3.18.21 \(\int \frac {(d+e x)^{11/2}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1721]

Optimal. Leaf size=346 \[ \frac {1155 e^4 (b d-a e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1155 e^4 (b d-a e)^{3/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

385/64*e^4*(b*x+a)*(e*x+d)^(3/2)/b^5/((b*x+a)^2)^(1/2)-231/64*e^3*(e*x+d)^(5/2)/b^4/((b*x+a)^2)^(1/2)-33/32*e^
2*(e*x+d)^(7/2)/b^3/(b*x+a)/((b*x+a)^2)^(1/2)-11/24*e*(e*x+d)^(9/2)/b^2/(b*x+a)^2/((b*x+a)^2)^(1/2)-1/4*(e*x+d
)^(11/2)/b/(b*x+a)^3/((b*x+a)^2)^(1/2)-1155/64*e^4*(-a*e+b*d)^(3/2)*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*
e+b*d)^(1/2))/b^(13/2)/((b*x+a)^2)^(1/2)+1155/64*e^4*(-a*e+b*d)*(b*x+a)*(e*x+d)^(1/2)/b^6/((b*x+a)^2)^(1/2)

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Rubi [A]
time = 0.13, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 43, 52, 65, 214} \begin {gather*} -\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1155 e^4 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1155 e^4 (a+b x) \sqrt {d+e x} (b d-a e)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(1155*e^4*(b*d - a*e)*(a + b*x)*Sqrt[d + e*x])/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (385*e^4*(a + b*x)*(d
+ e*x)^(3/2))/(64*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (231*e^3*(d + e*x)^(5/2))/(64*b^4*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) - (33*e^2*(d + e*x)^(7/2))/(32*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (11*e*(d + e*x)^(9/2))
/(24*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^(11/2)/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b
^2*x^2]) - (1155*e^4*(b*d - a*e)^(3/2)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(13/2
)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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]

Rule 660

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)^{11/2}}{\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)^{11/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (11 b^2 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (33 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^3} \, dx}{16 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (231 e^3 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 e^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{128 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 e^4 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{128 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {1155 e^4 (b d-a e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 e^4 \left (b^2 d-a b e\right )^2 \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^8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {1155 e^4 (b d-a e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 e^3 \left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \text {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^8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {1155 e^4 (b d-a e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1155 e^4 (b d-a e)^{3/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.30, size = 300, normalized size = 0.87 \begin {gather*} \frac {e^4 (a+b x)^5 \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (3465 a^5 e^5+1155 a^4 b e^4 (-4 d+11 e x)+231 a^3 b^2 e^3 \left (3 d^2-74 d e x+73 e^2 x^2\right )+99 a^2 b^3 e^2 \left (2 d^3+27 d^2 e x-232 d e^2 x^2+93 e^3 x^3\right )+11 a b^4 e \left (8 d^4+68 d^3 e x+345 d^2 e^2 x^2-1162 d e^3 x^3+128 e^4 x^4\right )+b^5 \left (48 d^5+328 d^4 e x+1030 d^3 e^2 x^2+2295 d^2 e^3 x^3-2048 d e^4 x^4-128 e^5 x^5\right )\right )}{e^4 (a+b x)^4}+3465 (-b d+a e)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{192 b^{13/2} \left ((a+b x)^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(e^4*(a + b*x)^5*(-((Sqrt[b]*Sqrt[d + e*x]*(3465*a^5*e^5 + 1155*a^4*b*e^4*(-4*d + 11*e*x) + 231*a^3*b^2*e^3*(3
*d^2 - 74*d*e*x + 73*e^2*x^2) + 99*a^2*b^3*e^2*(2*d^3 + 27*d^2*e*x - 232*d*e^2*x^2 + 93*e^3*x^3) + 11*a*b^4*e*
(8*d^4 + 68*d^3*e*x + 345*d^2*e^2*x^2 - 1162*d*e^3*x^3 + 128*e^4*x^4) + b^5*(48*d^5 + 328*d^4*e*x + 1030*d^3*e
^2*x^2 + 2295*d^2*e^3*x^3 - 2048*d*e^4*x^4 - 128*e^5*x^5)))/(e^4*(a + b*x)^4)) + 3465*(-(b*d) + a*e)^(3/2)*Arc
Tan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]]))/(192*b^(13/2)*((a + b*x)^2)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1470\) vs. \(2(233)=466\).
time = 0.87, size = 1471, normalized size = 4.25

method	result	size
risch	\(-\frac {2 e^{4} \left (-b e x +15 a e -16 b d \right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{3 b^{6} \left (b x +a \right )}+\frac {\left (\frac {765 e^{5} \left (e x +d \right )^{\frac {7}{2}} a d}{32 b^{2} \left (b e x +a e \right )^{4}}-\frac {765 e^{6} \left (e x +d \right )^{\frac {7}{2}} a^{2}}{64 b^{3} \left (b e x +a e \right )^{4}}-\frac {5855 e^{7} \left (e x +d \right )^{\frac {5}{2}} a^{3}}{192 b^{4} \left (b e x +a e \right )^{4}}-\frac {5153 e^{8} \left (e x +d \right )^{\frac {3}{2}} a^{4}}{192 b^{5} \left (b e x +a e \right )^{4}}+\frac {5855 e^{6} \left (e x +d \right )^{\frac {5}{2}} a^{2} d}{64 b^{3} \left (b e x +a e \right )^{4}}-\frac {5855 e^{5} \left (e x +d \right )^{\frac {5}{2}} a \,d^{2}}{64 b^{2} \left (b e x +a e \right )^{4}}+\frac {5153 e^{7} \left (e x +d \right )^{\frac {3}{2}} a^{3} d}{48 b^{4} \left (b e x +a e \right )^{4}}-\frac {5153 e^{6} \left (e x +d \right )^{\frac {3}{2}} a^{2} d^{2}}{32 b^{3} \left (b e x +a e \right )^{4}}+\frac {5153 e^{5} \left (e x +d \right )^{\frac {3}{2}} a \,d^{3}}{48 b^{2} \left (b e x +a e \right )^{4}}+\frac {2575 e^{8} \sqrt {e x +d}\, a^{4} d}{64 b^{5} \left (b e x +a e \right )^{4}}-\frac {2575 e^{7} \sqrt {e x +d}\, a^{3} d^{2}}{32 b^{4} \left (b e x +a e \right )^{4}}+\frac {2575 e^{6} \sqrt {e x +d}\, a^{2} d^{3}}{32 b^{3} \left (b e x +a e \right )^{4}}-\frac {2575 e^{5} \sqrt {e x +d}\, a \,d^{4}}{64 b^{2} \left (b e x +a e \right )^{4}}-\frac {1155 e^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a d}{32 b^{5} \sqrt {b \left (a e -b d \right )}}-\frac {765 e^{4} \left (e x +d \right )^{\frac {7}{2}} d^{2}}{64 b \left (b e x +a e \right )^{4}}+\frac {5855 e^{4} \left (e x +d \right )^{\frac {5}{2}} d^{3}}{192 b \left (b e x +a e \right )^{4}}-\frac {5153 e^{4} \left (e x +d \right )^{\frac {3}{2}} d^{4}}{192 b \left (b e x +a e \right )^{4}}-\frac {515 e^{9} \sqrt {e x +d}\, a^{5}}{64 b^{6} \left (b e x +a e \right )^{4}}+\frac {515 e^{4} \sqrt {e x +d}\, d^{5}}{64 b \left (b e x +a e \right )^{4}}+\frac {1155 e^{6} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2}}{64 b^{6} \sqrt {b \left (a e -b d \right )}}+\frac {1155 e^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) d^{2}}{64 b^{4} \sqrt {b \left (a e -b d \right )}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\)	\(717\)
default	\(\text {Expression too large to display}\)	\(1471\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/192*(-128*(e*x+d)^(3/2)*(b*(a*e-b*d))^(1/2)*b^5*e^4*x^4-3465*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^
2*b^4*e^6*x^4-3465*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*b^6*d^2*e^4*x^4-3465*arctan(b*(e*x+d)^(1/2)/(b*
(a*e-b*d))^(1/2))*a^6*e^6+6930*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a*b^5*d*e^5*x^4-512*(e*x+d)^(3/2)*(
b*(a*e-b*d))^(1/2)*a*b^4*e^4*x^3+1920*(e*x+d)^(1/2)*(b*(a*e-b*d))^(1/2)*a*b^4*e^5*x^4-1920*(e*x+d)^(1/2)*(b*(a
*e-b*d))^(1/2)*b^5*d*e^4*x^4+27720*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^2*b^4*d*e^5*x^3-13860*arctan(
b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a*b^5*d^2*e^4*x^3-4590*(e*x+d)^(7/2)*(b*(a*e-b*d))^(1/2)*a*b^4*d*e-768*(e
*x+d)^(3/2)*(b*(a*e-b*d))^(1/2)*a^2*b^3*e^4*x^2+7680*(e*x+d)^(1/2)*(b*(a*e-b*d))^(1/2)*a^2*b^3*e^5*x^3+41580*a
rctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^3*b^3*d*e^5*x^2-20612*(e*x+d)^(3/2)*(b*(a*e-b*d))^(1/2)*a*b^4*d^3
*e+7680*(e*x+d)^(1/2)*(b*(a*e-b*d))^(1/2)*a^4*b*e^5*x-9645*(e*x+d)^(1/2)*(b*(a*e-b*d))^(1/2)*a^4*b*d*e^4+15450
*(e*x+d)^(1/2)*(b*(a*e-b*d))^(1/2)*a^3*b^2*d^2*e^3-15450*(e*x+d)^(1/2)*(b*(a*e-b*d))^(1/2)*a^2*b^3*d^3*e^2+772
5*(e*x+d)^(1/2)*(b*(a*e-b*d))^(1/2)*a*b^4*d^4*e+2295*(e*x+d)^(7/2)*(b*(a*e-b*d))^(1/2)*b^5*d^2-5855*(e*x+d)^(5
/2)*(b*(a*e-b*d))^(1/2)*b^5*d^3+5153*(e*x+d)^(3/2)*(b*(a*e-b*d))^(1/2)*b^5*d^4+3465*(e*x+d)^(1/2)*(b*(a*e-b*d)
)^(1/2)*a^5*e^5-1545*(e*x+d)^(1/2)*(b*(a*e-b*d))^(1/2)*b^5*d^5-20790*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2
))*a^2*b^4*d^2*e^4*x^2-17565*(e*x+d)^(5/2)*(b*(a*e-b*d))^(1/2)*a^2*b^3*d*e^2+17565*(e*x+d)^(5/2)*(b*(a*e-b*d))
^(1/2)*a*b^4*d^2*e-512*(e*x+d)^(3/2)*(b*(a*e-b*d))^(1/2)*a^3*b^2*e^4*x+11520*(e*x+d)^(1/2)*(b*(a*e-b*d))^(1/2)
*a^3*b^2*e^5*x^2+27720*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^4*b^2*d*e^5*x-13860*arctan(b*(e*x+d)^(1/2
)/(b*(a*e-b*d))^(1/2))*a^3*b^3*d^2*e^4*x-20612*(e*x+d)^(3/2)*(b*(a*e-b*d))^(1/2)*a^3*b^2*d*e^3+30918*(e*x+d)^(
3/2)*(b*(a*e-b*d))^(1/2)*a^2*b^3*d^2*e^2-7680*(e*x+d)^(1/2)*(b*(a*e-b*d))^(1/2)*a*b^4*d*e^4*x^3-11520*(e*x+d)^
(1/2)*(b*(a*e-b*d))^(1/2)*a^2*b^3*d*e^4*x^2-7680*(e*x+d)^(1/2)*(b*(a*e-b*d))^(1/2)*a^3*b^2*d*e^4*x-13860*arcta
n(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^3*b^3*e^6*x^3+2295*(e*x+d)^(7/2)*(b*(a*e-b*d))^(1/2)*a^2*b^3*e^2-2079
0*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^4*b^2*e^6*x^2+5855*(e*x+d)^(5/2)*(b*(a*e-b*d))^(1/2)*a^3*b^2*e
^3-13860*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^5*b*e^6*x+5025*(e*x+d)^(3/2)*(b*(a*e-b*d))^(1/2)*a^4*b*
e^4+6930*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^5*b*d*e^5-3465*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/
2))*a^4*b^2*d^2*e^4)*(b*x+a)/(b*(a*e-b*d))^(1/2)/b^6/((b*x+a)^2)^(5/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((x*e + d)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), x)

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Fricas [A]
time = 2.09, size = 893, normalized size = 2.58 \begin {gather*} \left [\frac {3465 \, {\left ({\left (a b^{4} x^{4} + 4 \, a^{2} b^{3} x^{3} + 6 \, a^{3} b^{2} x^{2} + 4 \, a^{4} b x + a^{5}\right )} e^{5} - {\left (b^{5} d x^{4} + 4 \, a b^{4} d x^{3} + 6 \, a^{2} b^{3} d x^{2} + 4 \, a^{3} b^{2} d x + a^{4} b d\right )} e^{4}\right )} \sqrt {\frac {b d - a e}{b}} \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 (48 \, b^{5} d^{5} - {\left (128 \, b^{5} x^{5} - 1408 \, a b^{4} x^{4} - 9207 \, a^{2} b^{3} x^{3} - 16863 \, a^{3} b^{2} x^{2} - 12705 \, a^{4} b x - 3465 \, a^{5}\right )} e^{5} - 2 \, {\left (1024 \, b^{5} d x^{4} + 6391 \, a b^{4} d x^{3} + 11484 \, a^{2} b^{3} d x^{2} + 8547 \, a^{3} b^{2} d x + 2310 \, a^{4} b d\right )} e^{4} + 3 \, {\left (765 \, b^{5} d^{2} x^{3} + 1265 \, a b^{4} d^{2} x^{2} + 891 \, a^{2} b^{3} d^{2} x + 231 \, a^{3} b^{2} d^{2}\right )} e^{3} + 2 \, {\left (515 \, b^{5} d^{3} x^{2} + 374 \, a b^{4} d^{3} x + 99 \, a^{2} b^{3} d^{3}\right )} e^{2} + 8 \, {\left (41 \, b^{5} d^{4} x + 11 \, a b^{4} d^{4}\right )} e\right )} \sqrt {x e + d}}{384 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}}, \frac {3465 \, {\left ({\left (a b^{4} x^{4} + 4 \, a^{2} b^{3} x^{3} + 6 \, a^{3} b^{2} x^{2} + 4 \, a^{4} b x + a^{5}\right )} e^{5} - {\left (b^{5} d x^{4} + 4 \, a b^{4} d x^{3} + 6 \, a^{2} b^{3} d x^{2} + 4 \, a^{3} b^{2} d x + a^{4} b d\right )} e^{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 ) - {\left (48 \, b^{5} d^{5} - {\left (128 \, b^{5} x^{5} - 1408 \, a b^{4} x^{4} - 9207 \, a^{2} b^{3} x^{3} - 16863 \, a^{3} b^{2} x^{2} - 12705 \, a^{4} b x - 3465 \, a^{5}\right )} e^{5} - 2 \, {\left (1024 \, b^{5} d x^{4} + 6391 \, a b^{4} d x^{3} + 11484 \, a^{2} b^{3} d x^{2} + 8547 \, a^{3} b^{2} d x + 2310 \, a^{4} b d\right )} e^{4} + 3 \, {\left (765 \, b^{5} d^{2} x^{3} + 1265 \, a b^{4} d^{2} x^{2} + 891 \, a^{2} b^{3} d^{2} x + 231 \, a^{3} b^{2} d^{2}\right )} e^{3} + 2 \, {\left (515 \, b^{5} d^{3} x^{2} + 374 \, a b^{4} d^{3} x + 99 \, a^{2} b^{3} d^{3}\right )} e^{2} + 8 \, {\left (41 \, b^{5} d^{4} x + 11 \, a b^{4} d^{4}\right )} e\right )} \sqrt {x e + d}}{192 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/384*(3465*((a*b^4*x^4 + 4*a^2*b^3*x^3 + 6*a^3*b^2*x^2 + 4*a^4*b*x + a^5)*e^5 - (b^5*d*x^4 + 4*a*b^4*d*x^3 +
6*a^2*b^3*d*x^2 + 4*a^3*b^2*d*x + a^4*b*d)*e^4)*sqrt((b*d - a*e)/b)*log((2*b*d + 2*sqrt(x*e + d)*b*sqrt((b*d
- a*e)/b) + (b*x - a)*e)/(b*x + a)) - 2*(48*b^5*d^5 - (128*b^5*x^5 - 1408*a*b^4*x^4 - 9207*a^2*b^3*x^3 - 16863
*a^3*b^2*x^2 - 12705*a^4*b*x - 3465*a^5)*e^5 - 2*(1024*b^5*d*x^4 + 6391*a*b^4*d*x^3 + 11484*a^2*b^3*d*x^2 + 85
47*a^3*b^2*d*x + 2310*a^4*b*d)*e^4 + 3*(765*b^5*d^2*x^3 + 1265*a*b^4*d^2*x^2 + 891*a^2*b^3*d^2*x + 231*a^3*b^2
*d^2)*e^3 + 2*(515*b^5*d^3*x^2 + 374*a*b^4*d^3*x + 99*a^2*b^3*d^3)*e^2 + 8*(41*b^5*d^4*x + 11*a*b^4*d^4)*e)*sq
rt(x*e + d))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6), 1/192*(3465*((a*b^4*x^4 + 4*a^2
*b^3*x^3 + 6*a^3*b^2*x^2 + 4*a^4*b*x + a^5)*e^5 - (b^5*d*x^4 + 4*a*b^4*d*x^3 + 6*a^2*b^3*d*x^2 + 4*a^3*b^2*d*x
+ a^4*b*d)*e^4)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(x*e + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (48*b^5*d^5
- (128*b^5*x^5 - 1408*a*b^4*x^4 - 9207*a^2*b^3*x^3 - 16863*a^3*b^2*x^2 - 12705*a^4*b*x - 3465*a^5)*e^5 - 2*(10
24*b^5*d*x^4 + 6391*a*b^4*d*x^3 + 11484*a^2*b^3*d*x^2 + 8547*a^3*b^2*d*x + 2310*a^4*b*d)*e^4 + 3*(765*b^5*d^2*
x^3 + 1265*a*b^4*d^2*x^2 + 891*a^2*b^3*d^2*x + 231*a^3*b^2*d^2)*e^3 + 2*(515*b^5*d^3*x^2 + 374*a*b^4*d^3*x + 9
9*a^2*b^3*d^3)*e^2 + 8*(41*b^5*d^4*x + 11*a*b^4*d^4)*e)*sqrt(x*e + d))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2
+ 4*a^3*b^7*x + a^4*b^6)]

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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((e*x+d)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (239) = 478\).
time = 0.79, size = 500, normalized size = 1.45 \begin {gather*} \frac {1155 \, {\left (b^{2} d^{2} e^{4} - 2 \, a b d e^{5} + a^{2} e^{6}\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^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {2295 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} d^{2} e^{4} - 5855 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d^{3} e^{4} + 5153 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{4} e^{4} - 1545 \, \sqrt {x e + d} b^{5} d^{5} e^{4} - 4590 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{4} d e^{5} + 17565 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} d^{2} e^{5} - 20612 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d^{3} e^{5} + 7725 \, \sqrt {x e + d} a b^{4} d^{4} e^{5} + 2295 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{3} e^{6} - 17565 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{3} d e^{6} + 30918 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} d^{2} e^{6} - 15450 \, \sqrt {x e + d} a^{2} b^{3} d^{3} e^{6} + 5855 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{2} e^{7} - 20612 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{2} d e^{7} + 15450 \, \sqrt {x e + d} a^{3} b^{2} d^{2} e^{7} + 5153 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b e^{8} - 7725 \, \sqrt {x e + d} a^{4} b d e^{8} + 1545 \, \sqrt {x e + d} a^{5} e^{9}}{192 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{10} e^{4} + 15 \, \sqrt {x e + d} b^{10} d e^{4} - 15 \, \sqrt {x e + d} a b^{9} e^{5}\right )}}{3 \, b^{15} \mathrm {sgn}\left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1155/64*(b^2*d^2*e^4 - 2*a*b*d*e^5 + a^2*e^6)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*
e)*b^6*sgn(b*x + a)) - 1/192*(2295*(x*e + d)^(7/2)*b^5*d^2*e^4 - 5855*(x*e + d)^(5/2)*b^5*d^3*e^4 + 5153*(x*e
+ d)^(3/2)*b^5*d^4*e^4 - 1545*sqrt(x*e + d)*b^5*d^5*e^4 - 4590*(x*e + d)^(7/2)*a*b^4*d*e^5 + 17565*(x*e + d)^(
5/2)*a*b^4*d^2*e^5 - 20612*(x*e + d)^(3/2)*a*b^4*d^3*e^5 + 7725*sqrt(x*e + d)*a*b^4*d^4*e^5 + 2295*(x*e + d)^(
7/2)*a^2*b^3*e^6 - 17565*(x*e + d)^(5/2)*a^2*b^3*d*e^6 + 30918*(x*e + d)^(3/2)*a^2*b^3*d^2*e^6 - 15450*sqrt(x*
e + d)*a^2*b^3*d^3*e^6 + 5855*(x*e + d)^(5/2)*a^3*b^2*e^7 - 20612*(x*e + d)^(3/2)*a^3*b^2*d*e^7 + 15450*sqrt(x
*e + d)*a^3*b^2*d^2*e^7 + 5153*(x*e + d)^(3/2)*a^4*b*e^8 - 7725*sqrt(x*e + d)*a^4*b*d*e^8 + 1545*sqrt(x*e + d)
*a^5*e^9)/(((x*e + d)*b - b*d + a*e)^4*b^6*sgn(b*x + a)) + 2/3*((x*e + d)^(3/2)*b^10*e^4 + 15*sqrt(x*e + d)*b^
10*d*e^4 - 15*sqrt(x*e + d)*a*b^9*e^5)/(b^15*sgn(b*x + a))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{11/2}}{{\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)^(11/2)/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((d + e*x)^(11/2)/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

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