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

Optimal. Leaf size=270 \[ \frac {3 e^3 \sqrt {d+e x}}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}} \]

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Rubi [A]  time = 0.14, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {646, 47, 51, 63, 208} \begin {gather*} \frac {3 e^3 \sqrt {d+e x}}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {3 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, 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 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps

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

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Mathematica [C]  time = 0.03, size = 67, normalized size = 0.25 \begin {gather*} -\frac {2 e^4 (a+b x) (d+e x)^{5/2} \, _2F_1\left (\frac {5}{2},5;\frac {7}{2};\frac {b (d+e x)}{b d-a e}\right )}{5 \sqrt {(a+b x)^2} (b d-a e)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e^4*(a + b*x)*(d + e*x)^(5/2)*Hypergeometric2F1[5/2, 5, 7/2, (b*(d + e*x))/(b*d - a*e)])/(5*(b*d - a*e)^5*
Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [A]  time = 37.75, size = 258, normalized size = 0.96 \begin {gather*} \frac {(-a e-b e x) \left (\frac {3 e^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 b^{5/2} (a e-b d)^{5/2}}-\frac {e^4 \sqrt {d+e x} \left (-3 a^3 e^3-11 a^2 b e^2 (d+e x)+9 a^2 b d e^2-9 a b^2 d^2 e+11 a b^2 e (d+e x)^2+22 a b^2 d e (d+e x)+3 b^3 d^3-11 b^3 d^2 (d+e x)+3 b^3 (d+e x)^3-11 b^3 d (d+e x)^2\right )}{64 b^2 (b d-a e)^2 (-a e-b (d+e x)+b d)^4}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-(a*e) - b*e*x)*(-1/64*(e^4*Sqrt[d + e*x]*(3*b^3*d^3 - 9*a*b^2*d^2*e + 9*a^2*b*d*e^2 - 3*a^3*e^3 - 11*b^3*d^
2*(d + e*x) + 22*a*b^2*d*e*(d + e*x) - 11*a^2*b*e^2*(d + e*x) - 11*b^3*d*(d + e*x)^2 + 11*a*b^2*e*(d + e*x)^2
+ 3*b^3*(d + e*x)^3))/(b^2*(b*d - a*e)^2*(b*d - a*e - b*(d + e*x))^4) + (3*e^4*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a
*e]*Sqrt[d + e*x])/(b*d - a*e)])/(64*b^(5/2)*(-(b*d) + a*e)^(5/2))))/(e*Sqrt[(a*e + b*e*x)^2/e^2])

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fricas [B]  time = 0.43, size = 1043, normalized size = 3.86 \begin {gather*} \left [\frac {3 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (16 \, b^{5} d^{4} - 40 \, a b^{4} d^{3} e + 26 \, a^{2} b^{3} d^{2} e^{2} + a^{3} b^{2} d e^{3} - 3 \, a^{4} b e^{4} - 3 \, {\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} + {\left (2 \, b^{5} d^{2} e^{2} - 13 \, a b^{4} d e^{3} + 11 \, a^{2} b^{3} e^{4}\right )} x^{2} + {\left (24 \, b^{5} d^{3} e - 68 \, a b^{4} d^{2} e^{2} + 55 \, a^{2} b^{3} d e^{3} - 11 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{128 \, {\left (a^{4} b^{6} d^{3} - 3 \, a^{5} b^{5} d^{2} e + 3 \, a^{6} b^{4} d e^{2} - a^{7} b^{3} e^{3} + {\left (b^{10} d^{3} - 3 \, a b^{9} d^{2} e + 3 \, a^{2} b^{8} d e^{2} - a^{3} b^{7} e^{3}\right )} x^{4} + 4 \, {\left (a b^{9} d^{3} - 3 \, a^{2} b^{8} d^{2} e + 3 \, a^{3} b^{7} d e^{2} - a^{4} b^{6} e^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d^{3} - 3 \, a^{3} b^{7} d^{2} e + 3 \, a^{4} b^{6} d e^{2} - a^{5} b^{5} e^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d^{3} - 3 \, a^{4} b^{6} d^{2} e + 3 \, a^{5} b^{5} d e^{2} - a^{6} b^{4} e^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (16 \, b^{5} d^{4} - 40 \, a b^{4} d^{3} e + 26 \, a^{2} b^{3} d^{2} e^{2} + a^{3} b^{2} d e^{3} - 3 \, a^{4} b e^{4} - 3 \, {\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} + {\left (2 \, b^{5} d^{2} e^{2} - 13 \, a b^{4} d e^{3} + 11 \, a^{2} b^{3} e^{4}\right )} x^{2} + {\left (24 \, b^{5} d^{3} e - 68 \, a b^{4} d^{2} e^{2} + 55 \, a^{2} b^{3} d e^{3} - 11 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{64 \, {\left (a^{4} b^{6} d^{3} - 3 \, a^{5} b^{5} d^{2} e + 3 \, a^{6} b^{4} d e^{2} - a^{7} b^{3} e^{3} + {\left (b^{10} d^{3} - 3 \, a b^{9} d^{2} e + 3 \, a^{2} b^{8} d e^{2} - a^{3} b^{7} e^{3}\right )} x^{4} + 4 \, {\left (a b^{9} d^{3} - 3 \, a^{2} b^{8} d^{2} e + 3 \, a^{3} b^{7} d e^{2} - a^{4} b^{6} e^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d^{3} - 3 \, a^{3} b^{7} d^{2} e + 3 \, a^{4} b^{6} d e^{2} - a^{5} b^{5} e^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d^{3} - 3 \, a^{4} b^{6} d^{2} e + 3 \, a^{5} b^{5} d e^{2} - a^{6} b^{4} e^{3}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/128*(3*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*sqrt(b^2*d - a*b*e)*lo
g((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) - 2*(16*b^5*d^4 - 40*a*b^4*d^3*e + 26
*a^2*b^3*d^2*e^2 + a^3*b^2*d*e^3 - 3*a^4*b*e^4 - 3*(b^5*d*e^3 - a*b^4*e^4)*x^3 + (2*b^5*d^2*e^2 - 13*a*b^4*d*e
^3 + 11*a^2*b^3*e^4)*x^2 + (24*b^5*d^3*e - 68*a*b^4*d^2*e^2 + 55*a^2*b^3*d*e^3 - 11*a^3*b^2*e^4)*x)*sqrt(e*x +
d))/(a^4*b^6*d^3 - 3*a^5*b^5*d^2*e + 3*a^6*b^4*d*e^2 - a^7*b^3*e^3 + (b^10*d^3 - 3*a*b^9*d^2*e + 3*a^2*b^8*d*
e^2 - a^3*b^7*e^3)*x^4 + 4*(a*b^9*d^3 - 3*a^2*b^8*d^2*e + 3*a^3*b^7*d*e^2 - a^4*b^6*e^3)*x^3 + 6*(a^2*b^8*d^3
- 3*a^3*b^7*d^2*e + 3*a^4*b^6*d*e^2 - a^5*b^5*e^3)*x^2 + 4*(a^3*b^7*d^3 - 3*a^4*b^6*d^2*e + 3*a^5*b^5*d*e^2 -
a^6*b^4*e^3)*x), 1/64*(3*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*sqrt(-b
^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) - (16*b^5*d^4 - 40*a*b^4*d^3*e + 26*a^2
*b^3*d^2*e^2 + a^3*b^2*d*e^3 - 3*a^4*b*e^4 - 3*(b^5*d*e^3 - a*b^4*e^4)*x^3 + (2*b^5*d^2*e^2 - 13*a*b^4*d*e^3 +
11*a^2*b^3*e^4)*x^2 + (24*b^5*d^3*e - 68*a*b^4*d^2*e^2 + 55*a^2*b^3*d*e^3 - 11*a^3*b^2*e^4)*x)*sqrt(e*x + d))
/(a^4*b^6*d^3 - 3*a^5*b^5*d^2*e + 3*a^6*b^4*d*e^2 - a^7*b^3*e^3 + (b^10*d^3 - 3*a*b^9*d^2*e + 3*a^2*b^8*d*e^2
- a^3*b^7*e^3)*x^4 + 4*(a*b^9*d^3 - 3*a^2*b^8*d^2*e + 3*a^3*b^7*d*e^2 - a^4*b^6*e^3)*x^3 + 6*(a^2*b^8*d^3 - 3*
a^3*b^7*d^2*e + 3*a^4*b^6*d*e^2 - a^5*b^5*e^3)*x^2 + 4*(a^3*b^7*d^3 - 3*a^4*b^6*d^2*e + 3*a^5*b^5*d*e^2 - a^6*
b^4*e^3)*x)]

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giac [B]  time = 0.35, size = 421, normalized size = 1.56 \begin {gather*} \frac {3 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{4}}{64 \, {\left (b^{4} d^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 2 \, a b^{3} d e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{2} b^{2} e^{2} \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 {3 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} e^{4} - 11 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} d e^{4} - 11 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{4} + 3 \, \sqrt {x e + d} b^{3} d^{3} e^{4} + 11 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{2} e^{5} + 22 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} d e^{5} - 9 \, \sqrt {x e + d} a b^{2} d^{2} e^{5} - 11 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b e^{6} + 9 \, \sqrt {x e + d} a^{2} b d e^{6} - 3 \, \sqrt {x e + d} a^{3} e^{7}}{64 \, {\left (b^{4} d^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 2 \, a b^{3} d e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{2} b^{2} e^{2} \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 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

3/64*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^4*d^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 2*a*b^3*d
*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^2*b^2*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(-b^2*d + a*b*e))
+ 1/64*(3*(x*e + d)^(7/2)*b^3*e^4 - 11*(x*e + d)^(5/2)*b^3*d*e^4 - 11*(x*e + d)^(3/2)*b^3*d^2*e^4 + 3*sqrt(x*e
+ d)*b^3*d^3*e^4 + 11*(x*e + d)^(5/2)*a*b^2*e^5 + 22*(x*e + d)^(3/2)*a*b^2*d*e^5 - 9*sqrt(x*e + d)*a*b^2*d^2*
e^5 - 11*(x*e + d)^(3/2)*a^2*b*e^6 + 9*sqrt(x*e + d)*a^2*b*d*e^6 - 3*sqrt(x*e + d)*a^3*e^7)/((b^4*d^2*sgn((x*e
+ d)*b*e - b*d*e + a*e^2) - 2*a*b^3*d*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^2*b^2*e^2*sgn((x*e + d)*b*e -
b*d*e + a*e^2))*((x*e + d)*b - b*d + a*e)^4)

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maple [B]  time = 0.09, size = 477, normalized size = 1.77 \begin {gather*} \frac {\left (b x +a \right ) \left (3 b^{4} e^{4} x^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+12 a \,b^{3} e^{4} x^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+18 a^{2} b^{2} e^{4} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+12 a^{3} b \,e^{4} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+3 a^{4} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-3 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{3} e^{3}+9 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{2} b d \,e^{2}-9 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d^{2} e +3 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{3}-11 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a^{2} b \,e^{2}+22 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d e -11 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{2}+11 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} e -11 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{3} d +3 \left (e x +d \right )^{\frac {7}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{3}\right )}{64 \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{2} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(b*x+a)*(3*b^4*e^4*x^4*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+12*a*b^3*e^4*x^3*arctan((e*x+d)^(1/2)/
((a*e-b*d)*b)^(1/2)*b)+3*(e*x+d)^(7/2)*((a*e-b*d)*b)^(1/2)*b^3+18*a^2*b^2*e^4*x^2*arctan((e*x+d)^(1/2)/((a*e-b
*d)*b)^(1/2)*b)+11*(e*x+d)^(5/2)*((a*e-b*d)*b)^(1/2)*a*b^2*e-11*(e*x+d)^(5/2)*((a*e-b*d)*b)^(1/2)*b^3*d+12*a^3
*b*e^4*x*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-11*(e*x+d)^(3/2)*((a*e-b*d)*b)^(1/2)*a^2*b*e^2+22*(e*x+d)
^(3/2)*((a*e-b*d)*b)^(1/2)*a*b^2*d*e-11*(e*x+d)^(3/2)*((a*e-b*d)*b)^(1/2)*b^3*d^2+3*a^4*e^4*arctan((e*x+d)^(1/
2)/((a*e-b*d)*b)^(1/2)*b)-3*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a^3*e^3+9*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a^2*
b*d*e^2-9*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a*b^2*d^2*e+3*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*b^3*d^3)/((a*e-b*d
)*b)^(1/2)/b^2/(a*e-b*d)^2/((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 {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

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

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

[Out]

int((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((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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