3.15.32 \(\int \frac {1}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=435 \[ \frac {3003 b^2 e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^7}+\frac {1001 b e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}+\frac {3003 e^4 (a+b x)}{320 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}+\frac {429 e^3}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}-\frac {143 e^2}{96 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}+\frac {13 e}{24 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac {1}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}-\frac {3003 b^{5/2} e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{15/2}} \]

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Rubi [A]  time = 0.26, antiderivative size = 435, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {646, 51, 63, 208} \begin {gather*} \frac {3003 b^2 e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^7}+\frac {1001 b e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}+\frac {3003 e^4 (a+b x)}{320 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}+\frac {429 e^3}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}-\frac {143 e^2}{96 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac {3003 b^{5/2} e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{15/2}}+\frac {13 e}{24 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac {1}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(429*e^3)/(64*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(4*(b*d - a*e)*(a + b*x)^3*(d +
e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (13*e)/(24*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2]) - (143*e^2)/(96*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3
003*e^4*(a + b*x))/(320*(b*d - a*e)^5*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (1001*b*e^4*(a + b*x))/
(64*(b*d - a*e)^6*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3003*b^2*e^4*(a + b*x))/(64*(b*d - a*e)^7*
Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3003*b^(5/2)*e^4*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqr
t[b*d - a*e]])/(64*(b*d - a*e)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 {1}{(d+e x)^{7/2} \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 {1}{\left (a b+b^2 x\right )^5 (d+e x)^{7/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (13 b^3 e \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^4 (d+e x)^{7/2}} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (143 b^2 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 (d+e x)^{7/2}} \, dx}{48 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (429 b e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 (d+e x)^{7/2}} \, dx}{64 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \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 {1}{\left (a b+b^2 x\right ) (d+e x)^{7/2}} \, dx}{128 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 b e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{5/2}} \, dx}{128 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 b^2 e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{128 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 b^2 e^4 (a+b x)}{64 (b d-a e)^7 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 b^3 e^4 \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 d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 b^2 e^4 (a+b x)}{64 (b d-a e)^7 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 b^3 e^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 d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 b^2 e^4 (a+b x)}{64 (b d-a e)^7 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 b^{5/2} e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 67, normalized size = 0.15 \begin {gather*} \frac {2 e^4 (a+b x) \, _2F_1\left (-\frac {5}{2},5;-\frac {3}{2};\frac {b (d+e x)}{b d-a e}\right )}{5 \sqrt {(a+b x)^2} (d+e x)^{5/2} (b d-a e)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(2*e^4*(a + b*x)*Hypergeometric2F1[-5/2, 5, -3/2, (b*(d + e*x))/(b*d - a*e)])/(5*(b*d - a*e)^5*Sqrt[(a + b*x)^
2]*(d + e*x)^(5/2))

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IntegrateAlgebraic [A]  time = 72.02, size = 571, normalized size = 1.31 \begin {gather*} \frac {(-a e-b e x) \left (\frac {3003 b^{5/2} e^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 (b d-a e)^7 \sqrt {a e-b d}}-\frac {e^4 \left (384 a^6 e^6-1664 a^5 b e^5 (d+e x)-2304 a^5 b d e^5+5760 a^4 b^2 d^2 e^4+18304 a^4 b^2 e^4 (d+e x)^2+8320 a^4 b^2 d e^4 (d+e x)-7680 a^3 b^3 d^3 e^3-16640 a^3 b^3 d^2 e^3 (d+e x)+119691 a^3 b^3 e^3 (d+e x)^3-73216 a^3 b^3 d e^3 (d+e x)^2+5760 a^2 b^4 d^4 e^2+16640 a^2 b^4 d^3 e^2 (d+e x)+109824 a^2 b^4 d^2 e^2 (d+e x)^2+219219 a^2 b^4 e^2 (d+e x)^4-359073 a^2 b^4 d e^2 (d+e x)^3-2304 a b^5 d^5 e-8320 a b^5 d^4 e (d+e x)-73216 a b^5 d^3 e (d+e x)^2+359073 a b^5 d^2 e (d+e x)^3+165165 a b^5 e (d+e x)^5-438438 a b^5 d e (d+e x)^4+384 b^6 d^6+1664 b^6 d^5 (d+e x)+18304 b^6 d^4 (d+e x)^2-119691 b^6 d^3 (d+e x)^3+219219 b^6 d^2 (d+e x)^4+45045 b^6 (d+e x)^6-165165 b^6 d (d+e x)^5\right )}{960 (d+e x)^{5/2} (b d-a e)^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[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

((-(a*e) - b*e*x)*(-1/960*(e^4*(384*b^6*d^6 - 2304*a*b^5*d^5*e + 5760*a^2*b^4*d^4*e^2 - 7680*a^3*b^3*d^3*e^3 +
5760*a^4*b^2*d^2*e^4 - 2304*a^5*b*d*e^5 + 384*a^6*e^6 + 1664*b^6*d^5*(d + e*x) - 8320*a*b^5*d^4*e*(d + e*x) +
16640*a^2*b^4*d^3*e^2*(d + e*x) - 16640*a^3*b^3*d^2*e^3*(d + e*x) + 8320*a^4*b^2*d*e^4*(d + e*x) - 1664*a^5*b
*e^5*(d + e*x) + 18304*b^6*d^4*(d + e*x)^2 - 73216*a*b^5*d^3*e*(d + e*x)^2 + 109824*a^2*b^4*d^2*e^2*(d + e*x)^
2 - 73216*a^3*b^3*d*e^3*(d + e*x)^2 + 18304*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 + 219219*b^6*d^2*(d +
e*x)^4 - 438438*a*b^5*d*e*(d + e*x)^4 + 219219*a^2*b^4*e^2*(d + e*x)^4 - 165165*b^6*d*(d + e*x)^5 + 165165*a*b
^5*e*(d + e*x)^5 + 45045*b^6*(d + e*x)^6))/((b*d - a*e)^7*(d + e*x)^(5/2)*(b*d - a*e - b*(d + e*x))^4) + (3003
*b^(5/2)*e^4*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(64*(b*d - a*e)^7*Sqrt[-(b*d) + a
*e])))/(e*Sqrt[(a*e + b*e*x)^2/e^2])

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fricas [B]  time = 0.51, size = 3368, normalized size = 7.74

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/1920*(45045*(b^6*e^7*x^7 + a^4*b^2*d^3*e^4 + (3*b^6*d*e^6 + 4*a*b^5*e^7)*x^6 + 3*(b^6*d^2*e^5 + 4*a*b^5*d*
e^6 + 2*a^2*b^4*e^7)*x^5 + (b^6*d^3*e^4 + 12*a*b^5*d^2*e^5 + 18*a^2*b^4*d*e^6 + 4*a^3*b^3*e^7)*x^4 + (4*a*b^5*
d^3*e^4 + 18*a^2*b^4*d^2*e^5 + 12*a^3*b^3*d*e^6 + a^4*b^2*e^7)*x^3 + 3*(2*a^2*b^4*d^3*e^4 + 4*a^3*b^3*d^2*e^5
+ a^4*b^2*d*e^6)*x^2 + (4*a^3*b^3*d^3*e^4 + 3*a^4*b^2*d^2*e^5)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e
+ 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) - 2*(45045*b^6*e^6*x^6 - 240*b^6*d^6 + 1960*a*b
^5*d^5*e - 7630*a^2*b^4*d^4*e^2 + 22155*a^3*b^3*d^3*e^3 + 32384*a^4*b^2*d^2*e^4 - 3968*a^5*b*d*e^5 + 384*a^6*e
^6 + 15015*(7*b^6*d*e^5 + 11*a*b^5*e^6)*x^5 + 3003*(23*b^6*d^2*e^4 + 129*a*b^5*d*e^5 + 73*a^2*b^4*e^6)*x^4 + 4
29*(15*b^6*d^3*e^3 + 599*a*b^5*d^2*e^4 + 1207*a^2*b^4*d*e^5 + 279*a^3*b^3*e^6)*x^3 - 143*(10*b^6*d^4*e^2 - 175
*a*b^5*d^3*e^3 - 2433*a^2*b^4*d^2*e^4 - 1999*a^3*b^3*d*e^5 - 128*a^4*b^2*e^6)*x^2 + 13*(40*b^6*d^5*e - 420*a*b
^5*d^4*e^2 + 2765*a^2*b^4*d^3*e^3 + 15077*a^3*b^3*d^2*e^4 + 3456*a^4*b^2*d*e^5 - 128*a^5*b*e^6)*x)*sqrt(e*x +
d))/(a^4*b^7*d^10 - 7*a^5*b^6*d^9*e + 21*a^6*b^5*d^8*e^2 - 35*a^7*b^4*d^7*e^3 + 35*a^8*b^3*d^6*e^4 - 21*a^9*b^
2*d^5*e^5 + 7*a^10*b*d^4*e^6 - a^11*d^3*e^7 + (b^11*d^7*e^3 - 7*a*b^10*d^6*e^4 + 21*a^2*b^9*d^5*e^5 - 35*a^3*b
^8*d^4*e^6 + 35*a^4*b^7*d^3*e^7 - 21*a^5*b^6*d^2*e^8 + 7*a^6*b^5*d*e^9 - a^7*b^4*e^10)*x^7 + (3*b^11*d^8*e^2 -
17*a*b^10*d^7*e^3 + 35*a^2*b^9*d^6*e^4 - 21*a^3*b^8*d^5*e^5 - 35*a^4*b^7*d^4*e^6 + 77*a^5*b^6*d^3*e^7 - 63*a^
6*b^5*d^2*e^8 + 25*a^7*b^4*d*e^9 - 4*a^8*b^3*e^10)*x^6 + 3*(b^11*d^9*e - 3*a*b^10*d^8*e^2 - 5*a^2*b^9*d^7*e^3
+ 35*a^3*b^8*d^6*e^4 - 63*a^4*b^7*d^5*e^5 + 49*a^5*b^6*d^4*e^6 - 7*a^6*b^5*d^3*e^7 - 15*a^7*b^4*d^2*e^8 + 10*a
^8*b^3*d*e^9 - 2*a^9*b^2*e^10)*x^5 + (b^11*d^10 + 5*a*b^10*d^9*e - 45*a^2*b^9*d^8*e^2 + 95*a^3*b^8*d^7*e^3 - 3
5*a^4*b^7*d^6*e^4 - 147*a^5*b^6*d^5*e^5 + 245*a^6*b^5*d^4*e^6 - 155*a^7*b^4*d^3*e^7 + 30*a^8*b^3*d^2*e^8 + 10*
a^9*b^2*d*e^9 - 4*a^10*b*e^10)*x^4 + (4*a*b^10*d^10 - 10*a^2*b^9*d^9*e - 30*a^3*b^8*d^8*e^2 + 155*a^4*b^7*d^7*
e^3 - 245*a^5*b^6*d^6*e^4 + 147*a^6*b^5*d^5*e^5 + 35*a^7*b^4*d^4*e^6 - 95*a^8*b^3*d^3*e^7 + 45*a^9*b^2*d^2*e^8
- 5*a^10*b*d*e^9 - a^11*e^10)*x^3 + 3*(2*a^2*b^9*d^10 - 10*a^3*b^8*d^9*e + 15*a^4*b^7*d^8*e^2 + 7*a^5*b^6*d^7
*e^3 - 49*a^6*b^5*d^6*e^4 + 63*a^7*b^4*d^5*e^5 - 35*a^8*b^3*d^4*e^6 + 5*a^9*b^2*d^3*e^7 + 3*a^10*b*d^2*e^8 - a
^11*d*e^9)*x^2 + (4*a^3*b^8*d^10 - 25*a^4*b^7*d^9*e + 63*a^5*b^6*d^8*e^2 - 77*a^6*b^5*d^7*e^3 + 35*a^7*b^4*d^6
*e^4 + 21*a^8*b^3*d^5*e^5 - 35*a^9*b^2*d^4*e^6 + 17*a^10*b*d^3*e^7 - 3*a^11*d^2*e^8)*x), -1/960*(45045*(b^6*e^
7*x^7 + a^4*b^2*d^3*e^4 + (3*b^6*d*e^6 + 4*a*b^5*e^7)*x^6 + 3*(b^6*d^2*e^5 + 4*a*b^5*d*e^6 + 2*a^2*b^4*e^7)*x^
5 + (b^6*d^3*e^4 + 12*a*b^5*d^2*e^5 + 18*a^2*b^4*d*e^6 + 4*a^3*b^3*e^7)*x^4 + (4*a*b^5*d^3*e^4 + 18*a^2*b^4*d^
2*e^5 + 12*a^3*b^3*d*e^6 + a^4*b^2*e^7)*x^3 + 3*(2*a^2*b^4*d^3*e^4 + 4*a^3*b^3*d^2*e^5 + a^4*b^2*d*e^6)*x^2 +
(4*a^3*b^3*d^3*e^4 + 3*a^4*b^2*d^2*e^5)*x)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d
- a*e))/(b*e*x + b*d)) - (45045*b^6*e^6*x^6 - 240*b^6*d^6 + 1960*a*b^5*d^5*e - 7630*a^2*b^4*d^4*e^2 + 22155*a
^3*b^3*d^3*e^3 + 32384*a^4*b^2*d^2*e^4 - 3968*a^5*b*d*e^5 + 384*a^6*e^6 + 15015*(7*b^6*d*e^5 + 11*a*b^5*e^6)*x
^5 + 3003*(23*b^6*d^2*e^4 + 129*a*b^5*d*e^5 + 73*a^2*b^4*e^6)*x^4 + 429*(15*b^6*d^3*e^3 + 599*a*b^5*d^2*e^4 +
1207*a^2*b^4*d*e^5 + 279*a^3*b^3*e^6)*x^3 - 143*(10*b^6*d^4*e^2 - 175*a*b^5*d^3*e^3 - 2433*a^2*b^4*d^2*e^4 - 1
999*a^3*b^3*d*e^5 - 128*a^4*b^2*e^6)*x^2 + 13*(40*b^6*d^5*e - 420*a*b^5*d^4*e^2 + 2765*a^2*b^4*d^3*e^3 + 15077
*a^3*b^3*d^2*e^4 + 3456*a^4*b^2*d*e^5 - 128*a^5*b*e^6)*x)*sqrt(e*x + d))/(a^4*b^7*d^10 - 7*a^5*b^6*d^9*e + 21*
a^6*b^5*d^8*e^2 - 35*a^7*b^4*d^7*e^3 + 35*a^8*b^3*d^6*e^4 - 21*a^9*b^2*d^5*e^5 + 7*a^10*b*d^4*e^6 - a^11*d^3*e
^7 + (b^11*d^7*e^3 - 7*a*b^10*d^6*e^4 + 21*a^2*b^9*d^5*e^5 - 35*a^3*b^8*d^4*e^6 + 35*a^4*b^7*d^3*e^7 - 21*a^5*
b^6*d^2*e^8 + 7*a^6*b^5*d*e^9 - a^7*b^4*e^10)*x^7 + (3*b^11*d^8*e^2 - 17*a*b^10*d^7*e^3 + 35*a^2*b^9*d^6*e^4 -
21*a^3*b^8*d^5*e^5 - 35*a^4*b^7*d^4*e^6 + 77*a^5*b^6*d^3*e^7 - 63*a^6*b^5*d^2*e^8 + 25*a^7*b^4*d*e^9 - 4*a^8*
b^3*e^10)*x^6 + 3*(b^11*d^9*e - 3*a*b^10*d^8*e^2 - 5*a^2*b^9*d^7*e^3 + 35*a^3*b^8*d^6*e^4 - 63*a^4*b^7*d^5*e^5
+ 49*a^5*b^6*d^4*e^6 - 7*a^6*b^5*d^3*e^7 - 15*a^7*b^4*d^2*e^8 + 10*a^8*b^3*d*e^9 - 2*a^9*b^2*e^10)*x^5 + (b^1
1*d^10 + 5*a*b^10*d^9*e - 45*a^2*b^9*d^8*e^2 + 95*a^3*b^8*d^7*e^3 - 35*a^4*b^7*d^6*e^4 - 147*a^5*b^6*d^5*e^5 +
245*a^6*b^5*d^4*e^6 - 155*a^7*b^4*d^3*e^7 + 30*a^8*b^3*d^2*e^8 + 10*a^9*b^2*d*e^9 - 4*a^10*b*e^10)*x^4 + (4*a
*b^10*d^10 - 10*a^2*b^9*d^9*e - 30*a^3*b^8*d^8*e^2 + 155*a^4*b^7*d^7*e^3 - 245*a^5*b^6*d^6*e^4 + 147*a^6*b^5*d
^5*e^5 + 35*a^7*b^4*d^4*e^6 - 95*a^8*b^3*d^3*e^7 + 45*a^9*b^2*d^2*e^8 - 5*a^10*b*d*e^9 - a^11*e^10)*x^3 + 3*(2
*a^2*b^9*d^10 - 10*a^3*b^8*d^9*e + 15*a^4*b^7*d^8*e^2 + 7*a^5*b^6*d^7*e^3 - 49*a^6*b^5*d^6*e^4 + 63*a^7*b^4*d^
5*e^5 - 35*a^8*b^3*d^4*e^6 + 5*a^9*b^2*d^3*e^7 + 3*a^10*b*d^2*e^8 - a^11*d*e^9)*x^2 + (4*a^3*b^8*d^10 - 25*a^4
*b^7*d^9*e + 63*a^5*b^6*d^8*e^2 - 77*a^6*b^5*d^7*e^3 + 35*a^7*b^4*d^6*e^4 + 21*a^8*b^3*d^5*e^5 - 35*a^9*b^2*d^
4*e^6 + 17*a^10*b*d^3*e^7 - 3*a^11*d^2*e^8)*x)]

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giac [B]  time = 0.66, size = 1114, normalized size = 2.56

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

3003/64*b^3*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^7*d^7*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 7*
a*b^6*d^6*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 21*a^2*b^5*d^5*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 35*a^
3*b^4*d^4*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 35*a^4*b^3*d^3*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 21*
a^5*b^2*d^2*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 7*a^6*b*d*e^6*sgn((x*e + d)*b*e - b*d*e + a*e^2) - a^7*e^
7*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(-b^2*d + a*b*e)) + 2/15*(225*(x*e + d)^2*b^2*e^4 + 25*(x*e + d)*b^2
*d*e^4 + 3*b^2*d^2*e^4 - 25*(x*e + d)*a*b*e^5 - 6*a*b*d*e^5 + 3*a^2*e^6)/((b^7*d^7*sgn((x*e + d)*b*e - b*d*e +
a*e^2) - 7*a*b^6*d^6*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 21*a^2*b^5*d^5*e^2*sgn((x*e + d)*b*e - b*d*e + a*
e^2) - 35*a^3*b^4*d^4*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 35*a^4*b^3*d^3*e^4*sgn((x*e + d)*b*e - b*d*e +
a*e^2) - 21*a^5*b^2*d^2*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 7*a^6*b*d*e^6*sgn((x*e + d)*b*e - b*d*e + a*e
^2) - a^7*e^7*sgn((x*e + d)*b*e - b*d*e + a*e^2))*(x*e + d)^(5/2)) + 1/192*(3249*(x*e + d)^(7/2)*b^6*e^4 - 106
33*(x*e + d)^(5/2)*b^6*d*e^4 + 11767*(x*e + d)^(3/2)*b^6*d^2*e^4 - 4431*sqrt(x*e + d)*b^6*d^3*e^4 + 10633*(x*e
+ d)^(5/2)*a*b^5*e^5 - 23534*(x*e + d)^(3/2)*a*b^5*d*e^5 + 13293*sqrt(x*e + d)*a*b^5*d^2*e^5 + 11767*(x*e + d
)^(3/2)*a^2*b^4*e^6 - 13293*sqrt(x*e + d)*a^2*b^4*d*e^6 + 4431*sqrt(x*e + d)*a^3*b^3*e^7)/((b^7*d^7*sgn((x*e +
d)*b*e - b*d*e + a*e^2) - 7*a*b^6*d^6*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 21*a^2*b^5*d^5*e^2*sgn((x*e + d)
*b*e - b*d*e + a*e^2) - 35*a^3*b^4*d^4*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 35*a^4*b^3*d^3*e^4*sgn((x*e +
d)*b*e - b*d*e + a*e^2) - 21*a^5*b^2*d^2*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 7*a^6*b*d*e^6*sgn((x*e + d)*
b*e - b*d*e + a*e^2) - a^7*e^7*sgn((x*e + d)*b*e - b*d*e + a*e^2))*((x*e + d)*b - b*d + a*e)^4)

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maple [B]  time = 0.10, size = 951, normalized size = 2.19 \begin {gather*} -\frac {\left (45045 \sqrt {\left (a e -b d \right ) b}\, b^{6} e^{6} x^{6}+165165 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} e^{6} x^{5}+105105 \sqrt {\left (a e -b d \right ) b}\, b^{6} d \,e^{5} x^{5}+219219 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} e^{6} x^{4}+387387 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d \,e^{5} x^{4}+45045 \left (e x +d \right )^{\frac {5}{2}} b^{7} e^{4} x^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+69069 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{2} e^{4} x^{4}+119691 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} e^{6} x^{3}+517803 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d \,e^{5} x^{3}+180180 \left (e x +d \right )^{\frac {5}{2}} a \,b^{6} e^{4} x^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+256971 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{2} e^{4} x^{3}+6435 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{3} e^{3} x^{3}+18304 \sqrt {\left (a e -b d \right ) b}\, a^{4} b^{2} e^{6} x^{2}+285857 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} d \,e^{5} x^{2}+270270 \left (e x +d \right )^{\frac {5}{2}} a^{2} b^{5} e^{4} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+347919 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d^{2} e^{4} x^{2}+25025 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{3} e^{3} x^{2}-1430 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{4} e^{2} x^{2}-1664 \sqrt {\left (a e -b d \right ) b}\, a^{5} b \,e^{6} x +44928 \sqrt {\left (a e -b d \right ) b}\, a^{4} b^{2} d \,e^{5} x +180180 \left (e x +d \right )^{\frac {5}{2}} a^{3} b^{4} e^{4} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+196001 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} d^{2} e^{4} x +35945 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d^{3} e^{3} x -5460 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{4} e^{2} x +520 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{5} e x +384 \sqrt {\left (a e -b d \right ) b}\, a^{6} e^{6}-3968 \sqrt {\left (a e -b d \right ) b}\, a^{5} b d \,e^{5}+45045 \left (e x +d \right )^{\frac {5}{2}} a^{4} b^{3} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+32384 \sqrt {\left (a e -b d \right ) b}\, a^{4} b^{2} d^{2} e^{4}+22155 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} d^{3} e^{3}-7630 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d^{4} e^{2}+1960 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{5} e -240 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{6}\right ) \left (b x +a \right )}{960 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{7} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/960*(44928*((a*e-b*d)*b)^(1/2)*x*a^4*b^2*d*e^5+165165*((a*e-b*d)*b)^(1/2)*x^5*a*b^5*e^6+105105*((a*e-b*d)*b
)^(1/2)*x^5*b^6*d*e^5-1664*((a*e-b*d)*b)^(1/2)*x*a^5*b*e^6+45045*(e*x+d)^(5/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)
*b)^(1/2)*b)*a^4*b^3*e^4+219219*((a*e-b*d)*b)^(1/2)*x^4*a^2*b^4*e^6+69069*((a*e-b*d)*b)^(1/2)*x^4*b^6*d^2*e^4+
45045*((a*e-b*d)*b)^(1/2)*x^6*b^6*e^6-240*((a*e-b*d)*b)^(1/2)*b^6*d^6-3968*((a*e-b*d)*b)^(1/2)*a^5*b*d*e^5+323
84*((a*e-b*d)*b)^(1/2)*a^4*b^2*d^2*e^4+384*((a*e-b*d)*b)^(1/2)*a^6*e^6+22155*((a*e-b*d)*b)^(1/2)*a^3*b^3*d^3*e
^3-7630*((a*e-b*d)*b)^(1/2)*a^2*b^4*d^4*e^2+1960*((a*e-b*d)*b)^(1/2)*a*b^5*d^5*e+180180*(e*x+d)^(5/2)*arctan((
e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x*a^3*b^4*e^4+25025*((a*e-b*d)*b)^(1/2)*x^2*a*b^5*d^3*e^3+119691*((a*e-b*d
)*b)^(1/2)*x^3*a^3*b^3*e^6+18304*((a*e-b*d)*b)^(1/2)*x^2*a^4*b^2*e^6+6435*((a*e-b*d)*b)^(1/2)*x^3*b^6*d^3*e^3-
1430*((a*e-b*d)*b)^(1/2)*x^2*b^6*d^4*e^2+520*((a*e-b*d)*b)^(1/2)*x*b^6*d^5*e+45045*(e*x+d)^(5/2)*arctan((e*x+d
)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^4*b^7*e^4+35945*((a*e-b*d)*b)^(1/2)*x*a^2*b^4*d^3*e^3-5460*((a*e-b*d)*b)^(1/2
)*x*a*b^5*d^4*e^2+180180*(e*x+d)^(5/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^3*a*b^6*e^4+270270*(e*x+d
)^(5/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*x^2*a^2*b^5*e^4+285857*((a*e-b*d)*b)^(1/2)*x^2*a^3*b^3*d*e
^5+347919*((a*e-b*d)*b)^(1/2)*x^2*a^2*b^4*d^2*e^4+256971*((a*e-b*d)*b)^(1/2)*x^3*a*b^5*d^2*e^4+387387*((a*e-b*
d)*b)^(1/2)*x^4*a*b^5*d*e^5+517803*((a*e-b*d)*b)^(1/2)*x^3*a^2*b^4*d*e^5+196001*((a*e-b*d)*b)^(1/2)*x*a^3*b^3*
d^2*e^4)*(b*x+a)/(e*x+d)^(5/2)/((a*e-b*d)*b)^(1/2)/(a*e-b*d)^7/((b*x+a)^2)^(5/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^(7/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^{7/2}\,{\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(1/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)

[Out]

int(1/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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