3.21.84 \(\int \frac {(a+b x) (d+e x)^{9/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [2084]

Optimal. Leaf size=171 \[ \frac {315 e^4 \sqrt {d+e x}}{64 b^5}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}-\frac {315 e^4 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2}} \]

[Out]

-105/64*e^3*(e*x+d)^(3/2)/b^4/(b*x+a)-21/32*e^2*(e*x+d)^(5/2)/b^3/(b*x+a)^2-3/8*e*(e*x+d)^(7/2)/b^2/(b*x+a)^3-
1/4*(e*x+d)^(9/2)/b/(b*x+a)^4-315/64*e^4*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*(-a*e+b*d)^(1/2)/b^(1
1/2)+315/64*e^4*(e*x+d)^(1/2)/b^5

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Rubi [A]
time = 0.06, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 43, 52, 65, 214} \begin {gather*} -\frac {315 e^4 \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}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {315 e^4 \sqrt {d+e x}}{64 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(315*e^4*Sqrt[d + e*x])/(64*b^5) - (105*e^3*(d + e*x)^(3/2))/(64*b^4*(a + b*x)) - (21*e^2*(d + e*x)^(5/2))/(32
*b^3*(a + b*x)^2) - (3*e*(d + e*x)^(7/2))/(8*b^2*(a + b*x)^3) - (d + e*x)^(9/2)/(4*b*(a + b*x)^4) - (315*e^4*S
qrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(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 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

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 65

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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {(a+b x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^{9/2}}{(a+b x)^5} \, dx\\ &=-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {(9 e) \int \frac {(d+e x)^{7/2}}{(a+b x)^4} \, dx}{8 b}\\ &=-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {\left (21 e^2\right ) \int \frac {(d+e x)^{5/2}}{(a+b x)^3} \, dx}{16 b^2}\\ &=-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {\left (105 e^3\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^2} \, dx}{64 b^3}\\ &=-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {\left (315 e^4\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{128 b^4}\\ &=\frac {315 e^4 \sqrt {d+e x}}{64 b^5}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {\left (315 e^4 (b d-a e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{128 b^5}\\ &=\frac {315 e^4 \sqrt {d+e x}}{64 b^5}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}+\frac {\left (315 e^3 (b d-a e)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 b^5}\\ &=\frac {315 e^4 \sqrt {d+e x}}{64 b^5}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}-\frac {315 e^4 \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}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 212, normalized size = 1.24 \begin {gather*} -\frac {\sqrt {d+e x} \left (-315 a^4 e^4+105 a^3 b e^3 (d-11 e x)+21 a^2 b^2 e^2 \left (2 d^2+19 d e x-73 e^2 x^2\right )+3 a b^3 e \left (8 d^3+52 d^2 e x+185 d e^2 x^2-279 e^3 x^3\right )+b^4 \left (16 d^4+88 d^3 e x+210 d^2 e^2 x^2+325 d e^3 x^3-128 e^4 x^4\right )\right )}{64 b^5 (a+b x)^4}-\frac {315 e^4 \sqrt {-b d+a e} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{64 b^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-1/64*(Sqrt[d + e*x]*(-315*a^4*e^4 + 105*a^3*b*e^3*(d - 11*e*x) + 21*a^2*b^2*e^2*(2*d^2 + 19*d*e*x - 73*e^2*x^
2) + 3*a*b^3*e*(8*d^3 + 52*d^2*e*x + 185*d*e^2*x^2 - 279*e^3*x^3) + b^4*(16*d^4 + 88*d^3*e*x + 210*d^2*e^2*x^2
+ 325*d*e^3*x^3 - 128*e^4*x^4)))/(b^5*(a + b*x)^4) - (315*e^4*Sqrt[-(b*d) + a*e]*ArcTan[(Sqrt[b]*Sqrt[d + e*x
])/Sqrt[-(b*d) + a*e]])/(64*b^(11/2))

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Maple [A]
time = 0.08, size = 249, normalized size = 1.46

method	result	size
derivativedivides	\(2 e^{4} \left (\frac {\sqrt {e x +d}}{b^{5}}-\frac {\frac {\left (-\frac {325}{128} a \,b^{3} e +\frac {325}{128} b^{4} d \right ) \left (e x +d \right )^{\frac {7}{2}}-\frac {765 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2} \left (e x +d \right )^{\frac {5}{2}}}{128}+\left (-\frac {643}{128} a^{3} b \,e^{3}+\frac {1929}{128} a^{2} b^{2} d \,e^{2}-\frac {1929}{128} a \,b^{3} d^{2} e +\frac {643}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {187}{128} a^{4} e^{4}+\frac {187}{32} a^{3} b d \,e^{3}-\frac {561}{64} a^{2} b^{2} d^{2} e^{2}+\frac {187}{32} a \,b^{3} d^{3} e -\frac {187}{128} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {315 \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}}{b^{5}}\right )\)	\(249\)
default	\(2 e^{4} \left (\frac {\sqrt {e x +d}}{b^{5}}-\frac {\frac {\left (-\frac {325}{128} a \,b^{3} e +\frac {325}{128} b^{4} d \right ) \left (e x +d \right )^{\frac {7}{2}}-\frac {765 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2} \left (e x +d \right )^{\frac {5}{2}}}{128}+\left (-\frac {643}{128} a^{3} b \,e^{3}+\frac {1929}{128} a^{2} b^{2} d \,e^{2}-\frac {1929}{128} a \,b^{3} d^{2} e +\frac {643}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {187}{128} a^{4} e^{4}+\frac {187}{32} a^{3} b d \,e^{3}-\frac {561}{64} a^{2} b^{2} d^{2} e^{2}+\frac {187}{32} a \,b^{3} d^{3} e -\frac {187}{128} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {315 \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}}{b^{5}}\right )\)	\(249\)
risch	\(\frac {2 e^{4} \sqrt {e x +d}}{b^{5}}+\frac {325 e^{5} \left (e x +d \right )^{\frac {7}{2}} a}{64 b^{2} \left (b e x +a e \right )^{4}}-\frac {325 e^{4} \left (e x +d \right )^{\frac {7}{2}} d}{64 b \left (b e x +a e \right )^{4}}+\frac {765 e^{6} \left (e x +d \right )^{\frac {5}{2}} a^{2}}{64 b^{3} \left (b e x +a e \right )^{4}}-\frac {765 e^{5} \left (e x +d \right )^{\frac {5}{2}} a d}{32 b^{2} \left (b e x +a e \right )^{4}}+\frac {765 e^{4} \left (e x +d \right )^{\frac {5}{2}} d^{2}}{64 b \left (b e x +a e \right )^{4}}+\frac {643 e^{7} \left (e x +d \right )^{\frac {3}{2}} a^{3}}{64 b^{4} \left (b e x +a e \right )^{4}}-\frac {1929 e^{6} \left (e x +d \right )^{\frac {3}{2}} a^{2} d}{64 b^{3} \left (b e x +a e \right )^{4}}+\frac {1929 e^{5} \left (e x +d \right )^{\frac {3}{2}} a \,d^{2}}{64 b^{2} \left (b e x +a e \right )^{4}}-\frac {643 e^{4} \left (e x +d \right )^{\frac {3}{2}} d^{3}}{64 b \left (b e x +a e \right )^{4}}+\frac {187 e^{8} \sqrt {e x +d}\, a^{4}}{64 b^{5} \left (b e x +a e \right )^{4}}-\frac {187 e^{7} \sqrt {e x +d}\, a^{3} d}{16 b^{4} \left (b e x +a e \right )^{4}}+\frac {561 e^{6} \sqrt {e x +d}\, a^{2} d^{2}}{32 b^{3} \left (b e x +a e \right )^{4}}-\frac {187 e^{5} \sqrt {e x +d}\, a \,d^{3}}{16 b^{2} \left (b e x +a e \right )^{4}}+\frac {187 e^{4} \sqrt {e x +d}\, d^{4}}{64 b \left (b e x +a e \right )^{4}}-\frac {315 e^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a}{64 b^{5} \sqrt {\left (a e -b d \right ) b}}+\frac {315 e^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) d}{64 b^{4} \sqrt {\left (a e -b d \right ) b}}\)	\(497\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^4*(1/b^5*(e*x+d)^(1/2)-1/b^5*(((-325/128*a*b^3*e+325/128*b^4*d)*(e*x+d)^(7/2)-765/128*(a^2*e^2-2*a*b*d*e+b
^2*d^2)*b^2*(e*x+d)^(5/2)+(-643/128*a^3*b*e^3+1929/128*a^2*b^2*d*e^2-1929/128*a*b^3*d^2*e+643/128*b^4*d^3)*(e*
x+d)^(3/2)+(-187/128*a^4*e^4+187/32*a^3*b*d*e^3-561/64*a^2*b^2*d^2*e^2+187/32*a*b^3*d^3*e-187/128*b^4*d^4)*(e*
x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^4+315/128*(a*e-b*d)/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(
1/2))))

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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)^(9/2)/(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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (144) = 288\).
time = 1.33, size = 630, normalized size = 3.68 \begin {gather*} \left [\frac {315 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {\frac {b d - a e}{b}} e^{4} \log \left (\frac {2 \, b d - 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (16 \, b^{4} d^{4} - {\left (128 \, b^{4} x^{4} + 837 \, a b^{3} x^{3} + 1533 \, a^{2} b^{2} x^{2} + 1155 \, a^{3} b x + 315 \, a^{4}\right )} e^{4} + {\left (325 \, b^{4} d x^{3} + 555 \, a b^{3} d x^{2} + 399 \, a^{2} b^{2} d x + 105 \, a^{3} b d\right )} e^{3} + 6 \, {\left (35 \, b^{4} d^{2} x^{2} + 26 \, a b^{3} d^{2} x + 7 \, a^{2} b^{2} d^{2}\right )} e^{2} + 8 \, {\left (11 \, b^{4} d^{3} x + 3 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + 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} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) e^{4} + {\left (16 \, b^{4} d^{4} - {\left (128 \, b^{4} x^{4} + 837 \, a b^{3} x^{3} + 1533 \, a^{2} b^{2} x^{2} + 1155 \, a^{3} b x + 315 \, a^{4}\right )} e^{4} + {\left (325 \, b^{4} d x^{3} + 555 \, a b^{3} d x^{2} + 399 \, a^{2} b^{2} d x + 105 \, a^{3} b d\right )} e^{3} + 6 \, {\left (35 \, b^{4} d^{2} x^{2} + 26 \, a b^{3} d^{2} x + 7 \, a^{2} b^{2} d^{2}\right )} e^{2} + 8 \, {\left (11 \, b^{4} d^{3} x + 3 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + 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((b*x+a)*(e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/128*(315*(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*sqrt((b*d - a*e)/b)*e^4*log((2*b*d - 2*s
qrt(x*e + d)*b*sqrt((b*d - a*e)/b) + (b*x - a)*e)/(b*x + a)) - 2*(16*b^4*d^4 - (128*b^4*x^4 + 837*a*b^3*x^3 +
1533*a^2*b^2*x^2 + 1155*a^3*b*x + 315*a^4)*e^4 + (325*b^4*d*x^3 + 555*a*b^3*d*x^2 + 399*a^2*b^2*d*x + 105*a^3*
b*d)*e^3 + 6*(35*b^4*d^2*x^2 + 26*a*b^3*d^2*x + 7*a^2*b^2*d^2)*e^2 + 8*(11*b^4*d^3*x + 3*a*b^3*d^3)*e)*sqrt(x*
e + 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*x^4 + 4*a*b^3*x^3 + 6
*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(x*e + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e))
*e^4 + (16*b^4*d^4 - (128*b^4*x^4 + 837*a*b^3*x^3 + 1533*a^2*b^2*x^2 + 1155*a^3*b*x + 315*a^4)*e^4 + (325*b^4*
d*x^3 + 555*a*b^3*d*x^2 + 399*a^2*b^2*d*x + 105*a^3*b*d)*e^3 + 6*(35*b^4*d^2*x^2 + 26*a*b^3*d^2*x + 7*a^2*b^2*
d^2)*e^2 + 8*(11*b^4*d^3*x + 3*a*b^3*d^3)*e)*sqrt(x*e + 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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Sympy [F(-1)] Timed out
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)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (144) = 288\).
time = 2.66, size = 343, normalized size = 2.01 \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}} + \frac {2 \, \sqrt {x e + d} e^{4}}{b^{5}} - \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,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) + 2*sqrt(x*e
+ d)*e^4/b^5 - 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 + 748*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*sqrt(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)

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Mupad [B]
time = 2.25, size = 436, normalized size = 2.55 \begin {gather*} \frac {{\left (d+e\,x\right )}^{5/2}\,\left (\frac {765\,a^2\,b^2\,e^6}{64}-\frac {765\,a\,b^3\,d\,e^5}{32}+\frac {765\,b^4\,d^2\,e^4}{64}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {643\,a^3\,b\,e^7}{64}-\frac {1929\,a^2\,b^2\,d\,e^6}{64}+\frac {1929\,a\,b^3\,d^2\,e^5}{64}-\frac {643\,b^4\,d^3\,e^4}{64}\right )+\left (\frac {325\,a\,b^3\,e^5}{64}-\frac {325\,b^4\,d\,e^4}{64}\right )\,{\left (d+e\,x\right )}^{7/2}+\sqrt {d+e\,x}\,\left (\frac {187\,a^4\,e^8}{64}-\frac {187\,a^3\,b\,d\,e^7}{16}+\frac {561\,a^2\,b^2\,d^2\,e^6}{32}-\frac {187\,a\,b^3\,d^3\,e^5}{16}+\frac {187\,b^4\,d^4\,e^4}{64}\right )}{b^9\,{\left (d+e\,x\right )}^4-\left (4\,b^9\,d-4\,a\,b^8\,e\right )\,{\left (d+e\,x\right )}^3+b^9\,d^4+{\left (d+e\,x\right )}^2\,\left (6\,a^2\,b^7\,e^2-12\,a\,b^8\,d\,e+6\,b^9\,d^2\right )-\left (d+e\,x\right )\,\left (-4\,a^3\,b^6\,e^3+12\,a^2\,b^7\,d\,e^2-12\,a\,b^8\,d^2\,e+4\,b^9\,d^3\right )+a^4\,b^5\,e^4-4\,a^3\,b^6\,d\,e^3+6\,a^2\,b^7\,d^2\,e^2-4\,a\,b^8\,d^3\,e}+\frac {2\,e^4\,\sqrt {d+e\,x}}{b^5}-\frac {315\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {a\,e-b\,d}\,\sqrt {d+e\,x}}{a\,e^5-b\,d\,e^4}\right )\,\sqrt {a\,e-b\,d}}{64\,b^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(d + e*x)^(9/2))/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

((d + e*x)^(5/2)*((765*a^2*b^2*e^6)/64 + (765*b^4*d^2*e^4)/64 - (765*a*b^3*d*e^5)/32) + (d + e*x)^(3/2)*((643*
a^3*b*e^7)/64 - (643*b^4*d^3*e^4)/64 + (1929*a*b^3*d^2*e^5)/64 - (1929*a^2*b^2*d*e^6)/64) + ((325*a*b^3*e^5)/6
4 - (325*b^4*d*e^4)/64)*(d + e*x)^(7/2) + (d + e*x)^(1/2)*((187*a^4*e^8)/64 + (187*b^4*d^4*e^4)/64 - (187*a*b^
3*d^3*e^5)/16 + (561*a^2*b^2*d^2*e^6)/32 - (187*a^3*b*d*e^7)/16))/(b^9*(d + e*x)^4 - (4*b^9*d - 4*a*b^8*e)*(d
+ e*x)^3 + b^9*d^4 + (d + e*x)^2*(6*b^9*d^2 + 6*a^2*b^7*e^2 - 12*a*b^8*d*e) - (d + e*x)*(4*b^9*d^3 - 4*a^3*b^6
*e^3 + 12*a^2*b^7*d*e^2 - 12*a*b^8*d^2*e) + a^4*b^5*e^4 - 4*a^3*b^6*d*e^3 + 6*a^2*b^7*d^2*e^2 - 4*a*b^8*d^3*e)
+ (2*e^4*(d + e*x)^(1/2))/b^5 - (315*e^4*atan((b^(1/2)*e^4*(a*e - b*d)^(1/2)*(d + e*x)^(1/2))/(a*e^5 - b*d*e^
4))*(a*e - b*d)^(1/2))/(64*b^(11/2))

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