3.6.98 \(\int x^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx\) [598]

Optimal. Leaf size=315 \[ -\frac {(b c-a d)^2 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^4}+\frac {(b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^3 d^3}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {(b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{9/2}} \]

[Out]

-1/40*(5*a*d+7*b*c)*(b*x+a)^(5/2)*(d*x+c)^(3/2)/b^2/d^2+1/5*x*(b*x+a)^(5/2)*(d*x+c)^(3/2)/b/d+1/128*(-a*d+b*c)
^3*(3*a^2*d^2+6*a*b*c*d+7*b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(7/2)/d^(9/2)+1/192*
(-a*d+b*c)*(3*a^2*d^2+6*a*b*c*d+7*b^2*c^2)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^3/d^3+1/48*(3*a^2*d^2+6*a*b*c*d+7*b^2
*c^2)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/b^3/d^2-1/128*(-a*d+b*c)^2*(3*a^2*d^2+6*a*b*c*d+7*b^2*c^2)*(b*x+a)^(1/2)*(d*
x+c)^(1/2)/b^3/d^4

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Rubi [A]
time = 0.20, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {92, 81, 52, 65, 223, 212} \begin {gather*} \frac {\left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{9/2}}+\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right )}{48 b^3 d^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^2}{128 b^3 d^4}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)}{192 b^3 d^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2} (5 a d+7 b c)}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^(3/2)*Sqrt[c + d*x],x]

[Out]

-1/128*((b*c - a*d)^2*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(b^3*d^4) + ((b*c - a*d
)*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(192*b^3*d^3) + ((7*b^2*c^2 + 6*a*b*c*d +
3*a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(48*b^3*d^2) - ((7*b*c + 5*a*d)*(a + b*x)^(5/2)*(c + d*x)^(3/2))/(4
0*b^2*d^2) + (x*(a + b*x)^(5/2)*(c + d*x)^(3/2))/(5*b*d) + ((b*c - a*d)^3*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*
ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(7/2)*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 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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^2 (a+b x)^{3/2} \sqrt {c+d x} \, dx &=\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {\int (a+b x)^{3/2} \sqrt {c+d x} \left (-a c-\frac {1}{2} (7 b c+5 a d) x\right ) \, dx}{5 b d}\\ &=-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \int (a+b x)^{3/2} \sqrt {c+d x} \, dx}{16 b^2 d^2}\\ &=\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {\left ((b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{96 b^3 d^2}\\ &=\frac {(b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^3 d^3}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}-\frac {\left ((b c-a d)^2 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b^3 d^3}\\ &=-\frac {(b c-a d)^2 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^4}+\frac {(b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^3 d^3}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {\left ((b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^3 d^4}\\ &=-\frac {(b c-a d)^2 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^4}+\frac {(b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^3 d^3}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {\left ((b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\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^4 d^4}\\ &=-\frac {(b c-a d)^2 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^4}+\frac {(b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^3 d^3}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {\left ((b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\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^4 d^4}\\ &=-\frac {(b c-a d)^2 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^4}+\frac {(b c-a d) \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^3 d^3}+\frac {\left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{48 b^3 d^2}-\frac {(7 b c+5 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{3/2}}{5 b d}+\frac {(b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.60, size = 241, normalized size = 0.77 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (45 a^4 d^4-30 a^3 b d^3 (c+d x)+6 a^2 b^2 d^2 \left (-6 c^2+3 c d x+4 d^2 x^2\right )+2 a b^3 d \left (95 c^3-61 c^2 d x+48 c d^2 x^2+264 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^3 d^4}+\frac {(b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{128 b^{7/2} d^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^(3/2)*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*(c + d*x) + 6*a^2*b^2*d^2*(-6*c^2 + 3*c*d*x + 4*d^2*x^
2) + 2*a*b^3*d*(95*c^3 - 61*c^2*d*x + 48*c*d^2*x^2 + 264*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^3*d^4) + ((b*c - a*d)^3*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*ArcTanh[
(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(128*b^(7/2)*d^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(787\) vs. \(2(271)=542\).
time = 0.07, size = 788, normalized size = 2.50

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 )}-1056 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 )}-48 a^{2} b^{2} d^{4} x^{2} \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-192 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}-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^{4} b c \,d^{4}-30 \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}-90 \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}+225 \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 -36 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c \,d^{3} x +244 \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}+60 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b c \,d^{3}+72 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{2} d^{2}-380 \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^{3} \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^2*(b*x+a)^(3/2)*(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)-1056*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)-48*a^2*b^2*d^4*x^2*(b*
d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)-192*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^5-45*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^4*b*c*d^4-30*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^3-90*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+225*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-36*(b*d)^(1/2)*((d*x+c)*(b
*x+a))^(1/2)*a^2*b^2*c*d^3*x+244*(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+60*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/
2)*a^3*b*c*d^3+72*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*b^2*c^2*d^2-380*(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^3/((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^2*(b*x+a)^(3/2)*(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 = 1.51, size = 704, normalized size = 2.23 \begin {gather*} \left [-\frac {15 \, {\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, 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 + 190 \, a b^{4} c^{3} d^{2} - 36 \, a^{2} b^{3} c^{2} d^{3} - 30 \, a^{3} b^{2} c d^{4} + 45 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + 11 \, a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 12 \, a b^{4} c d^{4} - 3 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 61 \, a b^{4} c^{2} d^{3} + 9 \, 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^{4} d^{5}}, -\frac {15 \, {\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, 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 + 190 \, a b^{4} c^{3} d^{2} - 36 \, a^{2} b^{3} c^{2} d^{3} - 30 \, a^{3} b^{2} c d^{4} + 45 \, a^{4} b d^{5} + 48 \, {\left (b^{5} c d^{4} + 11 \, a b^{4} d^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{5} c^{2} d^{3} - 12 \, a b^{4} c d^{4} - 3 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{5} c^{3} d^{2} - 61 \, a b^{4} c^{2} d^{3} + 9 \, 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^{4} d^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/7680*(15*(7*b^5*c^5 - 15*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 + 3*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 + 190*a*b^4*c^3*d^2 - 36*a^2*b^3
*c^2*d^3 - 30*a^3*b^2*c*d^4 + 45*a^4*b*d^5 + 48*(b^5*c*d^4 + 11*a*b^4*d^5)*x^3 - 8*(7*b^5*c^2*d^3 - 12*a*b^4*c
*d^4 - 3*a^2*b^3*d^5)*x^2 + 2*(35*b^5*c^3*d^2 - 61*a*b^4*c^2*d^3 + 9*a^2*b^3*c*d^4 - 15*a^3*b^2*d^5)*x)*sqrt(b
*x + a)*sqrt(d*x + c))/(b^4*d^5), -1/3840*(15*(7*b^5*c^5 - 15*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*
d^3 + 3*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 + 190*a*b^4*c^3*d^
2 - 36*a^2*b^3*c^2*d^3 - 30*a^3*b^2*c*d^4 + 45*a^4*b*d^5 + 48*(b^5*c*d^4 + 11*a*b^4*d^5)*x^3 - 8*(7*b^5*c^2*d^
3 - 12*a*b^4*c*d^4 - 3*a^2*b^3*d^5)*x^2 + 2*(35*b^5*c^3*d^2 - 61*a*b^4*c^2*d^3 + 9*a^2*b^3*c*d^4 - 15*a^3*b^2*
d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*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**2*(b*x+a)**(3/2)*(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. 878 vs. \(2 (271) = 542\).
time = 1.99, size = 878, normalized size = 2.79 \begin {gather*} \frac {{\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 )} {\left | b \right |} + \frac {80 \, {\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^{2}} + \frac {20 \, {\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 |}}{b}}{1920 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1920*((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 + 26*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 - 447*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))*abs(b) + 80*(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^2 +
20*(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)/b)/b

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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