3.7.48 \(\int x (a+b x)^{5/2} \sqrt {c+d x} \, dx\) [648]

Optimal. Leaf size=268 \[ -\frac {(b c-a d)^3 (7 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^4}+\frac {(b c-a d)^2 (7 b c+3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^3}-\frac {(b c-a d) (7 b c+3 a d) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^2}-\frac {(7 b c+3 a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b d}+\frac {(b c-a d)^4 (7 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{5/2} d^{9/2}} \]

[Out]

1/5*(b*x+a)^(7/2)*(d*x+c)^(3/2)/b/d+1/128*(-a*d+b*c)^4*(3*a*d+7*b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*
x+c)^(1/2))/b^(5/2)/d^(9/2)+1/192*(-a*d+b*c)^2*(3*a*d+7*b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^2/d^3-1/240*(-a*d+b
*c)*(3*a*d+7*b*c)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/b^2/d^2-1/40*(3*a*d+7*b*c)*(b*x+a)^(7/2)*(d*x+c)^(1/2)/b^2/d-1/1
28*(-a*d+b*c)^3*(3*a*d+7*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^2/d^4

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Rubi [A]
time = 0.11, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \begin {gather*} \frac {(3 a d+7 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{5/2} d^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+7 b c) (b c-a d)^3}{128 b^2 d^4}+\frac {(a+b x)^{3/2} \sqrt {c+d x} (3 a d+7 b c) (b c-a d)^2}{192 b^2 d^3}-\frac {(a+b x)^{5/2} \sqrt {c+d x} (3 a d+7 b c) (b c-a d)}{240 b^2 d^2}-\frac {(a+b x)^{7/2} \sqrt {c+d x} (3 a d+7 b c)}{40 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^(5/2)*Sqrt[c + d*x],x]

[Out]

-1/128*((b*c - a*d)^3*(7*b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(b^2*d^4) + ((b*c - a*d)^2*(7*b*c + 3*a*d)*
(a + b*x)^(3/2)*Sqrt[c + d*x])/(192*b^2*d^3) - ((b*c - a*d)*(7*b*c + 3*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24
0*b^2*d^2) - ((7*b*c + 3*a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(40*b^2*d) + ((a + b*x)^(7/2)*(c + d*x)^(3/2))/(5
*b*d) + ((b*c - a*d)^4*(7*b*c + 3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(5/2)*
d^(9/2))

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]

Rubi steps

\begin {align*} \int x (a+b x)^{5/2} \sqrt {c+d x} \, dx &=\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b d}-\frac {(7 b c+3 a d) \int (a+b x)^{5/2} \sqrt {c+d x} \, dx}{10 b d}\\ &=-\frac {(7 b c+3 a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b d}-\frac {((b c-a d) (7 b c+3 a d)) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{80 b^2 d}\\ &=-\frac {(b c-a d) (7 b c+3 a d) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^2}-\frac {(7 b c+3 a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b d}+\frac {\left ((b c-a d)^2 (7 b c+3 a d)\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{96 b^2 d^2}\\ &=\frac {(b c-a d)^2 (7 b c+3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^3}-\frac {(b c-a d) (7 b c+3 a d) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^2}-\frac {(7 b c+3 a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b d}-\frac {\left ((b c-a d)^3 (7 b c+3 a d)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b^2 d^3}\\ &=-\frac {(b c-a d)^3 (7 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^4}+\frac {(b c-a d)^2 (7 b c+3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^3}-\frac {(b c-a d) (7 b c+3 a d) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^2}-\frac {(7 b c+3 a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b d}+\frac {\left ((b c-a d)^4 (7 b c+3 a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^2 d^4}\\ &=-\frac {(b c-a d)^3 (7 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^4}+\frac {(b c-a d)^2 (7 b c+3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^3}-\frac {(b c-a d) (7 b c+3 a d) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^2}-\frac {(7 b c+3 a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b d}+\frac {\left ((b c-a d)^4 (7 b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{128 b^3 d^4}\\ &=-\frac {(b c-a d)^3 (7 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^4}+\frac {(b c-a d)^2 (7 b c+3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^3}-\frac {(b c-a d) (7 b c+3 a d) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^2}-\frac {(7 b c+3 a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b d}+\frac {\left ((b c-a d)^4 (7 b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 b^3 d^4}\\ &=-\frac {(b c-a d)^3 (7 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^4}+\frac {(b c-a d)^2 (7 b c+3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^3}-\frac {(b c-a d) (7 b c+3 a d) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^2}-\frac {(7 b c+3 a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b d}+\frac {(b c-a d)^4 (7 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{5/2} d^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.51, size = 229, normalized size = 0.85 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4+30 a^3 b d^3 (2 c+d x)+2 a^2 b^2 d^2 \left (-173 c^2+109 c d x+372 d^2 x^2\right )+2 a b^3 d \left (170 c^3-111 c^2 d x+88 c d^2 x^2+504 d^3 x^3\right )+b^4 \left (-105 c^4+70 c^3 d x-56 c^2 d^2 x^2+48 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^2 d^4}+\frac {(b c-a d)^4 (7 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{128 b^{5/2} d^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^(5/2)*Sqrt[c + d*x],x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-45*a^4*d^4 + 30*a^3*b*d^3*(2*c + d*x) + 2*a^2*b^2*d^2*(-173*c^2 + 109*c*d*x + 3
72*d^2*x^2) + 2*a*b^3*d*(170*c^3 - 111*c^2*d*x + 88*c*d^2*x^2 + 504*d^3*x^3) + b^4*(-105*c^4 + 70*c^3*d*x - 56
*c^2*d^2*x^2 + 48*c*d^3*x^3 + 384*d^4*x^4)))/(1920*b^2*d^4) + ((b*c - a*d)^4*(7*b*c + 3*a*d)*ArcTanh[(Sqrt[b]*
Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(128*b^(5/2)*d^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(787\) vs. \(2(224)=448\).
time = 0.07, size = 788, normalized size = 2.94

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (768 b^{4} d^{4} x^{4} \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2016 a \,b^{3} d^{4} x^{3} \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+96 b^{4} c \,d^{3} x^{3} \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+1488 a^{2} b^{2} d^{4} x^{2} \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+352 a \,b^{3} c \,d^{3} x^{2} \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-112 b^{4} c^{2} d^{2} x^{2} \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+45 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{5} d^{5}-75 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} b c \,d^{4}-150 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b^{2} c^{2} d^{3}+450 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{3} c^{3} d^{2}-375 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{4} c^{4} d +105 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{5} c^{5}+60 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,d^{4} x +436 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c \,d^{3} x -444 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{2} d^{2} x +140 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{4} c^{3} d x -90 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} d^{4}+120 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b c \,d^{3}-692 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{2} d^{2}+680 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{3} d -210 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{4} c^{4}\right )}{3840 b^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, d^{4} \sqrt {b d}}\)	\(788\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(768*b^4*d^4*x^4*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2016*a*b^3*d^4*x^3*(b*
d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+96*b^4*c*d^3*x^3*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+1488*a^2*b^2*d^4*x^2*(b*
d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+352*a*b^3*c*d^3*x^2*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-112*b^4*c^2*d^2*x^2*(
b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+45*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/
2))*a^5*d^5-75*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b*c*d^4-150*ln(
1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^2*c^2*d^3+450*ln(1/2*(2*b*d*x+2
*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^3*c^3*d^2-375*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x
+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^4*c^4*d+105*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(
1/2)+a*d+b*c)/(b*d)^(1/2))*b^5*c^5+60*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^3*b*d^4*x+436*(b*d)^(1/2)*((d*x+c)
*(b*x+a))^(1/2)*a^2*b^2*c*d^3*x-444*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b^3*c^2*d^2*x+140*(b*d)^(1/2)*((d*x+
c)*(b*x+a))^(1/2)*b^4*c^3*d*x-90*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^4*d^4+120*(b*d)^(1/2)*((d*x+c)*(b*x+a))
^(1/2)*a^3*b*c*d^3-692*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^2*c^2*d^2+680*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(
1/2)*a*b^3*c^3*d-210*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^4*c^4)/b^2/((d*x+c)*(b*x+a))^(1/2)/d^4/(b*d)^(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(x*(b*x+a)^(5/2)*(d*x+c)^(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*d-b*c>0)', see `assume?` for
more detail

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Fricas [A]
time = 1.14, size = 704, normalized size = 2.63 \begin {gather*} \left [\frac {15 \, {\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 340 \, a b^{4} c^{3} d^{2} - 346 \, a^{2} b^{3} c^{2} d^{3} + 60 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + 21 \, a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 22 \, a b^{4} c d^{4} - 93 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 111 \, a b^{4} c^{2} d^{3} + 109 \, a^{2} b^{3} c d^{4} + 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{3} d^{5}}, -\frac {15 \, {\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 340 \, a b^{4} c^{3} d^{2} - 346 \, a^{2} b^{3} c^{2} d^{3} + 60 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + 21 \, a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 22 \, a b^{4} c d^{4} - 93 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 111 \, a b^{4} c^{2} d^{3} + 109 \, a^{2} b^{3} c d^{4} + 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{3} d^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/7680*(15*(7*b^5*c^5 - 25*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 3*a^5*d^5)
*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)
*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(384*b^5*d^5*x^4 - 105*b^5*c^4*d + 340*a*b^4*c^3*d^2 - 346*a^2*b
^3*c^2*d^3 + 60*a^3*b^2*c*d^4 - 45*a^4*b*d^5 + 48*(b^5*c*d^4 + 21*a*b^4*d^5)*x^3 - 8*(7*b^5*c^2*d^3 - 22*a*b^4
*c*d^4 - 93*a^2*b^3*d^5)*x^2 + 2*(35*b^5*c^3*d^2 - 111*a*b^4*c^2*d^3 + 109*a^2*b^3*c*d^4 + 15*a^3*b^2*d^5)*x)*
sqrt(b*x + a)*sqrt(d*x + c))/(b^3*d^5), -1/3840*(15*(7*b^5*c^5 - 25*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2 - 10*a^3*
b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*
sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 2*(384*b^5*d^5*x^4 - 105*b^5*c^4*d + 340*a*b^
4*c^3*d^2 - 346*a^2*b^3*c^2*d^3 + 60*a^3*b^2*c*d^4 - 45*a^4*b*d^5 + 48*(b^5*c*d^4 + 21*a*b^4*d^5)*x^3 - 8*(7*b
^5*c^2*d^3 - 22*a*b^4*c*d^4 - 93*a^2*b^3*d^5)*x^2 + 2*(35*b^5*c^3*d^2 - 111*a*b^4*c^2*d^3 + 109*a^2*b^3*c*d^4
+ 15*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^3*d^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(x*(b*x+a)**(5/2)*(d*x+c)**(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. 1011 vs. \(2 (224) = 448\).
time = 1.07, size = 1011, normalized size = 3.77 \begin {gather*} \frac {30 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\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} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac {5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{3}}\right )} a {\left | b \right |} + \frac {240 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{2}}\right )} a^{2} {\left | b \right |}}{b} + {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (6 \, {\left (b x + a\right )} {\left (\frac {8 \, {\left (b x + a\right )}}{b^{4}} + \frac {b^{20} c d^{7} - 41 \, a b^{19} d^{8}}{b^{23} d^{8}}\right )} - \frac {7 \, b^{21} c^{2} d^{6} + 26 \, a b^{20} c d^{7} - 513 \, a^{2} b^{19} d^{8}}{b^{23} d^{8}}\right )} + \frac {5 \, {\left (7 \, b^{22} c^{3} d^{5} + 19 \, a b^{21} c^{2} d^{6} + 37 \, a^{2} b^{20} c d^{7} - 447 \, a^{3} b^{19} d^{8}\right )}}{b^{23} d^{8}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (7 \, b^{23} c^{4} d^{4} + 12 \, a b^{22} c^{3} d^{5} + 18 \, a^{2} b^{21} c^{2} d^{6} + 28 \, a^{3} b^{20} c d^{7} - 193 \, a^{4} b^{19} d^{8}\right )}}{b^{23} d^{8}}\right )} \sqrt {b x + a} - \frac {15 \, {\left (7 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 10 \, a^{3} b^{2} c^{2} d^{3} + 35 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{3} d^{4}}\right )} b {\left | b \right |} + \frac {480 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d}\right )} a^{3} {\left | b \right |}}{b^{3}}}{1920 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1920*(30*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*c*d^5 - 25*
a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^
3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3
*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3))*a*abs(b) + 240*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2
*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*c*d^3 - 13*a*b^5*d^4)/(b^7*d^4)) - 3*(b^7*c^2*d^2 + 2*a*b^6*c*d^3 - 11*a^2*
b^5*d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*log(abs(-sqrt(b*d)*sqrt(b*x + a) +
sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^2))*a^2*abs(b)/b + (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*
(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^4 + (b^20*c*d^7 - 41*a*b^19*d^8)/(b^23*d^8)) - (7*b^21*c^2*d^6 + 2
6*a*b^20*c*d^7 - 513*a^2*b^19*d^8)/(b^23*d^8)) + 5*(7*b^22*c^3*d^5 + 19*a*b^21*c^2*d^6 + 37*a^2*b^20*c*d^7 - 4
47*a^3*b^19*d^8)/(b^23*d^8))*(b*x + a) - 15*(7*b^23*c^4*d^4 + 12*a*b^22*c^3*d^5 + 18*a^2*b^21*c^2*d^6 + 28*a^3
*b^20*c*d^7 - 193*a^4*b^19*d^8)/(b^23*d^8))*sqrt(b*x + a) - 15*(7*b^5*c^5 + 5*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2
+ 10*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 - 63*a^5*d^5)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*
b*d - a*b*d)))/(sqrt(b*d)*b^3*d^4))*b*abs(b) + 480*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*b*x + 2*a + (b*c*d
- 5*a*d^2)/d^2)*sqrt(b*x + a) + (b^3*c^2 + 2*a*b^2*c*d - 3*a^2*b*d^2)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(
b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d))*a^3*abs(b)/b^3)/b

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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