3.18.20 \(\int \frac {(d+e x)^{13/2}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1720]

Optimal. Leaf size=400 \[ \frac {3003 e^4 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 e^4 (b d-a e)^{5/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

1001/64*e^4*(-a*e+b*d)*(b*x+a)*(e*x+d)^(3/2)/b^6/((b*x+a)^2)^(1/2)+3003/320*e^4*(b*x+a)*(e*x+d)^(5/2)/b^5/((b*
x+a)^2)^(1/2)-429/64*e^3*(e*x+d)^(7/2)/b^4/((b*x+a)^2)^(1/2)-143/96*e^2*(e*x+d)^(9/2)/b^3/(b*x+a)/((b*x+a)^2)^
(1/2)-13/24*e*(e*x+d)^(11/2)/b^2/(b*x+a)^2/((b*x+a)^2)^(1/2)-1/4*(e*x+d)^(13/2)/b/(b*x+a)^3/((b*x+a)^2)^(1/2)-
3003/64*e^4*(-a*e+b*d)^(5/2)*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(15/2)/((b*x+a)^2)^(1/2
)+3003/64*e^4*(-a*e+b*d)^2*(b*x+a)*(e*x+d)^(1/2)/b^7/((b*x+a)^2)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 10, 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)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 e^4 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) \sqrt {d+e x} (b d-a e)^2}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (a+b x) (d+e x)^{3/2} (b d-a e)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 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)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(3003*e^4*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x])/(64*b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (1001*e^4*(b*d - a*e
)*(a + b*x)*(d + e*x)^(3/2))/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3003*e^4*(a + b*x)*(d + e*x)^(5/2))/(32
0*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (429*e^3*(d + e*x)^(7/2))/(64*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (143
*e^2*(d + e*x)^(9/2))/(96*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (13*e*(d + e*x)^(11/2))/(24*b^2*(a +
b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^(13/2)/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (30
03*e^4*(b*d - a*e)^(5/2)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(15/2)*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)^{13/2}}{\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)^{13/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (13 b^2 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{11/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (143 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^3} \, dx}{48 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (429 e^3 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{a b+b^2 x} \, dx}{128 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^4 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{128 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^4 \left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{128 b^8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3003 e^4 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^4 \left (b^2 d-a b e\right )^3 \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^{10} \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3003 e^4 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^3 \left (b^2 d-a b e\right )^3 \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^{10} \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3003 e^4 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{64 b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 e^4 (b d-a e) (a+b x) (d+e x)^{3/2}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 e^4 (b d-a e)^{5/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.47, size = 374, normalized size = 0.94 \begin {gather*} \frac {e^4 (a+b x)^5 \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (-45045 a^6 e^6+15015 a^5 b e^5 (7 d-11 e x)-3003 a^4 b^2 e^4 \left (23 d^2-129 d e x+73 e^2 x^2\right )+429 a^3 b^3 e^3 \left (15 d^3-599 d^2 e x+1207 d e^2 x^2-279 e^3 x^3\right )+143 a^2 b^4 e^2 \left (10 d^4+175 d^3 e x-2433 d^2 e^2 x^2+1999 d e^3 x^3-128 e^4 x^4\right )+13 a b^5 e \left (40 d^5+420 d^4 e x+2765 d^3 e^2 x^2-15077 d^2 e^3 x^3+3456 d e^4 x^4+128 e^5 x^5\right )+b^6 \left (240 d^6+1960 d^5 e x+7630 d^4 e^2 x^2+22155 d^3 e^3 x^3-32384 d^2 e^4 x^4-3968 d e^5 x^5-384 e^6 x^6\right )\right )}{e^4 (a+b x)^4}-45045 (-b d+a e)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{960 b^{15/2} \left ((a+b x)^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(13/2)/(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]*(-45045*a^6*e^6 + 15015*a^5*b*e^5*(7*d - 11*e*x) - 3003*a^4*b^2*e^4
*(23*d^2 - 129*d*e*x + 73*e^2*x^2) + 429*a^3*b^3*e^3*(15*d^3 - 599*d^2*e*x + 1207*d*e^2*x^2 - 279*e^3*x^3) + 1
43*a^2*b^4*e^2*(10*d^4 + 175*d^3*e*x - 2433*d^2*e^2*x^2 + 1999*d*e^3*x^3 - 128*e^4*x^4) + 13*a*b^5*e*(40*d^5 +
420*d^4*e*x + 2765*d^3*e^2*x^2 - 15077*d^2*e^3*x^3 + 3456*d*e^4*x^4 + 128*e^5*x^5) + b^6*(240*d^6 + 1960*d^5*
e*x + 7630*d^4*e^2*x^2 + 22155*d^3*e^3*x^3 - 32384*d^2*e^4*x^4 - 3968*d*e^5*x^5 - 384*e^6*x^6)))/(e^4*(a + b*x
)^4)) - 45045*(-(b*d) + a*e)^(5/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]]))/(960*b^(15/2)*((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. \(2191\) vs. \(2(272)=544\).
time = 0.77, size = 2192, normalized size = 5.48

method	result	size
risch	\(\frac {2 e^{4} \left (3 b^{2} x^{2} e^{2}-25 a b \,e^{2} x +31 b^{2} d e x +225 a^{2} e^{2}-475 a b d e +253 b^{2} d^{2}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{15 b^{7} \left (b x +a \right )}+\frac {\left (\frac {1477 e^{7} \left (e x +d \right )^{\frac {7}{2}} a^{3}}{64 b^{4} \left (b e x +a e \right )^{4}}-\frac {9009 e^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,d^{2}}{64 b^{5} \sqrt {b \left (a e -b d \right )}}-\frac {4431 e^{6} \left (e x +d \right )^{\frac {7}{2}} a^{2} d}{64 b^{3} \left (b e x +a e \right )^{4}}+\frac {4431 e^{5} \left (e x +d \right )^{\frac {7}{2}} a \,d^{2}}{64 b^{2} \left (b e x +a e \right )^{4}}-\frac {11767 e^{7} \left (e x +d \right )^{\frac {5}{2}} a^{3} d}{48 b^{4} \left (b e x +a e \right )^{4}}-\frac {53165 e^{8} \left (e x +d \right )^{\frac {3}{2}} a^{4} d}{192 b^{5} \left (b e x +a e \right )^{4}}+\frac {53165 e^{7} \left (e x +d \right )^{\frac {3}{2}} a^{3} d^{2}}{96 b^{4} \left (b e x +a e \right )^{4}}-\frac {53165 e^{6} \left (e x +d \right )^{\frac {3}{2}} a^{2} d^{3}}{96 b^{3} \left (b e x +a e \right )^{4}}+\frac {53165 e^{5} \left (e x +d \right )^{\frac {3}{2}} a \,d^{4}}{192 b^{2} \left (b e x +a e \right )^{4}}+\frac {11767 e^{6} \left (e x +d \right )^{\frac {5}{2}} a^{2} d^{2}}{32 b^{3} \left (b e x +a e \right )^{4}}-\frac {11767 e^{5} \left (e x +d \right )^{\frac {5}{2}} a \,d^{3}}{48 b^{2} \left (b e x +a e \right )^{4}}-\frac {3249 e^{9} \sqrt {e x +d}\, a^{5} d}{32 b^{6} \left (b e x +a e \right )^{4}}+\frac {16245 e^{8} \sqrt {e x +d}\, a^{4} d^{2}}{64 b^{5} \left (b e x +a e \right )^{4}}-\frac {5415 e^{7} \sqrt {e x +d}\, a^{3} d^{3}}{16 b^{4} \left (b e x +a e \right )^{4}}+\frac {16245 e^{6} \sqrt {e x +d}\, a^{2} d^{4}}{64 b^{3} \left (b e x +a e \right )^{4}}-\frac {3249 e^{5} \sqrt {e x +d}\, a \,d^{5}}{32 b^{2} \left (b e x +a e \right )^{4}}+\frac {3003 e^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) d^{3}}{64 b^{4} \sqrt {b \left (a e -b d \right )}}+\frac {11767 e^{8} \left (e x +d \right )^{\frac {5}{2}} a^{4}}{192 b^{5} \left (b e x +a e \right )^{4}}+\frac {10633 e^{9} \left (e x +d \right )^{\frac {3}{2}} a^{5}}{192 b^{6} \left (b e x +a e \right )^{4}}-\frac {1477 e^{4} \left (e x +d \right )^{\frac {7}{2}} d^{3}}{64 b \left (b e x +a e \right )^{4}}+\frac {11767 e^{4} \left (e x +d \right )^{\frac {5}{2}} d^{4}}{192 b \left (b e x +a e \right )^{4}}-\frac {10633 e^{4} \left (e x +d \right )^{\frac {3}{2}} d^{5}}{192 b \left (b e x +a e \right )^{4}}+\frac {1083 e^{10} \sqrt {e x +d}\, a^{6}}{64 b^{7} \left (b e x +a e \right )^{4}}+\frac {1083 e^{4} \sqrt {e x +d}\, d^{6}}{64 b \left (b e x +a e \right )^{4}}-\frac {3003 e^{7} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3}}{64 b^{7} \sqrt {b \left (a e -b d \right )}}+\frac {9009 e^{6} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} d}{64 b^{6} \sqrt {b \left (a e -b d \right )}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\)	\(925\)
default	\(\text {Expression too large to display}\)	\(2192\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/960*(1536*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^5*e^4*x^3-3200*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^5*e^5*x
^4+3200*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^6*d*e^4*x^4+135135*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^2
*b^5*d*e^6*x^4-135135*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a*b^6*d^2*e^5*x^4+2304*(b*(a*e-b*d))^(1/2)*(
e*x+d)^(5/2)*a^2*b^4*e^4*x^2-12800*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b^4*e^5*x^3+12800*(b*(a*e-b*d))^(1/2)
*(e*x+d)^(3/2)*a*b^5*d*e^4*x^3-57600*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^5*d*e^5*x^4+19200*(b*(a*e-b*d))^(1/
2)*(e*x+d)^(3/2)*a^2*b^4*d*e^4*x^2+45045*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*b^7*d^3*e^4*x^4-180180*ar
ctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^4*b^3*e^7*x^3+22155*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*a^3*b^3*e^3-
270270*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^5*b^2*e^7*x^2+59219*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a^4
*b^2*e^4-180180*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^6*b*e^7*x+49965*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2
)*a^5*b*e^5+135135*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^6*b*d*e^6-135135*arctan(b*(e*x+d)^(1/2)/(b*(a
*e-b*d))^(1/2))*a^5*b^2*d^2*e^5+45045*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^4*b^3*d^3*e^4-810810*arcta
n(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^3*b^4*d^2*e^5*x^2+270270*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*
a^2*b^5*d^3*e^4*x^2-235340*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a^3*b^3*d*e^3+353010*(b*(a*e-b*d))^(1/2)*(e*x+d)^
(5/2)*a^2*b^4*d^2*e^2-235340*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^5*d^3*e-12800*(b*(a*e-b*d))^(1/2)*(e*x+d)^(
3/2)*a^4*b^2*e^5*x+172800*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^4*b^2*e^6*x^2+540540*arctan(b*(e*x+d)^(1/2)/(b*(
a*e-b*d))^(1/2))*a^5*b^2*d*e^6*x-22155*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*b^6*d^3+58835*(b*(a*e-b*d))^(1/2)*(e*
x+d)^(5/2)*b^6*d^4-53165*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^6*d^5+45045*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^6
*e^6+16245*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^6*d^6-45045*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^7*e^7
-230400*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^4*d*e^5*x^3+115200*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^5*d^2
*e^4*x^3+12800*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^3*b^3*d*e^4*x-345600*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*
b^3*d*e^5*x^2+172800*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^4*d^2*e^4*x^2-230400*(b*(a*e-b*d))^(1/2)*(e*x+d)^
(1/2)*a^4*b^2*d*e^5*x+115200*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*b^3*d^2*e^4*x+28800*(b*(a*e-b*d))^(1/2)*(e*
x+d)^(1/2)*a^2*b^4*e^6*x^4+28800*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^6*d^2*e^4*x^4+540540*arctan(b*(e*x+d)^(1/
2)/(b*(a*e-b*d))^(1/2))*a^3*b^4*d*e^6*x^3-540540*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^2*b^5*d^2*e^5*x
^3+180180*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a*b^6*d^3*e^4*x^3-66465*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2
)*a^2*b^4*d*e^2+66465*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*a*b^5*d^2*e+1536*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a^3
*b^3*e^4*x-19200*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^3*b^3*e^5*x^2+115200*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^
3*b^3*e^6*x^3+810810*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^4*b^3*d*e^6*x^2-540540*arctan(b*(e*x+d)^(1/
2)/(b*(a*e-b*d))^(1/2))*a^4*b^3*d^2*e^5*x+180180*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^3*b^4*d^3*e^4*x
-262625*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^4*b^2*d*e^4+531650*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^3*b^3*d^2*e
^3-531650*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b^4*d^3*e^2+265825*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^5*d^4
*e+115200*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^5*b*e^6*x-155070*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^5*b*d*e^5+2
72475*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^4*b^2*d^2*e^4-324900*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*b^3*d^3*e
^3+243675*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^4*d^4*e^2-97470*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^5*d^5*
e+384*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^6*e^4*x^4-45045*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^3*b^4*
e^7*x^4)*(b*x+a)/(b*(a*e-b*d))^(1/2)/b^7/((b*x+a)^2)^(5/2)

________________________________________________________________________________________

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)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((x*e + d)^(13/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 595 vs. \(2 (279) = 558\).
time = 2.22, size = 1202, normalized size = 3.00 \begin {gather*} \left [\frac {45045 \, {\left ({\left (a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + 6 \, a^{4} b^{2} x^{2} + 4 \, a^{5} b x + a^{6}\right )} e^{6} - 2 \, {\left (a b^{5} d x^{4} + 4 \, a^{2} b^{4} d x^{3} + 6 \, a^{3} b^{3} d x^{2} + 4 \, a^{4} b^{2} d x + a^{5} b d\right )} e^{5} + {\left (b^{6} d^{2} x^{4} + 4 \, a b^{5} d^{2} x^{3} + 6 \, a^{2} b^{4} d^{2} x^{2} + 4 \, a^{3} b^{3} d^{2} x + a^{4} b^{2} d^{2}\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 (240 \, b^{6} d^{6} - {\left (384 \, b^{6} x^{6} - 1664 \, a b^{5} x^{5} + 18304 \, a^{2} b^{4} x^{4} + 119691 \, a^{3} b^{3} x^{3} + 219219 \, a^{4} b^{2} x^{2} + 165165 \, a^{5} b x + 45045 \, a^{6}\right )} e^{6} - {\left (3968 \, b^{6} d x^{5} - 44928 \, a b^{5} d x^{4} - 285857 \, a^{2} b^{4} d x^{3} - 517803 \, a^{3} b^{3} d x^{2} - 387387 \, a^{4} b^{2} d x - 105105 \, a^{5} b d\right )} e^{5} - {\left (32384 \, b^{6} d^{2} x^{4} + 196001 \, a b^{5} d^{2} x^{3} + 347919 \, a^{2} b^{4} d^{2} x^{2} + 256971 \, a^{3} b^{3} d^{2} x + 69069 \, a^{4} b^{2} d^{2}\right )} e^{4} + 5 \, {\left (4431 \, b^{6} d^{3} x^{3} + 7189 \, a b^{5} d^{3} x^{2} + 5005 \, a^{2} b^{4} d^{3} x + 1287 \, a^{3} b^{3} d^{3}\right )} e^{3} + 10 \, {\left (763 \, b^{6} d^{4} x^{2} + 546 \, a b^{5} d^{4} x + 143 \, a^{2} b^{4} d^{4}\right )} e^{2} + 40 \, {\left (49 \, b^{6} d^{5} x + 13 \, a b^{5} d^{5}\right )} e\right )} \sqrt {x e + d}}{1920 \, {\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}}, -\frac {45045 \, {\left ({\left (a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + 6 \, a^{4} b^{2} x^{2} + 4 \, a^{5} b x + a^{6}\right )} e^{6} - 2 \, {\left (a b^{5} d x^{4} + 4 \, a^{2} b^{4} d x^{3} + 6 \, a^{3} b^{3} d x^{2} + 4 \, a^{4} b^{2} d x + a^{5} b d\right )} e^{5} + {\left (b^{6} d^{2} x^{4} + 4 \, a b^{5} d^{2} x^{3} + 6 \, a^{2} b^{4} d^{2} x^{2} + 4 \, a^{3} b^{3} d^{2} x + a^{4} b^{2} d^{2}\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 (240 \, b^{6} d^{6} - {\left (384 \, b^{6} x^{6} - 1664 \, a b^{5} x^{5} + 18304 \, a^{2} b^{4} x^{4} + 119691 \, a^{3} b^{3} x^{3} + 219219 \, a^{4} b^{2} x^{2} + 165165 \, a^{5} b x + 45045 \, a^{6}\right )} e^{6} - {\left (3968 \, b^{6} d x^{5} - 44928 \, a b^{5} d x^{4} - 285857 \, a^{2} b^{4} d x^{3} - 517803 \, a^{3} b^{3} d x^{2} - 387387 \, a^{4} b^{2} d x - 105105 \, a^{5} b d\right )} e^{5} - {\left (32384 \, b^{6} d^{2} x^{4} + 196001 \, a b^{5} d^{2} x^{3} + 347919 \, a^{2} b^{4} d^{2} x^{2} + 256971 \, a^{3} b^{3} d^{2} x + 69069 \, a^{4} b^{2} d^{2}\right )} e^{4} + 5 \, {\left (4431 \, b^{6} d^{3} x^{3} + 7189 \, a b^{5} d^{3} x^{2} + 5005 \, a^{2} b^{4} d^{3} x + 1287 \, a^{3} b^{3} d^{3}\right )} e^{3} + 10 \, {\left (763 \, b^{6} d^{4} x^{2} + 546 \, a b^{5} d^{4} x + 143 \, a^{2} b^{4} d^{4}\right )} e^{2} + 40 \, {\left (49 \, b^{6} d^{5} x + 13 \, a b^{5} d^{5}\right )} e\right )} \sqrt {x e + d}}{960 \, {\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/1920*(45045*((a^2*b^4*x^4 + 4*a^3*b^3*x^3 + 6*a^4*b^2*x^2 + 4*a^5*b*x + a^6)*e^6 - 2*(a*b^5*d*x^4 + 4*a^2*b
^4*d*x^3 + 6*a^3*b^3*d*x^2 + 4*a^4*b^2*d*x + a^5*b*d)*e^5 + (b^6*d^2*x^4 + 4*a*b^5*d^2*x^3 + 6*a^2*b^4*d^2*x^2
+ 4*a^3*b^3*d^2*x + a^4*b^2*d^2)*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*(240*b^6*d^6 - (384*b^6*x^6 - 1664*a*b^5*x^5 + 18304*a^2*b^4*x^4 + 119691*a^3*b^
3*x^3 + 219219*a^4*b^2*x^2 + 165165*a^5*b*x + 45045*a^6)*e^6 - (3968*b^6*d*x^5 - 44928*a*b^5*d*x^4 - 285857*a^
2*b^4*d*x^3 - 517803*a^3*b^3*d*x^2 - 387387*a^4*b^2*d*x - 105105*a^5*b*d)*e^5 - (32384*b^6*d^2*x^4 + 196001*a*
b^5*d^2*x^3 + 347919*a^2*b^4*d^2*x^2 + 256971*a^3*b^3*d^2*x + 69069*a^4*b^2*d^2)*e^4 + 5*(4431*b^6*d^3*x^3 + 7
189*a*b^5*d^3*x^2 + 5005*a^2*b^4*d^3*x + 1287*a^3*b^3*d^3)*e^3 + 10*(763*b^6*d^4*x^2 + 546*a*b^5*d^4*x + 143*a
^2*b^4*d^4)*e^2 + 40*(49*b^6*d^5*x + 13*a*b^5*d^5)*e)*sqrt(x*e + d))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2
+ 4*a^3*b^8*x + a^4*b^7), -1/960*(45045*((a^2*b^4*x^4 + 4*a^3*b^3*x^3 + 6*a^4*b^2*x^2 + 4*a^5*b*x + a^6)*e^6 -
2*(a*b^5*d*x^4 + 4*a^2*b^4*d*x^3 + 6*a^3*b^3*d*x^2 + 4*a^4*b^2*d*x + a^5*b*d)*e^5 + (b^6*d^2*x^4 + 4*a*b^5*d^
2*x^3 + 6*a^2*b^4*d^2*x^2 + 4*a^3*b^3*d^2*x + a^4*b^2*d^2)*e^4)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(x*e + d)*b*s
qrt(-(b*d - a*e)/b)/(b*d - a*e)) + (240*b^6*d^6 - (384*b^6*x^6 - 1664*a*b^5*x^5 + 18304*a^2*b^4*x^4 + 119691*a
^3*b^3*x^3 + 219219*a^4*b^2*x^2 + 165165*a^5*b*x + 45045*a^6)*e^6 - (3968*b^6*d*x^5 - 44928*a*b^5*d*x^4 - 2858
57*a^2*b^4*d*x^3 - 517803*a^3*b^3*d*x^2 - 387387*a^4*b^2*d*x - 105105*a^5*b*d)*e^5 - (32384*b^6*d^2*x^4 + 1960
01*a*b^5*d^2*x^3 + 347919*a^2*b^4*d^2*x^2 + 256971*a^3*b^3*d^2*x + 69069*a^4*b^2*d^2)*e^4 + 5*(4431*b^6*d^3*x^
3 + 7189*a*b^5*d^3*x^2 + 5005*a^2*b^4*d^3*x + 1287*a^3*b^3*d^3)*e^3 + 10*(763*b^6*d^4*x^2 + 546*a*b^5*d^4*x +
143*a^2*b^4*d^4)*e^2 + 40*(49*b^6*d^5*x + 13*a*b^5*d^5)*e)*sqrt(x*e + d))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9
*x^2 + 4*a^3*b^8*x + a^4*b^7)]

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(13/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3655 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (279) = 558\).
time = 1.00, size = 652, normalized size = 1.63 \begin {gather*} \frac {3003 \, {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\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^{7} \mathrm {sgn}\left (b x + a\right )} - \frac {4431 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{6} d^{3} e^{4} - 11767 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} d^{4} e^{4} + 10633 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} d^{5} e^{4} - 3249 \, \sqrt {x e + d} b^{6} d^{6} e^{4} - 13293 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{5} d^{2} e^{5} + 47068 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{5} d^{3} e^{5} - 53165 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{5} d^{4} e^{5} + 19494 \, \sqrt {x e + d} a b^{5} d^{5} e^{5} + 13293 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{4} d e^{6} - 70602 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{4} d^{2} e^{6} + 106330 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{4} d^{3} e^{6} - 48735 \, \sqrt {x e + d} a^{2} b^{4} d^{4} e^{6} - 4431 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{3} b^{3} e^{7} + 47068 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{3} d e^{7} - 106330 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{3} d^{2} e^{7} + 64980 \, \sqrt {x e + d} a^{3} b^{3} d^{3} e^{7} - 11767 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{4} b^{2} e^{8} + 53165 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b^{2} d e^{8} - 48735 \, \sqrt {x e + d} a^{4} b^{2} d^{2} e^{8} - 10633 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{5} b e^{9} + 19494 \, \sqrt {x e + d} a^{5} b d e^{9} - 3249 \, \sqrt {x e + d} a^{6} e^{10}}{192 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{20} e^{4} + 25 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{20} d e^{4} + 225 \, \sqrt {x e + d} b^{20} d^{2} e^{4} - 25 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{19} e^{5} - 450 \, \sqrt {x e + d} a b^{19} d e^{5} + 225 \, \sqrt {x e + d} a^{2} b^{18} e^{6}\right )}}{15 \, b^{25} \mathrm {sgn}\left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

3003/64*(b^3*d^3*e^4 - 3*a*b^2*d^2*e^5 + 3*a^2*b*d*e^6 - a^3*e^7)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))
/(sqrt(-b^2*d + a*b*e)*b^7*sgn(b*x + a)) - 1/192*(4431*(x*e + d)^(7/2)*b^6*d^3*e^4 - 11767*(x*e + d)^(5/2)*b^6
*d^4*e^4 + 10633*(x*e + d)^(3/2)*b^6*d^5*e^4 - 3249*sqrt(x*e + d)*b^6*d^6*e^4 - 13293*(x*e + d)^(7/2)*a*b^5*d^
2*e^5 + 47068*(x*e + d)^(5/2)*a*b^5*d^3*e^5 - 53165*(x*e + d)^(3/2)*a*b^5*d^4*e^5 + 19494*sqrt(x*e + d)*a*b^5*
d^5*e^5 + 13293*(x*e + d)^(7/2)*a^2*b^4*d*e^6 - 70602*(x*e + d)^(5/2)*a^2*b^4*d^2*e^6 + 106330*(x*e + d)^(3/2)
*a^2*b^4*d^3*e^6 - 48735*sqrt(x*e + d)*a^2*b^4*d^4*e^6 - 4431*(x*e + d)^(7/2)*a^3*b^3*e^7 + 47068*(x*e + d)^(5
/2)*a^3*b^3*d*e^7 - 106330*(x*e + d)^(3/2)*a^3*b^3*d^2*e^7 + 64980*sqrt(x*e + d)*a^3*b^3*d^3*e^7 - 11767*(x*e
+ d)^(5/2)*a^4*b^2*e^8 + 53165*(x*e + d)^(3/2)*a^4*b^2*d*e^8 - 48735*sqrt(x*e + d)*a^4*b^2*d^2*e^8 - 10633*(x*
e + d)^(3/2)*a^5*b*e^9 + 19494*sqrt(x*e + d)*a^5*b*d*e^9 - 3249*sqrt(x*e + d)*a^6*e^10)/(((x*e + d)*b - b*d +
a*e)^4*b^7*sgn(b*x + a)) + 2/15*(3*(x*e + d)^(5/2)*b^20*e^4 + 25*(x*e + d)^(3/2)*b^20*d*e^4 + 225*sqrt(x*e + d
)*b^20*d^2*e^4 - 25*(x*e + d)^(3/2)*a*b^19*e^5 - 450*sqrt(x*e + d)*a*b^19*d*e^5 + 225*sqrt(x*e + d)*a^2*b^18*e
^6)/(b^25*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 )}^{13/2}}{{\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)^(13/2)/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((d + e*x)^(13/2)/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

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