\(\int \frac {(d+e x)^{3/2} (15 d^2+20 d e x+8 e^2 x^2)}{\sqrt {a+b x}} \, dx\) [843]

Optimal result

Integrand size = 38, antiderivative size = 240 \[ \int \frac {(d+e x)^{3/2} \left (15 d^2+20 d e x+8 e^2 x^2\right )}{\sqrt {a+b x}} \, dx=\frac {(b d-a e) \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt {a+b x} \sqrt {d+e x}}{8 b^4}+\frac {\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(17 b d-13 a e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2}+\frac {2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac {(b d-a e)^2 \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{9/2} \sqrt {e}} \]

[Out]

2*e*(b*x+a)^(3/2)*(e*x+d)^(5/2)/b^2+1/8*(-a*e+b*d)^2*(35*a^2*e^2-90*a*b*d*e+73*b^2*d^2)*arctanh(e^(1/2)*(b*x+a
)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(9/2)/e^(1/2)+1/12*(35*a^2*e^2-90*a*b*d*e+73*b^2*d^2)*(e*x+d)^(3/2)*(b*x+a)^(
1/2)/b^3+1/3*(-13*a*e+17*b*d)*(e*x+d)^(5/2)*(b*x+a)^(1/2)/b^2+1/8*(-a*e+b*d)*(35*a^2*e^2-90*a*b*d*e+73*b^2*d^2
)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^4

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {965, 81, 52, 65, 223, 212} \[ \int \frac {(d+e x)^{3/2} \left (15 d^2+20 d e x+8 e^2 x^2\right )}{\sqrt {a+b x}} \, dx=\frac {(b d-a e)^2 \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{9/2} \sqrt {e}}+\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e) \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right )}{8 b^4}+\frac {\sqrt {a+b x} (d+e x)^{3/2} \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right )}{12 b^3}+\frac {2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac {\sqrt {a+b x} (d+e x)^{5/2} (17 b d-13 a e)}{3 b^2} \]

[In]

Int[((d + e*x)^(3/2)*(15*d^2 + 20*d*e*x + 8*e^2*x^2))/Sqrt[a + b*x],x]

[Out]

((b*d - a*e)*(73*b^2*d^2 - 90*a*b*d*e + 35*a^2*e^2)*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^4) + ((73*b^2*d^2 - 90*a
*b*d*e + 35*a^2*e^2)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(12*b^3) + ((17*b*d - 13*a*e)*Sqrt[a + b*x]*(d + e*x)^(5/2
))/(3*b^2) + (2*e*(a + b*x)^(3/2)*(d + e*x)^(5/2))/b^2 + ((b*d - a*e)^2*(73*b^2*d^2 - 90*a*b*d*e + 35*a^2*e^2)
*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(9/2)*Sqrt[e])

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 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 965

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x)^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Dist[1/(g*e^(2*p)*(m +
n + 2*p + 1)), Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x + c*x^2)^p - c^p*
(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x
] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && NeQ[m + n + 2*
p + 1, 0] && (IntegerQ[n] ||  !IntegerQ[m])

Rubi steps \begin{align*} \text {integral}& = \frac {2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac {\int \frac {(d+e x)^{3/2} \left (4 e \left (15 b^2 d^2-3 a b d e-5 a^2 e^2\right )+4 b e^2 (17 b d-13 a e) x\right )}{\sqrt {a+b x}} \, dx}{4 b^2 e} \\ & = \frac {(17 b d-13 a e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2}+\frac {2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac {\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \int \frac {(d+e x)^{3/2}}{\sqrt {a+b x}} \, dx}{6 b^2} \\ & = \frac {\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(17 b d-13 a e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2}+\frac {2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac {\left ((b d-a e) \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}} \, dx}{8 b^3} \\ & = \frac {(b d-a e) \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt {a+b x} \sqrt {d+e x}}{8 b^4}+\frac {\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(17 b d-13 a e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2}+\frac {2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac {\left ((b d-a e)^2 \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{16 b^4} \\ & = \frac {(b d-a e) \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt {a+b x} \sqrt {d+e x}}{8 b^4}+\frac {\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(17 b d-13 a e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2}+\frac {2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac {\left ((b d-a e)^2 \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{8 b^5} \\ & = \frac {(b d-a e) \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt {a+b x} \sqrt {d+e x}}{8 b^4}+\frac {\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(17 b d-13 a e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2}+\frac {2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac {\left ((b d-a e)^2 \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{8 b^5} \\ & = \frac {(b d-a e) \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt {a+b x} \sqrt {d+e x}}{8 b^4}+\frac {\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(17 b d-13 a e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2}+\frac {2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac {(b d-a e)^2 \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{9/2} \sqrt {e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.79 \[ \int \frac {(d+e x)^{3/2} \left (15 d^2+20 d e x+8 e^2 x^2\right )}{\sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {d+e x} \left (-105 a^3 e^3+5 a^2 b e^2 (89 d+14 e x)-a b^2 e \left (725 d^2+292 d e x+56 e^2 x^2\right )+b^3 \left (501 d^3+466 d^2 e x+232 d e^2 x^2+48 e^3 x^3\right )\right )}{24 b^4}+\frac {(b d-a e)^2 \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{9/2} \sqrt {e}} \]

[In]

Integrate[((d + e*x)^(3/2)*(15*d^2 + 20*d*e*x + 8*e^2*x^2))/Sqrt[a + b*x],x]

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-105*a^3*e^3 + 5*a^2*b*e^2*(89*d + 14*e*x) - a*b^2*e*(725*d^2 + 292*d*e*x + 56*e
^2*x^2) + b^3*(501*d^3 + 466*d^2*e*x + 232*d*e^2*x^2 + 48*e^3*x^3)))/(24*b^4) + ((b*d - a*e)^2*(73*b^2*d^2 - 9
0*a*b*d*e + 35*a^2*e^2)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(9/2)*Sqrt[e])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(570\) vs. \(2(204)=408\).

Time = 0.44 (sec) , antiderivative size = 571, normalized size of antiderivative = 2.38

method	result	size
default	\(\frac {\sqrt {e x +d}\, \sqrt {b x +a}\, \left (96 b^{3} e^{3} x^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-112 a \,b^{2} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+464 b^{3} d \,e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+105 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{4} e^{4}-480 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} b d \,e^{3}+864 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b^{2} d^{2} e^{2}-708 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{3} e +219 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{4}+140 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b \,e^{3} x -584 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} d \,e^{2} x +932 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{2} e x -210 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{3} e^{3}+890 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b d \,e^{2}-1450 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} d^{2} e +1002 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{3}\right )}{48 b^{4} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}\)	\(571\)

[In]

int((e*x+d)^(3/2)*(8*e^2*x^2+20*d*e*x+15*d^2)/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/48*(e*x+d)^(1/2)*(b*x+a)^(1/2)*(96*b^3*e^3*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-112*a*b^2*e^3*x^2*((b*x+a
)*(e*x+d))^(1/2)*(b*e)^(1/2)+464*b^3*d*e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+105*ln(1/2*(2*b*e*x+2*((b*x
+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*e^4-480*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*
e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b*d*e^3+864*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/
(b*e)^(1/2))*a^2*b^2*d^2*e^2-708*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a
*b^3*d^3*e+219*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d^4+140*((b*x+a
)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^2*b*e^3*x-584*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a*b^2*d*e^2*x+932*((b*x+a)*(e
*x+d))^(1/2)*(b*e)^(1/2)*b^3*d^2*e*x-210*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^3*e^3+890*((b*x+a)*(e*x+d))^(1/
2)*(b*e)^(1/2)*a^2*b*d*e^2-1450*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a*b^2*d^2*e+1002*((b*x+a)*(e*x+d))^(1/2)*(
b*e)^(1/2)*b^3*d^3)/b^4/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.28 \[ \int \frac {(d+e x)^{3/2} \left (15 d^2+20 d e x+8 e^2 x^2\right )}{\sqrt {a+b x}} \, dx=\left [\frac {3 \, {\left (73 \, b^{4} d^{4} - 236 \, a b^{3} d^{3} e + 288 \, a^{2} b^{2} d^{2} e^{2} - 160 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (48 \, b^{4} e^{4} x^{3} + 501 \, b^{4} d^{3} e - 725 \, a b^{3} d^{2} e^{2} + 445 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4} + 8 \, {\left (29 \, b^{4} d e^{3} - 7 \, a b^{3} e^{4}\right )} x^{2} + 2 \, {\left (233 \, b^{4} d^{2} e^{2} - 146 \, a b^{3} d e^{3} + 35 \, a^{2} b^{2} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{96 \, b^{5} e}, -\frac {3 \, {\left (73 \, b^{4} d^{4} - 236 \, a b^{3} d^{3} e + 288 \, a^{2} b^{2} d^{2} e^{2} - 160 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - 2 \, {\left (48 \, b^{4} e^{4} x^{3} + 501 \, b^{4} d^{3} e - 725 \, a b^{3} d^{2} e^{2} + 445 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4} + 8 \, {\left (29 \, b^{4} d e^{3} - 7 \, a b^{3} e^{4}\right )} x^{2} + 2 \, {\left (233 \, b^{4} d^{2} e^{2} - 146 \, a b^{3} d e^{3} + 35 \, a^{2} b^{2} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{48 \, b^{5} e}\right ] \]

[In]

integrate((e*x+d)^(3/2)*(8*e^2*x^2+20*d*e*x+15*d^2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/96*(3*(73*b^4*d^4 - 236*a*b^3*d^3*e + 288*a^2*b^2*d^2*e^2 - 160*a^3*b*d*e^3 + 35*a^4*e^4)*sqrt(b*e)*log(8*b
^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8
*(b^2*d*e + a*b*e^2)*x) + 4*(48*b^4*e^4*x^3 + 501*b^4*d^3*e - 725*a*b^3*d^2*e^2 + 445*a^2*b^2*d*e^3 - 105*a^3*
b*e^4 + 8*(29*b^4*d*e^3 - 7*a*b^3*e^4)*x^2 + 2*(233*b^4*d^2*e^2 - 146*a*b^3*d*e^3 + 35*a^2*b^2*e^4)*x)*sqrt(b*
x + a)*sqrt(e*x + d))/(b^5*e), -1/48*(3*(73*b^4*d^4 - 236*a*b^3*d^3*e + 288*a^2*b^2*d^2*e^2 - 160*a^3*b*d*e^3
+ 35*a^4*e^4)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2
+ a*b*d*e + (b^2*d*e + a*b*e^2)*x)) - 2*(48*b^4*e^4*x^3 + 501*b^4*d^3*e - 725*a*b^3*d^2*e^2 + 445*a^2*b^2*d*e^
3 - 105*a^3*b*e^4 + 8*(29*b^4*d*e^3 - 7*a*b^3*e^4)*x^2 + 2*(233*b^4*d^2*e^2 - 146*a*b^3*d*e^3 + 35*a^2*b^2*e^4
)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*e)]

Sympy [F]

\[ \int \frac {(d+e x)^{3/2} \left (15 d^2+20 d e x+8 e^2 x^2\right )}{\sqrt {a+b x}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (15 d^{2} + 20 d e x + 8 e^{2} x^{2}\right )}{\sqrt {a + b x}}\, dx \]

[In]

integrate((e*x+d)**(3/2)*(8*e**2*x**2+20*d*e*x+15*d**2)/(b*x+a)**(1/2),x)

[Out]

Integral((d + e*x)**(3/2)*(15*d**2 + 20*d*e*x + 8*e**2*x**2)/sqrt(a + b*x), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{3/2} \left (15 d^2+20 d e x+8 e^2 x^2\right )}{\sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^(3/2)*(8*e^2*x^2+20*d*e*x+15*d^2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 739 vs. \(2 (204) = 408\).

Time = 0.37 (sec) , antiderivative size = 739, normalized size of antiderivative = 3.08 \[ \int \frac {(d+e x)^{3/2} \left (15 d^2+20 d e x+8 e^2 x^2\right )}{\sqrt {a+b x}} \, dx=-\frac {\frac {360 \, {\left (\frac {{\left (b^{2} d - a b e\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a}\right )} d^{3} {\left | b \right |}}{b^{2}} - \frac {28 \, {\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} d e^{3} - 13 \, a b^{5} e^{4}}{b^{7} e^{4}}\right )} - \frac {3 \, {\left (b^{7} d^{2} e^{2} + 2 \, a b^{6} d e^{3} - 11 \, a^{2} b^{5} e^{4}\right )}}{b^{7} e^{4}}\right )} - \frac {3 \, {\left (b^{3} d^{3} + a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} b e^{2}}\right )} d e^{2} {\left | b \right |}}{b^{2}} - \frac {{\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} + \frac {b^{12} d e^{5} - 25 \, a b^{11} e^{6}}{b^{14} e^{6}}\right )} - \frac {5 \, b^{13} d^{2} e^{4} + 14 \, a b^{12} d e^{5} - 163 \, a^{2} b^{11} e^{6}}{b^{14} e^{6}}\right )} + \frac {3 \, {\left (5 \, b^{14} d^{3} e^{3} + 9 \, a b^{13} d^{2} e^{4} + 15 \, a^{2} b^{12} d e^{5} - 93 \, a^{3} b^{11} e^{6}\right )}}{b^{14} e^{6}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (5 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} - 35 \, a^{4} e^{4}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} b^{2} e^{3}}\right )} e^{3} {\left | b \right |}}{b^{2}} - \frac {210 \, {\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + 2 \, a + \frac {b d e - 5 \, a e^{2}}{e^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} e}\right )} d^{2} e {\left | b \right |}}{b^{3}}}{24 \, b} \]

[In]

integrate((e*x+d)^(3/2)*(8*e^2*x^2+20*d*e*x+15*d^2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/24*(360*((b^2*d - a*b*e)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b*e)
- sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a))*d^3*abs(b)/b^2 - 28*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)
*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a*b^5*e^4)/(b^7*e^4)) - 3*(b^7*d^2*e^2 + 2*a*b^
6*d*e^3 - 11*a^2*b^5*e^4)/(b^7*e^4)) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*log(abs(-sqrt(b*e
)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*b*e^2))*d*e^2*abs(b)/b^2 - (sqrt(b^2*d + (b
*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)/(b^14*e^6)) - (
5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)/(b^14*e^6)) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a
^2*b^12*d*e^5 - 93*a^3*b^11*e^6)/(b^14*e^6))*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2
+ 20*a^3*b*d*e^3 - 35*a^4*e^4)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(
b*e)*b^2*e^3))*e^3*abs(b)/b^2 - 210*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + 2*a + (b*d*e - 5*a*e^2)/e^2)
*sqrt(b*x + a) + (b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x +
a)*b*e - a*b*e)))/(sqrt(b*e)*e))*d^2*e*abs(b)/b^3)/b

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2} \left (15 d^2+20 d e x+8 e^2 x^2\right )}{\sqrt {a+b x}} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}\,\left (15\,d^2+20\,d\,e\,x+8\,e^2\,x^2\right )}{\sqrt {a+b\,x}} \,d x \]

[In]

int(((d + e*x)^(3/2)*(15*d^2 + 8*e^2*x^2 + 20*d*e*x))/(a + b*x)^(1/2),x)

[Out]

int(((d + e*x)^(3/2)*(15*d^2 + 8*e^2*x^2 + 20*d*e*x))/(a + b*x)^(1/2), x)