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

Optimal. Leaf size=240 \[ \frac {(b d-a e) \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt {a+b x} \sqrt {d+e x}}{8 b^4}+\frac {\left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^3}+\frac {(17 b d-13 a e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^2}+\frac {2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac {(b d-a e)^2 \left (73 b^2 d^2-90 a b d e+35 a^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{9/2} \sqrt {e}} \]

[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

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Rubi [A]
time = 0.14, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {965, 81, 52, 65, 223, 212} \begin {gather*} \frac {(b d-a e)^2 \left (35 a^2 e^2-90 a b d e+73 b^2 d^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}}+\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {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*}

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Mathematica [A]
time = 0.43, size = 189, normalized size = 0.79 \begin {gather*} \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 ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{9/2} \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[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])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(570\) vs. \(2(204)=408\).
time = 0.08, size = 571, normalized size = 2.38

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

Verification of antiderivative is not currently implemented for this CAS.

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 3.13, size = 514, normalized size = 2.14 \begin {gather*} \left [\frac {{\left (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^{\frac {1}{2}} \log \left (b^{2} d^{2} + 4 \, {\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {b} e^{\frac {1}{2}} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right ) + 4 \, {\left (501 \, b^{4} d^{3} e + {\left (48 \, b^{4} x^{3} - 56 \, a b^{3} x^{2} + 70 \, a^{2} b^{2} x - 105 \, a^{3} b\right )} e^{4} + {\left (232 \, b^{4} d x^{2} - 292 \, a b^{3} d x + 445 \, a^{2} b^{2} d\right )} e^{3} + {\left (466 \, b^{4} d^{2} x - 725 \, a b^{3} d^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{96 \, b^{5}}, -\frac {{\left (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 (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {-b e} \sqrt {x e + d}}{2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} e^{2} + {\left (b^{2} d x + a b d\right )} e\right )}}\right ) - 2 \, {\left (501 \, b^{4} d^{3} e + {\left (48 \, b^{4} x^{3} - 56 \, a b^{3} x^{2} + 70 \, a^{2} b^{2} x - 105 \, a^{3} b\right )} e^{4} + {\left (232 \, b^{4} d x^{2} - 292 \, a b^{3} d x + 445 \, a^{2} b^{2} d\right )} e^{3} + {\left (466 \, b^{4} d^{2} x - 725 \, a b^{3} d^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{48 \, b^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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^(1/2)*l
og(b^2*d^2 + 4*(b*d + (2*b*x + a)*e)*sqrt(b*x + a)*sqrt(x*e + d)*sqrt(b)*e^(1/2) + (8*b^2*x^2 + 8*a*b*x + a^2)
*e^2 + 2*(4*b^2*d*x + 3*a*b*d)*e) + 4*(501*b^4*d^3*e + (48*b^4*x^3 - 56*a*b^3*x^2 + 70*a^2*b^2*x - 105*a^3*b)*
e^4 + (232*b^4*d*x^2 - 292*a*b^3*d*x + 445*a^2*b^2*d)*e^3 + (466*b^4*d^2*x - 725*a*b^3*d^2)*e^2)*sqrt(b*x + a)
*sqrt(x*e + d))*e^(-1)/b^5, -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 + 3
5*a^4*e^4)*sqrt(-b*e)*arctan(1/2*(b*d + (2*b*x + a)*e)*sqrt(b*x + a)*sqrt(-b*e)*sqrt(x*e + d)/((b^2*x^2 + a*b*
x)*e^2 + (b^2*d*x + a*b*d)*e)) - 2*(501*b^4*d^3*e + (48*b^4*x^3 - 56*a*b^3*x^2 + 70*a^2*b^2*x - 105*a^3*b)*e^4
+ (232*b^4*d*x^2 - 292*a*b^3*d*x + 445*a^2*b^2*d)*e^3 + (466*b^4*d^2*x - 725*a*b^3*d^2)*e^2)*sqrt(b*x + a)*sq
rt(x*e + d))*e^(-1)/b^5]

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Sympy [F(-1)] Timed out
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)**(3/2)*(8*e**2*x**2+20*d*e*x+15*d**2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 717 vs. \(2 (211) = 422\).
time = 3.58, size = 717, normalized size = 2.99 \begin {gather*} -\frac {\frac {360 \, {\left (\frac {{\left (b^{2} d - a b e\right )} e^{\left (-\frac {1}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b}} - \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 {{\left (b^{6} d e^{3} - 13 \, a b^{5} e^{4}\right )} e^{\left (-4\right )}}{b^{7}}\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 )} e^{\left (-4\right )}}{b^{7}}\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 )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac {3}{2}}}\right )} d {\left | b \right |} e^{2}}{b^{2}} - \frac {210 \, {\left (\frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + {\left (b d e - 5 \, a e^{2}\right )} e^{\left (-2\right )} + 2 \, a\right )} \sqrt {b x + a}\right )} d^{2} {\left | b \right |} e}{b^{3}} - \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 {{\left (b^{12} d e^{5} - 25 \, a b^{11} e^{6}\right )} e^{\left (-6\right )}}{b^{14}}\right )} - \frac {{\left (5 \, b^{13} d^{2} e^{4} + 14 \, a b^{12} d e^{5} - 163 \, a^{2} b^{11} e^{6}\right )} e^{\left (-6\right )}}{b^{14}}\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 )} e^{\left (-6\right )}}{b^{14}}\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 )} e^{\left (-\frac {7}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac {5}{2}}}\right )} {\left | b \right |} e^{3}}{b^{2}}}{24 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)*e^(-1/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b
*e)))/sqrt(b) - 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)*e^(-4)/b^7) - 3*(b^7*d^
2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)*e^(-4)/b^7) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*e^
(-5/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*d*abs(b)*e^2/b^
2 - 210*((b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*e^(-3/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d +
(b*x + a)*b*e - a*b*e)))/sqrt(b) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + (b*d*e - 5*a*e^2)*e^(-2) + 2*a
)*sqrt(b*x + a))*d^2*abs(b)*e/b^3 - (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)*e^(-6)/b^14) - (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)*e^(-
6)/b^14) + 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)*e^(-6)/b^14)*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)*e^(-7/2)*log(abs(-sqrt(b
*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*abs(b)*e^3/b^2)/b

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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