3.18.28 \(\int \frac {1}{(d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1728]

Optimal. Leaf size=329 \[ \frac {105 e^3}{64 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {315 e^4 (a+b x)}{64 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {315 \sqrt {b} e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

-315/64*e^4*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*b^(1/2)/(-a*e+b*d)^(11/2)/((b*x+a)^2)^(1/2
)+105/64*e^3/(-a*e+b*d)^4/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)-1/4/(-a*e+b*d)/(b*x+a)^3/(e*x+d)^(1/2)/((b*x+a)^2)^(
1/2)+3/8*e/(-a*e+b*d)^2/(b*x+a)^2/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)-21/32*e^2/(-a*e+b*d)^3/(b*x+a)/(e*x+d)^(1/2)
/((b*x+a)^2)^(1/2)+315/64*e^4*(b*x+a)/(-a*e+b*d)^5/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 44, 53, 65, 214} \begin {gather*} \frac {315 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)^5}-\frac {315 \sqrt {b} 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)^{11/2}}+\frac {105 e^3}{64 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}-\frac {21 e^2}{32 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}+\frac {3 e}{8 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2}-\frac {1}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(105*e^3)/(64*(b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(4*(b*d - a*e)*(a + b*x)^3*Sqrt[d
+ e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*e)/(8*(b*d - a*e)^2*(a + b*x)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) - (21*e^2)/(32*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (315*e^4*(a +
b*x))/(64*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (315*Sqrt[b]*e^4*(a + b*x)*ArcTanh[(Sq
rt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*(b*d - a*e)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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 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 {1}{(d+e x)^{3/2} \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)^{3/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (9 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)^{3/2}} \, 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 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (21 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)^{3/2}} \, dx}{16 (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 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (105 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)^{3/2}} \, dx}{64 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {105 e^3}{64 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 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)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {105 e^3}{64 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {315 e^4 (a+b x)}{64 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 b 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)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {105 e^3}{64 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {315 e^4 (a+b x)}{64 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 b e^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 d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {105 e^3}{64 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {315 e^4 (a+b x)}{64 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {315 \sqrt {b} 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)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.06, size = 247, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {-b d+a e} \left (128 a^4 e^4+a^3 b e^3 (325 d+837 e x)+3 a^2 b^2 e^2 \left (-70 d^2+185 d e x+511 e^2 x^2\right )+a b^3 e \left (88 d^3-156 d^2 e x+399 d e^2 x^2+1155 e^3 x^3\right )+b^4 \left (-16 d^4+24 d^3 e x-42 d^2 e^2 x^2+105 d e^3 x^3+315 e^4 x^4\right )\right )+315 \sqrt {b} e^4 (a+b x)^4 \sqrt {d+e x} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{64 (-b d+a e)^{11/2} (a+b x)^3 \sqrt {(a+b x)^2} \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

-1/64*(Sqrt[-(b*d) + a*e]*(128*a^4*e^4 + a^3*b*e^3*(325*d + 837*e*x) + 3*a^2*b^2*e^2*(-70*d^2 + 185*d*e*x + 51
1*e^2*x^2) + a*b^3*e*(88*d^3 - 156*d^2*e*x + 399*d*e^2*x^2 + 1155*e^3*x^3) + b^4*(-16*d^4 + 24*d^3*e*x - 42*d^
2*e^2*x^2 + 105*d*e^3*x^3 + 315*e^4*x^4)) + 315*Sqrt[b]*e^4*(a + b*x)^4*Sqrt[d + e*x]*ArcTan[(Sqrt[b]*Sqrt[d +
e*x])/Sqrt[-(b*d) + a*e]])/((-(b*d) + a*e)^(11/2)*(a + b*x)^3*Sqrt[(a + b*x)^2]*Sqrt[d + e*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(601\) vs. \(2(231)=462\).
time = 0.70, size = 602, normalized size = 1.83

method	result	size
default	\(-\frac {\left (315 \sqrt {e x +d}\, \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{5} e^{4} x^{4}+1260 \sqrt {e x +d}\, \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{4} e^{4} x^{3}+1890 \sqrt {e x +d}\, \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{3} e^{4} x^{2}+315 \sqrt {b \left (a e -b d \right )}\, b^{4} e^{4} x^{4}+1260 \sqrt {e x +d}\, \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3} b^{2} e^{4} x +1155 \sqrt {b \left (a e -b d \right )}\, a \,b^{3} e^{4} x^{3}+105 \sqrt {b \left (a e -b d \right )}\, b^{4} d \,e^{3} x^{3}+315 \sqrt {e x +d}\, \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{4} b \,e^{4}+1533 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} e^{4} x^{2}+399 \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d \,e^{3} x^{2}-42 \sqrt {b \left (a e -b d \right )}\, b^{4} d^{2} e^{2} x^{2}+837 \sqrt {b \left (a e -b d \right )}\, a^{3} b \,e^{4} x +555 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} d \,e^{3} x -156 \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d^{2} e^{2} x +24 \sqrt {b \left (a e -b d \right )}\, b^{4} d^{3} e x +128 \sqrt {b \left (a e -b d \right )}\, a^{4} e^{4}+325 \sqrt {b \left (a e -b d \right )}\, a^{3} b d \,e^{3}-210 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} d^{2} e^{2}+88 \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d^{3} e -16 \sqrt {b \left (a e -b d \right )}\, b^{4} d^{4}\right ) \left (b x +a \right )}{64 \sqrt {e x +d}\, \sqrt {b \left (a e -b d \right )}\, \left (a e -b d \right )^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(602\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/64*(315*(e*x+d)^(1/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*b^5*e^4*x^4+1260*(e*x+d)^(1/2)*arctan(b*(
e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a*b^4*e^4*x^3+1890*(e*x+d)^(1/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))
*a^2*b^3*e^4*x^2+315*(b*(a*e-b*d))^(1/2)*b^4*e^4*x^4+1260*(e*x+d)^(1/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(
1/2))*a^3*b^2*e^4*x+1155*(b*(a*e-b*d))^(1/2)*a*b^3*e^4*x^3+105*(b*(a*e-b*d))^(1/2)*b^4*d*e^3*x^3+315*(e*x+d)^(
1/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^4*b*e^4+1533*(b*(a*e-b*d))^(1/2)*a^2*b^2*e^4*x^2+399*(b*(a*
e-b*d))^(1/2)*a*b^3*d*e^3*x^2-42*(b*(a*e-b*d))^(1/2)*b^4*d^2*e^2*x^2+837*(b*(a*e-b*d))^(1/2)*a^3*b*e^4*x+555*(
b*(a*e-b*d))^(1/2)*a^2*b^2*d*e^3*x-156*(b*(a*e-b*d))^(1/2)*a*b^3*d^2*e^2*x+24*(b*(a*e-b*d))^(1/2)*b^4*d^3*e*x+
128*(b*(a*e-b*d))^(1/2)*a^4*e^4+325*(b*(a*e-b*d))^(1/2)*a^3*b*d*e^3-210*(b*(a*e-b*d))^(1/2)*a^2*b^2*d^2*e^2+88
*(b*(a*e-b*d))^(1/2)*a*b^3*d^3*e-16*(b*(a*e-b*d))^(1/2)*b^4*d^4)*(b*x+a)/(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2)/(a*
e-b*d)^5/((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(1/(e*x+d)^(3/2)/(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)*(x*e + d)^(3/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 825 vs. \(2 (241) = 482\).
time = 2.40, size = 1662, normalized size = 5.05 \begin {gather*} \left [-\frac {315 \, {\left ({\left (b^{4} x^{5} + 4 \, a b^{3} x^{4} + 6 \, a^{2} b^{2} x^{3} + 4 \, a^{3} b x^{2} + a^{4} x\right )} e^{5} + {\left (b^{4} d x^{4} + 4 \, a b^{3} d x^{3} + 6 \, a^{2} b^{2} d x^{2} + 4 \, a^{3} b d x + a^{4} d\right )} e^{4}\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {2 \, b d + 2 \, {\left (b d - a e\right )} \sqrt {x e + d} \sqrt {\frac {b}{b d - a e}} + {\left (b x - a\right )} e}{b x + a}\right ) + 2 \, {\left (16 \, b^{4} d^{4} - {\left (315 \, b^{4} x^{4} + 1155 \, a b^{3} x^{3} + 1533 \, a^{2} b^{2} x^{2} + 837 \, a^{3} b x + 128 \, a^{4}\right )} e^{4} - {\left (105 \, b^{4} d x^{3} + 399 \, a b^{3} d x^{2} + 555 \, a^{2} b^{2} d x + 325 \, a^{3} b d\right )} e^{3} + 6 \, {\left (7 \, b^{4} d^{2} x^{2} + 26 \, a b^{3} d^{2} x + 35 \, a^{2} b^{2} d^{2}\right )} e^{2} - 8 \, {\left (3 \, b^{4} d^{3} x + 11 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d}}{128 \, {\left (b^{9} d^{6} x^{4} + 4 \, a b^{8} d^{6} x^{3} + 6 \, a^{2} b^{7} d^{6} x^{2} + 4 \, a^{3} b^{6} d^{6} x + a^{4} b^{5} d^{6} - {\left (a^{5} b^{4} x^{5} + 4 \, a^{6} b^{3} x^{4} + 6 \, a^{7} b^{2} x^{3} + 4 \, a^{8} b x^{2} + a^{9} x\right )} e^{6} + {\left (5 \, a^{4} b^{5} d x^{5} + 19 \, a^{5} b^{4} d x^{4} + 26 \, a^{6} b^{3} d x^{3} + 14 \, a^{7} b^{2} d x^{2} + a^{8} b d x - a^{9} d\right )} e^{5} - 5 \, {\left (2 \, a^{3} b^{6} d^{2} x^{5} + 7 \, a^{4} b^{5} d^{2} x^{4} + 8 \, a^{5} b^{4} d^{2} x^{3} + 2 \, a^{6} b^{3} d^{2} x^{2} - 2 \, a^{7} b^{2} d^{2} x - a^{8} b d^{2}\right )} e^{4} + 10 \, {\left (a^{2} b^{7} d^{3} x^{5} + 3 \, a^{3} b^{6} d^{3} x^{4} + 2 \, a^{4} b^{5} d^{3} x^{3} - 2 \, a^{5} b^{4} d^{3} x^{2} - 3 \, a^{6} b^{3} d^{3} x - a^{7} b^{2} d^{3}\right )} e^{3} - 5 \, {\left (a b^{8} d^{4} x^{5} + 2 \, a^{2} b^{7} d^{4} x^{4} - 2 \, a^{3} b^{6} d^{4} x^{3} - 8 \, a^{4} b^{5} d^{4} x^{2} - 7 \, a^{5} b^{4} d^{4} x - 2 \, a^{6} b^{3} d^{4}\right )} e^{2} + {\left (b^{9} d^{5} x^{5} - a b^{8} d^{5} x^{4} - 14 \, a^{2} b^{7} d^{5} x^{3} - 26 \, a^{3} b^{6} d^{5} x^{2} - 19 \, a^{4} b^{5} d^{5} x - 5 \, a^{5} b^{4} d^{5}\right )} e\right )}}, -\frac {315 \, {\left ({\left (b^{4} x^{5} + 4 \, a b^{3} x^{4} + 6 \, a^{2} b^{2} x^{3} + 4 \, a^{3} b x^{2} + a^{4} x\right )} e^{5} + {\left (b^{4} d x^{4} + 4 \, a b^{3} d x^{3} + 6 \, a^{2} b^{2} d x^{2} + 4 \, a^{3} b d x + a^{4} d\right )} e^{4}\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {x e + d} \sqrt {-\frac {b}{b d - a e}}}{b x e + b d}\right ) + {\left (16 \, b^{4} d^{4} - {\left (315 \, b^{4} x^{4} + 1155 \, a b^{3} x^{3} + 1533 \, a^{2} b^{2} x^{2} + 837 \, a^{3} b x + 128 \, a^{4}\right )} e^{4} - {\left (105 \, b^{4} d x^{3} + 399 \, a b^{3} d x^{2} + 555 \, a^{2} b^{2} d x + 325 \, a^{3} b d\right )} e^{3} + 6 \, {\left (7 \, b^{4} d^{2} x^{2} + 26 \, a b^{3} d^{2} x + 35 \, a^{2} b^{2} d^{2}\right )} e^{2} - 8 \, {\left (3 \, b^{4} d^{3} x + 11 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d}}{64 \, {\left (b^{9} d^{6} x^{4} + 4 \, a b^{8} d^{6} x^{3} + 6 \, a^{2} b^{7} d^{6} x^{2} + 4 \, a^{3} b^{6} d^{6} x + a^{4} b^{5} d^{6} - {\left (a^{5} b^{4} x^{5} + 4 \, a^{6} b^{3} x^{4} + 6 \, a^{7} b^{2} x^{3} + 4 \, a^{8} b x^{2} + a^{9} x\right )} e^{6} + {\left (5 \, a^{4} b^{5} d x^{5} + 19 \, a^{5} b^{4} d x^{4} + 26 \, a^{6} b^{3} d x^{3} + 14 \, a^{7} b^{2} d x^{2} + a^{8} b d x - a^{9} d\right )} e^{5} - 5 \, {\left (2 \, a^{3} b^{6} d^{2} x^{5} + 7 \, a^{4} b^{5} d^{2} x^{4} + 8 \, a^{5} b^{4} d^{2} x^{3} + 2 \, a^{6} b^{3} d^{2} x^{2} - 2 \, a^{7} b^{2} d^{2} x - a^{8} b d^{2}\right )} e^{4} + 10 \, {\left (a^{2} b^{7} d^{3} x^{5} + 3 \, a^{3} b^{6} d^{3} x^{4} + 2 \, a^{4} b^{5} d^{3} x^{3} - 2 \, a^{5} b^{4} d^{3} x^{2} - 3 \, a^{6} b^{3} d^{3} x - a^{7} b^{2} d^{3}\right )} e^{3} - 5 \, {\left (a b^{8} d^{4} x^{5} + 2 \, a^{2} b^{7} d^{4} x^{4} - 2 \, a^{3} b^{6} d^{4} x^{3} - 8 \, a^{4} b^{5} d^{4} x^{2} - 7 \, a^{5} b^{4} d^{4} x - 2 \, a^{6} b^{3} d^{4}\right )} e^{2} + {\left (b^{9} d^{5} x^{5} - a b^{8} d^{5} x^{4} - 14 \, a^{2} b^{7} d^{5} x^{3} - 26 \, a^{3} b^{6} d^{5} x^{2} - 19 \, a^{4} b^{5} d^{5} x - 5 \, a^{5} b^{4} d^{5}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/128*(315*((b^4*x^5 + 4*a*b^3*x^4 + 6*a^2*b^2*x^3 + 4*a^3*b*x^2 + a^4*x)*e^5 + (b^4*d*x^4 + 4*a*b^3*d*x^3 +
6*a^2*b^2*d*x^2 + 4*a^3*b*d*x + a^4*d)*e^4)*sqrt(b/(b*d - a*e))*log((2*b*d + 2*(b*d - a*e)*sqrt(x*e + d)*sqrt
(b/(b*d - a*e)) + (b*x - a)*e)/(b*x + a)) + 2*(16*b^4*d^4 - (315*b^4*x^4 + 1155*a*b^3*x^3 + 1533*a^2*b^2*x^2 +
837*a^3*b*x + 128*a^4)*e^4 - (105*b^4*d*x^3 + 399*a*b^3*d*x^2 + 555*a^2*b^2*d*x + 325*a^3*b*d)*e^3 + 6*(7*b^4
*d^2*x^2 + 26*a*b^3*d^2*x + 35*a^2*b^2*d^2)*e^2 - 8*(3*b^4*d^3*x + 11*a*b^3*d^3)*e)*sqrt(x*e + d))/(b^9*d^6*x^
4 + 4*a*b^8*d^6*x^3 + 6*a^2*b^7*d^6*x^2 + 4*a^3*b^6*d^6*x + a^4*b^5*d^6 - (a^5*b^4*x^5 + 4*a^6*b^3*x^4 + 6*a^7
*b^2*x^3 + 4*a^8*b*x^2 + a^9*x)*e^6 + (5*a^4*b^5*d*x^5 + 19*a^5*b^4*d*x^4 + 26*a^6*b^3*d*x^3 + 14*a^7*b^2*d*x^
2 + a^8*b*d*x - a^9*d)*e^5 - 5*(2*a^3*b^6*d^2*x^5 + 7*a^4*b^5*d^2*x^4 + 8*a^5*b^4*d^2*x^3 + 2*a^6*b^3*d^2*x^2
- 2*a^7*b^2*d^2*x - a^8*b*d^2)*e^4 + 10*(a^2*b^7*d^3*x^5 + 3*a^3*b^6*d^3*x^4 + 2*a^4*b^5*d^3*x^3 - 2*a^5*b^4*d
^3*x^2 - 3*a^6*b^3*d^3*x - a^7*b^2*d^3)*e^3 - 5*(a*b^8*d^4*x^5 + 2*a^2*b^7*d^4*x^4 - 2*a^3*b^6*d^4*x^3 - 8*a^4
*b^5*d^4*x^2 - 7*a^5*b^4*d^4*x - 2*a^6*b^3*d^4)*e^2 + (b^9*d^5*x^5 - a*b^8*d^5*x^4 - 14*a^2*b^7*d^5*x^3 - 26*a
^3*b^6*d^5*x^2 - 19*a^4*b^5*d^5*x - 5*a^5*b^4*d^5)*e), -1/64*(315*((b^4*x^5 + 4*a*b^3*x^4 + 6*a^2*b^2*x^3 + 4*
a^3*b*x^2 + a^4*x)*e^5 + (b^4*d*x^4 + 4*a*b^3*d*x^3 + 6*a^2*b^2*d*x^2 + 4*a^3*b*d*x + a^4*d)*e^4)*sqrt(-b/(b*d
- a*e))*arctan(-(b*d - a*e)*sqrt(x*e + d)*sqrt(-b/(b*d - a*e))/(b*x*e + b*d)) + (16*b^4*d^4 - (315*b^4*x^4 +
1155*a*b^3*x^3 + 1533*a^2*b^2*x^2 + 837*a^3*b*x + 128*a^4)*e^4 - (105*b^4*d*x^3 + 399*a*b^3*d*x^2 + 555*a^2*b^
2*d*x + 325*a^3*b*d)*e^3 + 6*(7*b^4*d^2*x^2 + 26*a*b^3*d^2*x + 35*a^2*b^2*d^2)*e^2 - 8*(3*b^4*d^3*x + 11*a*b^3
*d^3)*e)*sqrt(x*e + d))/(b^9*d^6*x^4 + 4*a*b^8*d^6*x^3 + 6*a^2*b^7*d^6*x^2 + 4*a^3*b^6*d^6*x + a^4*b^5*d^6 - (
a^5*b^4*x^5 + 4*a^6*b^3*x^4 + 6*a^7*b^2*x^3 + 4*a^8*b*x^2 + a^9*x)*e^6 + (5*a^4*b^5*d*x^5 + 19*a^5*b^4*d*x^4 +
26*a^6*b^3*d*x^3 + 14*a^7*b^2*d*x^2 + a^8*b*d*x - a^9*d)*e^5 - 5*(2*a^3*b^6*d^2*x^5 + 7*a^4*b^5*d^2*x^4 + 8*a
^5*b^4*d^2*x^3 + 2*a^6*b^3*d^2*x^2 - 2*a^7*b^2*d^2*x - a^8*b*d^2)*e^4 + 10*(a^2*b^7*d^3*x^5 + 3*a^3*b^6*d^3*x^
4 + 2*a^4*b^5*d^3*x^3 - 2*a^5*b^4*d^3*x^2 - 3*a^6*b^3*d^3*x - a^7*b^2*d^3)*e^3 - 5*(a*b^8*d^4*x^5 + 2*a^2*b^7*
d^4*x^4 - 2*a^3*b^6*d^4*x^3 - 8*a^4*b^5*d^4*x^2 - 7*a^5*b^4*d^4*x - 2*a^6*b^3*d^4)*e^2 + (b^9*d^5*x^5 - a*b^8*
d^5*x^4 - 14*a^2*b^7*d^5*x^3 - 26*a^3*b^6*d^5*x^2 - 19*a^4*b^5*d^5*x - 5*a^5*b^4*d^5)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x\right )^{\frac {3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(1/((d + e*x)**(3/2)*((a + b*x)**2)**(5/2)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (241) = 482\).
time = 1.25, size = 548, normalized size = 1.67 \begin {gather*} \frac {315 \, b \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{4}}{64 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, e^{4}}{{\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {x e + d}} + \frac {187 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} e^{4} - 643 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d e^{4} + 765 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{2} e^{4} - 325 \, \sqrt {x e + d} b^{4} d^{3} e^{4} + 643 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} e^{5} - 1530 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d e^{5} + 975 \, \sqrt {x e + d} a b^{3} d^{2} e^{5} + 765 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} e^{6} - 975 \, \sqrt {x e + d} a^{2} b^{2} d e^{6} + 325 \, \sqrt {x e + d} a^{3} b e^{7}}{64 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

315/64*b*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^5*d^5*sgn(b*x + a) - 5*a*b^4*d^4*e*sgn(b*x + a)
+ 10*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10*a^3*b^2*d^2*e^3*sgn(b*x + a) + 5*a^4*b*d*e^4*sgn(b*x + a) - a^5*e^5*sgn
(b*x + a))*sqrt(-b^2*d + a*b*e)) + 2*e^4/((b^5*d^5*sgn(b*x + a) - 5*a*b^4*d^4*e*sgn(b*x + a) + 10*a^2*b^3*d^3*
e^2*sgn(b*x + a) - 10*a^3*b^2*d^2*e^3*sgn(b*x + a) + 5*a^4*b*d*e^4*sgn(b*x + a) - a^5*e^5*sgn(b*x + a))*sqrt(x
*e + d)) + 1/64*(187*(x*e + d)^(7/2)*b^4*e^4 - 643*(x*e + d)^(5/2)*b^4*d*e^4 + 765*(x*e + d)^(3/2)*b^4*d^2*e^4
- 325*sqrt(x*e + d)*b^4*d^3*e^4 + 643*(x*e + d)^(5/2)*a*b^3*e^5 - 1530*(x*e + d)^(3/2)*a*b^3*d*e^5 + 975*sqrt
(x*e + d)*a*b^3*d^2*e^5 + 765*(x*e + d)^(3/2)*a^2*b^2*e^6 - 975*sqrt(x*e + d)*a^2*b^2*d*e^6 + 325*sqrt(x*e + d
)*a^3*b*e^7)/((b^5*d^5*sgn(b*x + a) - 5*a*b^4*d^4*e*sgn(b*x + a) + 10*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10*a^3*b^
2*d^2*e^3*sgn(b*x + a) + 5*a^4*b*d*e^4*sgn(b*x + a) - a^5*e^5*sgn(b*x + a))*((x*e + d)*b - b*d + a*e)^4)

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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 )}^{3/2}\,{\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)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)

[Out]

int(1/((d + e*x)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)

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