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

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

________________________________________________________________________________________

Rubi [A]  time = 0.24, 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 = {646, 47, 50, 63, 208} \begin {gather*} -\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 {429 e^3 (d+e x)^{7/2}}{64 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 {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) \sqrt {d+e x} (b d-a e)^2}{64 b^7 \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 {(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}} \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 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)^{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 ) \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^{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*}

________________________________________________________________________________________

Mathematica [C]  time = 0.06, size = 67, normalized size = 0.17 \begin {gather*} -\frac {2 e^4 (a+b x) (d+e x)^{15/2} \, _2F_1\left (5,\frac {15}{2};\frac {17}{2};\frac {b (d+e x)}{b d-a e}\right )}{15 \sqrt {(a+b x)^2} (b d-a e)^5} \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]

(-2*e^4*(a + b*x)*(d + e*x)^(15/2)*Hypergeometric2F1[5, 15/2, 17/2, (b*(d + e*x))/(b*d - a*e)])/(15*(b*d - a*e
)^5*Sqrt[(a + b*x)^2])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 62.45, size = 563, normalized size = 1.41 \begin {gather*} \frac {(-a e-b e x) \left (\frac {3003 e^4 (b d-a e)^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 b^{15/2} \sqrt {a e-b d}}-\frac {e^4 \sqrt {d+e x} \left (45045 a^6 e^6+165165 a^5 b e^5 (d+e x)-270270 a^5 b d e^5+675675 a^4 b^2 d^2 e^4+219219 a^4 b^2 e^4 (d+e x)^2-825825 a^4 b^2 d e^4 (d+e x)-900900 a^3 b^3 d^3 e^3+1651650 a^3 b^3 d^2 e^3 (d+e x)+119691 a^3 b^3 e^3 (d+e x)^3-876876 a^3 b^3 d e^3 (d+e x)^2+675675 a^2 b^4 d^4 e^2-1651650 a^2 b^4 d^3 e^2 (d+e x)+1315314 a^2 b^4 d^2 e^2 (d+e x)^2+18304 a^2 b^4 e^2 (d+e x)^4-359073 a^2 b^4 d e^2 (d+e x)^3-270270 a b^5 d^5 e+825825 a b^5 d^4 e (d+e x)-876876 a b^5 d^3 e (d+e x)^2+359073 a b^5 d^2 e (d+e x)^3-1664 a b^5 e (d+e x)^5-36608 a b^5 d e (d+e x)^4+45045 b^6 d^6-165165 b^6 d^5 (d+e x)+219219 b^6 d^4 (d+e x)^2-119691 b^6 d^3 (d+e x)^3+18304 b^6 d^2 (d+e x)^4+384 b^6 (d+e x)^6+1664 b^6 d (d+e x)^5\right )}{960 b^7 (a e+b (d+e x)-b d)^4}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((-(a*e) - b*e*x)*(-1/960*(e^4*Sqrt[d + e*x]*(45045*b^6*d^6 - 270270*a*b^5*d^5*e + 675675*a^2*b^4*d^4*e^2 - 90
0900*a^3*b^3*d^3*e^3 + 675675*a^4*b^2*d^2*e^4 - 270270*a^5*b*d*e^5 + 45045*a^6*e^6 - 165165*b^6*d^5*(d + e*x)
+ 825825*a*b^5*d^4*e*(d + e*x) - 1651650*a^2*b^4*d^3*e^2*(d + e*x) + 1651650*a^3*b^3*d^2*e^3*(d + e*x) - 82582
5*a^4*b^2*d*e^4*(d + e*x) + 165165*a^5*b*e^5*(d + e*x) + 219219*b^6*d^4*(d + e*x)^2 - 876876*a*b^5*d^3*e*(d +
e*x)^2 + 1315314*a^2*b^4*d^2*e^2*(d + e*x)^2 - 876876*a^3*b^3*d*e^3*(d + e*x)^2 + 219219*a^4*b^2*e^4*(d + e*x)
^2 - 119691*b^6*d^3*(d + e*x)^3 + 359073*a*b^5*d^2*e*(d + e*x)^3 - 359073*a^2*b^4*d*e^2*(d + e*x)^3 + 119691*a
^3*b^3*e^3*(d + e*x)^3 + 18304*b^6*d^2*(d + e*x)^4 - 36608*a*b^5*d*e*(d + e*x)^4 + 18304*a^2*b^4*e^2*(d + e*x)
^4 + 1664*b^6*d*(d + e*x)^5 - 1664*a*b^5*e*(d + e*x)^5 + 384*b^6*(d + e*x)^6))/(b^7*(-(b*d) + a*e + b*(d + e*x
))^4) + (3003*e^4*(b*d - a*e)^3*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(64*b^(15/2)*S
qrt[-(b*d) + a*e])))/(e*Sqrt[(a*e + b*e*x)^2/e^2])

________________________________________________________________________________________

fricas [B]  time = 0.44, size = 1286, normalized size = 3.22

result too large to display

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^4*b^2*d^2*e^4 - 2*a^5*b*d*e^5 + a^6*e^6 + (b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 +
4*(a*b^5*d^2*e^4 - 2*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 6*(a^2*b^4*d^2*e^4 - 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^
2 + 4*(a^3*b^3*d^2*e^4 - 2*a^4*b^2*d*e^5 + a^5*b*e^6)*x)*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*(384*b^6*e^6*x^6 - 240*b^6*d^6 - 520*a*b^5*d^5*e - 1430*a^2*b^
4*d^4*e^2 - 6435*a^3*b^3*d^3*e^3 + 69069*a^4*b^2*d^2*e^4 - 105105*a^5*b*d*e^5 + 45045*a^6*e^6 + 128*(31*b^6*d*
e^5 - 13*a*b^5*e^6)*x^5 + 128*(253*b^6*d^2*e^4 - 351*a*b^5*d*e^5 + 143*a^2*b^4*e^6)*x^4 - (22155*b^6*d^3*e^3 -
196001*a*b^5*d^2*e^4 + 285857*a^2*b^4*d*e^5 - 119691*a^3*b^3*e^6)*x^3 - (7630*b^6*d^4*e^2 + 35945*a*b^5*d^3*e
^3 - 347919*a^2*b^4*d^2*e^4 + 517803*a^3*b^3*d*e^5 - 219219*a^4*b^2*e^6)*x^2 - (1960*b^6*d^5*e + 5460*a*b^5*d^
4*e^2 + 25025*a^2*b^4*d^3*e^3 - 256971*a^3*b^3*d^2*e^4 + 387387*a^4*b^2*d*e^5 - 165165*a^5*b*e^6)*x)*sqrt(e*x
+ 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^4*b^2*d^2*e^4 - 2*a^
5*b*d*e^5 + a^6*e^6 + (b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 - 2*a^2*b^4*d*e^5 + a
^3*b^3*e^6)*x^3 + 6*(a^2*b^4*d^2*e^4 - 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 4*(a^3*b^3*d^2*e^4 - 2*a^4*b^2*d*e
^5 + a^5*b*e^6)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (384*b^6*e
^6*x^6 - 240*b^6*d^6 - 520*a*b^5*d^5*e - 1430*a^2*b^4*d^4*e^2 - 6435*a^3*b^3*d^3*e^3 + 69069*a^4*b^2*d^2*e^4 -
105105*a^5*b*d*e^5 + 45045*a^6*e^6 + 128*(31*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 + 128*(253*b^6*d^2*e^4 - 351*a*b^5
*d*e^5 + 143*a^2*b^4*e^6)*x^4 - (22155*b^6*d^3*e^3 - 196001*a*b^5*d^2*e^4 + 285857*a^2*b^4*d*e^5 - 119691*a^3*
b^3*e^6)*x^3 - (7630*b^6*d^4*e^2 + 35945*a*b^5*d^3*e^3 - 347919*a^2*b^4*d^2*e^4 + 517803*a^3*b^3*d*e^5 - 21921
9*a^4*b^2*e^6)*x^2 - (1960*b^6*d^5*e + 5460*a*b^5*d^4*e^2 + 25025*a^2*b^4*d^3*e^3 - 256971*a^3*b^3*d^2*e^4 + 3
87387*a^4*b^2*d*e^5 - 165165*a^5*b*e^6)*x)*sqrt(e*x + 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)]

________________________________________________________________________________________

giac [B]  time = 0.45, size = 700, normalized size = 1.75 \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 ({\left (x e + d\right )} b e - b d e + a e^{2}\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 ({\left (x e + d\right )} b e - b d e + a e^{2}\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 ({\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)^(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((x*e + d)*b*e - b*d*e + a*e^2)) - 1/192*(4431*(x*e + d)^(7/2)*b^6*d^3*e^4 - 117
67*(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 + 1949
4*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((x*e + d)*b*e - b*d*e + a*e^2)) + 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((x*e + d)*b*e - b*d*e + a*e^2))

________________________________________________________________________________________

maple [B]  time = 0.08, size = 2192, normalized size = 5.48

result too large to display

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)

[Out]

1/960*(58835*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*b^6*d^4-53165*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*b^6*d^5+45045*(
(a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^6*e^6+16245*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*b^6*d^6-22155*((a*e-b*d)*b)^(
1/2)*(e*x+d)^(7/2)*b^6*d^3+12800*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x*a^3*b^3*d*e^4-345600*((a*e-b*d)*b)^(1/2)*
(e*x+d)^(1/2)*x^2*a^3*b^3*d*e^5+172800*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^2*b^4*d^2*e^4-230400*((a*e-b*d)
*b)^(1/2)*(e*x+d)^(1/2)*x*a^4*b^2*d*e^5+115200*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^3*b^3*d^2*e^4-135135*arct
an((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*a*b^6*d^2*e^5+2304*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*x^2*a^2*b^4*e
^4-45045*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^7*e^7+115200*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^3*a*b^
5*d^2*e^4+12800*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^3*a*b^5*d*e^4-57600*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^4*
a*b^5*d*e^5+19200*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^2*a^2*b^4*d*e^4-230400*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)
*x^3*a^2*b^4*d*e^5-3200*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^4*a*b^5*e^5+3200*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)
*x^4*b^6*d*e^4+45045*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*b^7*d^3*e^4-180180*arctan((e*x+d)^(1/2)/(
(a*e-b*d)*b)^(1/2)*b)*x^3*a^4*b^3*e^7+22155*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a^3*b^3*e^3-270270*arctan((e*x+d
)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^5*b^2*e^7+59219*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a^4*b^2*e^4-180180*arct
an((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^6*b*e^7+49965*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^5*b*e^5+135135*a
rctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^6*b*d*e^6-135135*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^5*
b^2*d^2*e^5+45045*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a^4*b^3*d^3*e^4+384*((a*e-b*d)*b)^(1/2)*(e*x+d)^
(5/2)*x^4*b^6*e^4-45045*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*a^3*b^4*e^7+540540*arctan((e*x+d)^(1/2
)/((a*e-b*d)*b)^(1/2)*b)*x^3*a^3*b^4*d*e^6-531650*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b^4*d^3*e^2-12800*((a*
e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^3*a^2*b^4*e^5+1536*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*x^3*a*b^5*e^4+28800*((a*e
-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^4*a^2*b^4*e^6+28800*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^4*b^6*d^2*e^4+810810*ar
ctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^4*b^3*d*e^6-19200*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^2*a^3*b^
3*e^5-810810*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^3*b^4*d^2*e^5+265825*((a*e-b*d)*b)^(1/2)*(e*x+d
)^(3/2)*a*b^5*d^4*e+115200*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^5*b*e^6+115200*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1
/2)*x^3*a^3*b^3*e^6-540540*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a^2*b^5*d^2*e^5+135135*arctan((e*x+
d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*a^2*b^5*d*e^6+180180*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a*b^6
*d^3*e^4-66465*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a^2*b^4*d*e^2+66465*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a*b^5*d
^2*e+1536*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*x*a^3*b^3*e^4+270270*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x
^2*a^2*b^5*d^3*e^4-235340*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a^3*b^3*d*e^3+353010*((a*e-b*d)*b)^(1/2)*(e*x+d)^(
5/2)*a^2*b^4*d^2*e^2-235340*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a*b^5*d^3*e-12800*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3
/2)*x*a^4*b^2*e^5+172800*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^4*b^2*e^6+540540*arctan((e*x+d)^(1/2)/((a*e-b
*d)*b)^(1/2)*b)*x*a^5*b^2*d*e^6-540540*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^4*b^3*d^2*e^5+180180*ar
ctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^3*b^4*d^3*e^4-262625*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^4*b^2*d
*e^4+531650*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^3*b^3*d^2*e^3-155070*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^5*b*d
*e^5+272475*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^4*b^2*d^2*e^4-324900*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^3*b^3
*d^3*e^3+243675*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^2*b^4*d^4*e^2-97470*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^
5*d^5*e)*(b*x+a)/((a*e-b*d)*b)^(1/2)/b^7/((b*x+a)^2)^(5/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + d\right )}^{\frac {13}{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)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________