^2*e + 87*a^2*b*d*e^2 + 48*a^3*e^3 + 35*(b^3*
d*e^2 + 8*a*b^2*e^3)*x^2 - 7*(2*b^3*d^2*e - 14*a*b^2*d*e^2 - 33*a^2*b*e^3)*x)*sqrt(e*x + d))/(a^3*b^4*d^5 - 4*
a^4*b^3*d^4*e + 6*a^5*b^2*d^3*e^2 - 4*a^6*b*d^2*e^3 + a^7*d*e^4 + (b^7*d^4*e - 4*a*b^6*d^3*e^2 + 6*a^2*b^5*d^2
*e^3 - 4*a^3*b^4*d*e^4 + a^4*b^3*e^5)*x^4 + (b^7*d^5 - a*b^6*d^4*e - 6*a^2*b^5*d^3*e^2 + 14*a^3*b^4*d^2*e^3 -
11*a^4*b^3*d*e^4 + 3*a^5*b^2*e^5)*x^3 + 3*(a*b^6*d^5 - 3*a^2*b^5*d^4*e + 2*a^3*b^4*d^3*e^2 + 2*a^4*b^3*d^2*e^3
- 3*a^5*b^2*d*e^4 + a^6*b*e^5)*x^2 + (3*a^2*b^5*d^5 - 11*a^3*b^4*d^4*e + 14*a^4*b^3*d^3*e^2 - 6*a^5*b^2*d^2*e
^3 - a^6*b*d*e^4 + a^7*e^5)*x), 1/24*(105*(b^3*e^4*x^4 + a^3*d*e^3 + (b^3*d*e^3 + 3*a*b^2*e^4)*x^3 + 3*(a*b^2*
d*e^3 + a^2*b*e^4)*x^2 + (3*a^2*b*d*e^3 + a^3*e^4)*x)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(e*x + d)*s
qrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (105*b^3*e^3*x^3 + 8*b^3*d^3 - 38*a*b^2*d^2*e + 87*a^2*b*d*e^2 + 48*a^3*e
^3 + 35*(b^3*d*e^2 + 8*a*b^2*e^3)*x^2 - 7*(2*b^3*d^2*e - 14*a*b^2*d*e^2 - 33*a^2*b*e^3)*x)*sqrt(e*x + d))/(a^3
*b^4*d^5 - 4*a^4*b^3*d^4*e + 6*a^5*b^2*d^3*e^2 - 4*a^6*b*d^2*e^3 + a^7*d*e^4 + (b^7*d^4*e - 4*a*b^6*d^3*e^2 +
6*a^2*b^5*d^2*e^3 - 4*a^3*b^4*d*e^4 + a^4*b^3*e^5)*x^4 + (b^7*d^5 - a*b^6*d^4*e - 6*a^2*b^5*d^3*e^2 + 14*a^3*b
^4*d^2*e^3 - 11*a^4*b^3*d*e^4 + 3*a^5*b^2*e^5)*x^3 + 3*(a*b^6*d^5 - 3*a^2*b^5*d^4*e + 2*a^3*b^4*d^3*e^2 + 2*a^
4*b^3*d^2*e^3 - 3*a^5*b^2*d*e^4 + a^6*b*e^5)*x^2 + (3*a^2*b^5*d^5 - 11*a^3*b^4*d^4*e + 14*a^4*b^3*d^3*e^2 - 6*
a^5*b^2*d^2*e^3 - a^6*b*d*e^4 + a^7*e^5)*x)]

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giac [B]  time = 0.41, size = 654, normalized size = 2.37 \begin {gather*} -\frac {35 \, b \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{3}}{8 \, {\left (b^{4} d^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e \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^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{4} \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 \, e^{3}}{{\left (b^{4} d^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e \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^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} \sqrt {x e + d}} - \frac {57 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} e^{3} - 136 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d e^{3} + 87 \, \sqrt {x e + d} b^{3} d^{2} e^{3} + 136 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} e^{4} - 174 \, \sqrt {x e + d} a b^{2} d e^{4} + 87 \, \sqrt {x e + d} a^{2} b e^{5}}{24 \, {\left (b^{4} d^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e \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^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{4} \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)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

-35/8*b*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((b^4*d^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a*b^
3*d^3*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 6*a^2*b^2*d^2*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a^3*b*d*
e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^4*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(-b^2*d + a*b*e)) - 2
*e^3/((b^4*d^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a*b^3*d^3*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 6*a^2*b
^2*d^2*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a^3*b*d*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^4*e^4*sgn
((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(x*e + d)) - 1/24*(57*(x*e + d)^(5/2)*b^3*e^3 - 136*(x*e + d)^(3/2)*b^3*d
*e^3 + 87*sqrt(x*e + d)*b^3*d^2*e^3 + 136*(x*e + d)^(3/2)*a*b^2*e^4 - 174*sqrt(x*e + d)*a*b^2*d*e^4 + 87*sqrt(
x*e + d)*a^2*b*e^5)/((b^4*d^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a*b^3*d^3*e*sgn((x*e + d)*b*e - b*d*e + a
*e^2) + 6*a^2*b^2*d^2*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a^3*b*d*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2
) + a^4*e^4*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 = 0.07, size = 431, normalized size = 1.56 \begin {gather*} -\frac {\left (105 \sqrt {e x +d}\, b^{4} e^{3} x^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+315 \sqrt {e x +d}\, a \,b^{3} e^{3} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+315 \sqrt {e x +d}\, a^{2} b^{2} e^{3} x \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}\, b^{3} e^{3} x^{3}+105 \sqrt {e x +d}\, a^{3} b \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+280 \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} e^{3} x^{2}+35 \sqrt {\left (a e -b d \right ) b}\, b^{3} d \,e^{2} x^{2}+231 \sqrt {\left (a e -b d \right ) b}\, a^{2} b \,e^{3} x +98 \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d \,e^{2} x -14 \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{2} e x +48 \sqrt {\left (a e -b d \right ) b}\, a^{3} e^{3}+87 \sqrt {\left (a e -b d \right ) b}\, a^{2} b d \,e^{2}-38 \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d^{2} e +8 \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{3}\right ) \left (b x +a \right )^{2}}{24 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{4} \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)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/24*(105*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*x^3*b^4*e^3+315*arctan((e*x+d)^(1/2)/((a*
e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*x^2*a*b^3*e^3+105*((a*e-b*d)*b)^(1/2)*b^3*e^3*x^3+315*arctan((e*x+d)^(1/2)/((
a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*x*a^2*b^2*e^3+280*((a*e-b*d)*b)^(1/2)*a*b^2*e^3*x^2+35*((a*e-b*d)*b)^(1/2)*
b^3*d*e^2*x^2+105*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(1/2)*a^3*b*e^3+231*((a*e-b*d)*b)^(1/2)*
a^2*b*e^3*x+98*((a*e-b*d)*b)^(1/2)*a*b^2*d*e^2*x-14*((a*e-b*d)*b)^(1/2)*b^3*d^2*e*x+48*((a*e-b*d)*b)^(1/2)*a^3
*e^3+87*((a*e-b*d)*b)^(1/2)*a^2*b*d*e^2-38*((a*e-b*d)*b)^(1/2)*a*b^2*d^2*e+8*((a*e-b*d)*b)^(1/2)*b^3*d^3)*(b*x
+a)^2/(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)/(a*e-b*d)^4/((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 {3}{2}}}\,{d x} \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)^(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)^(3/2)), 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 )}^{3/2}\,{\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)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)

[Out]

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

[Out]

Timed out

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