3.15.24 \(\int \frac {(d+e x)^{9/2}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=294 \[ -\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {315 e^4 (a+b x) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.14, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {646, 47, 50, 63, 208} \begin {gather*} -\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {315 e^4 (a+b x) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(315*e^4*(a + b*x)*Sqrt[d + e*x])/(64*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (105*e^3*(d + e*x)^(3/2))/(64*b^4*S
qrt[a^2 + 2*a*b*x + b^2*x^2]) - (21*e^2*(d + e*x)^(5/2))/(32*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3
*e*(d + e*x)^(7/2))/(8*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^(9/2)/(4*b*(a + b*x)^3*Sqrt[
a^2 + 2*a*b*x + b^2*x^2]) - (315*e^4*Sqrt[b*d - a*e]*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]
])/(64*b^(11/2)*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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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)^{9/2}}{\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)^{9/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (9 b^2 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (21 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a b+b^2 x\right )^3} \, dx}{16 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^3 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 e^4 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{128 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 e^4 \left (b^2 d-a b e\right ) \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^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 e^3 \left (b^2 d-a b e\right ) \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^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {315 e^4 \sqrt {b d-a e} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 67, normalized size = 0.23 \begin {gather*} -\frac {2 e^4 (a+b x) (d+e x)^{11/2} \, _2F_1\left (5,\frac {11}{2};\frac {13}{2};\frac {b (d+e x)}{b d-a e}\right )}{11 \sqrt {(a+b x)^2} (b d-a e)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-2*e^4*(a + b*x)*(d + e*x)^(11/2)*Hypergeometric2F1[5, 11/2, 13/2, (b*(d + e*x))/(b*d - a*e)])/(11*(b*d - a*e
)^5*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [A]  time = 55.62, size = 328, normalized size = 1.12 \begin {gather*} \frac {(-a e-b e x) \left (-\frac {e^4 \sqrt {d+e x} \left (315 a^4 e^4+1155 a^3 b e^3 (d+e x)-1260 a^3 b d e^3+1890 a^2 b^2 d^2 e^2+1533 a^2 b^2 e^2 (d+e x)^2-3465 a^2 b^2 d e^2 (d+e x)-1260 a b^3 d^3 e+3465 a b^3 d^2 e (d+e x)+837 a b^3 e (d+e x)^3-3066 a b^3 d e (d+e x)^2+315 b^4 d^4-1155 b^4 d^3 (d+e x)+1533 b^4 d^2 (d+e x)^2+128 b^4 (d+e x)^4-837 b^4 d (d+e x)^3\right )}{64 b^5 (a e+b (d+e x)-b d)^4}-\frac {315 e^4 \sqrt {a e-b d} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 b^{11/2}}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(9/2)/(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]*(315*b^4*d^4 - 1260*a*b^3*d^3*e + 1890*a^2*b^2*d^2*e^2 - 1260*a^3*
b*d*e^3 + 315*a^4*e^4 - 1155*b^4*d^3*(d + e*x) + 3465*a*b^3*d^2*e*(d + e*x) - 3465*a^2*b^2*d*e^2*(d + e*x) + 1
155*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
- 837*b^4*d*(d + e*x)^3 + 837*a*b^3*e*(d + e*x)^3 + 128*b^4*(d + e*x)^4))/(b^5*(-(b*d) + a*e + b*(d + e*x))^4
) - (315*e^4*Sqrt[-(b*d) + a*e]*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(64*b^(11/2)))
)/(e*Sqrt[(a*e + b*e*x)^2/e^2])

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fricas [A]  time = 0.43, size = 680, normalized size = 2.31 \begin {gather*} \left [\frac {315 \, {\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 {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - {\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \, {\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{128 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}, -\frac {315 \, {\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 {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - {\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \, {\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{64 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/128*(315*(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*d - a*e)/b)*
log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) + 2*(128*b^4*e^4*x^4 - 16*b^4*d^4
- 24*a*b^3*d^3*e - 42*a^2*b^2*d^2*e^2 - 105*a^3*b*d*e^3 + 315*a^4*e^4 - (325*b^4*d*e^3 - 837*a*b^3*e^4)*x^3 -
3*(70*b^4*d^2*e^2 + 185*a*b^3*d*e^3 - 511*a^2*b^2*e^4)*x^2 - (88*b^4*d^3*e + 156*a*b^3*d^2*e^2 + 399*a^2*b^2*
d*e^3 - 1155*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5), -1/
64*(315*(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*d - a*e)/b)*arc
tan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (128*b^4*e^4*x^4 - 16*b^4*d^4 - 24*a*b^3*d^3*e - 42*a
^2*b^2*d^2*e^2 - 105*a^3*b*d*e^3 + 315*a^4*e^4 - (325*b^4*d*e^3 - 837*a*b^3*e^4)*x^3 - 3*(70*b^4*d^2*e^2 + 185
*a*b^3*d*e^3 - 511*a^2*b^2*e^4)*x^2 - (88*b^4*d^3*e + 156*a*b^3*d^2*e^2 + 399*a^2*b^2*d*e^3 - 1155*a^3*b*e^4)*
x)*sqrt(e*x + d))/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5)]

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giac [B]  time = 0.37, size = 415, normalized size = 1.41 \begin {gather*} \frac {315 \, {\left (b d e^{4} - a e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac {2 \, \sqrt {x e + d} e^{4}}{b^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac {325 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} d e^{4} - 765 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{4} + 643 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{4} - 187 \, \sqrt {x e + d} b^{4} d^{4} e^{4} - 325 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{3} e^{5} + 1530 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} d e^{5} - 1929 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{5} + 748 \, \sqrt {x e + d} a b^{3} d^{3} e^{5} - 765 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{6} + 1929 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{6} - 1122 \, \sqrt {x e + d} a^{2} b^{2} d^{2} e^{6} - 643 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b e^{7} + 748 \, \sqrt {x e + d} a^{3} b d e^{7} - 187 \, \sqrt {x e + d} a^{4} e^{8}}{64 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{5} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

315/64*(b*d*e^4 - a*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5*sgn((x*e + d)*
b*e - b*d*e + a*e^2)) + 2*sqrt(x*e + d)*e^4/(b^5*sgn((x*e + d)*b*e - b*d*e + a*e^2)) - 1/64*(325*(x*e + d)^(7/
2)*b^4*d*e^4 - 765*(x*e + d)^(5/2)*b^4*d^2*e^4 + 643*(x*e + d)^(3/2)*b^4*d^3*e^4 - 187*sqrt(x*e + d)*b^4*d^4*e
^4 - 325*(x*e + d)^(7/2)*a*b^3*e^5 + 1530*(x*e + d)^(5/2)*a*b^3*d*e^5 - 1929*(x*e + d)^(3/2)*a*b^3*d^2*e^5 + 7
48*sqrt(x*e + d)*a*b^3*d^3*e^5 - 765*(x*e + d)^(5/2)*a^2*b^2*e^6 + 1929*(x*e + d)^(3/2)*a^2*b^2*d*e^6 - 1122*s
qrt(x*e + d)*a^2*b^2*d^2*e^6 - 643*(x*e + d)^(3/2)*a^3*b*e^7 + 748*sqrt(x*e + d)*a^3*b*d*e^7 - 187*sqrt(x*e +
d)*a^4*e^8)/(((x*e + d)*b - b*d + a*e)^4*b^5*sgn((x*e + d)*b*e - b*d*e + a*e^2))

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maple [B]  time = 0.08, size = 892, normalized size = 3.03 \begin {gather*} \frac {\left (-315 a \,b^{4} e^{5} x^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+315 b^{5} d \,e^{4} x^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-1260 a^{2} b^{3} e^{5} x^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+1260 a \,b^{4} d \,e^{4} x^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-1890 a^{3} b^{2} e^{5} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+1890 a^{2} b^{3} d \,e^{4} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-1260 a^{4} b \,e^{5} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+1260 a^{3} b^{2} d \,e^{4} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+128 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{4} e^{4} x^{4}-315 a^{5} e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+315 a^{4} b d \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+512 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} e^{4} x^{3}+768 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} e^{4} x^{2}+512 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{3} b \,e^{4} x +315 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{4} e^{4}-748 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{3} b d \,e^{3}+1122 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} d^{2} e^{2}-748 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} d^{3} e +187 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{4} d^{4}+643 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a^{3} b \,e^{3}-1929 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} d \,e^{2}+1929 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} d^{2} e -643 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{4} d^{3}+765 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} e^{2}-1530 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} d e +765 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{4} d^{2}+325 \left (e x +d \right )^{\frac {7}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} e -325 \left (e x +d \right )^{\frac {7}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{4} d \right ) \left (b x +a \right )}{64 \sqrt {\left (a e -b d \right ) b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/64*(-1530*(e*x+d)^(5/2)*((a*e-b*d)*b)^(1/2)*a*b^3*d*e+1260*a^3*b^2*d*e^4*x*arctan((e*x+d)^(1/2)/((a*e-b*d)*b
)^(1/2)*b)-1929*(e*x+d)^(3/2)*((a*e-b*d)*b)^(1/2)*a^2*b^2*d*e^2+1929*(e*x+d)^(3/2)*((a*e-b*d)*b)^(1/2)*a*b^3*d
^2*e+512*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a^3*b*e^4*x-748*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a^3*b*d*e^3+1122*
(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a^2*b^2*d^2*e^2-748*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a*b^3*d^3*e+1890*a^2*b
^3*d*e^4*x^2*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-325*(e*x+d)^(7/2)*((a*e-b*d)*b)^(1/2)*b^4*d+765*(e*x+
d)^(5/2)*((a*e-b*d)*b)^(1/2)*b^4*d^2+187*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*b^4*d^4-643*(e*x+d)^(3/2)*((a*e-b*d
)*b)^(1/2)*b^4*d^3+315*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a^4*e^4+768*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a^2*b^2
*e^4*x^2-315*a^5*e^5*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+315*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*
b)*x^4*b^5*d*e^4+128*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*x^4*b^4*e^4-1260*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/
2)*b)*x^3*a^2*b^3*e^5+315*a^4*b*d*e^4*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+512*(e*x+d)^(1/2)*((a*e-b*d)
*b)^(1/2)*x^3*a*b^3*e^4+1260*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a*b^4*d*e^4+765*(e*x+d)^(5/2)*((a
*e-b*d)*b)^(1/2)*a^2*b^2*e^2-1890*a^3*b^2*e^5*x^2*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-1260*a^4*b*e^5*x
*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+643*(e*x+d)^(3/2)*((a*e-b*d)*b)^(1/2)*a^3*b*e^3+325*(e*x+d)^(7/2)
*((a*e-b*d)*b)^(1/2)*a*b^3*e-315*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*a*b^4*e^5)*(b*x+a)/((a*e-b*d)
*b)^(1/2)/b^5/((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 {9}{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)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(9/2)/(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 )}^{9/2}}{{\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)^(9/2)/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((d + e*x)^(9/2)/(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)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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