\(\int \frac {x^2 (c+d x)^{3/2}}{\sqrt {a+b x}} \, dx\) [706]

Optimal result

Integrand size = 22, antiderivative size = 254 \[ \int \frac {x^2 (c+d x)^{3/2}}{\sqrt {a+b x}} \, dx=\frac {(b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {(3 b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{5/2}} \]

[Out]

1/64*(-a*d+b*c)^2*(35*a^2*d^2+10*a*b*c*d+3*b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(9/
2)/d^(5/2)+1/96*(35*a^2*d^2+10*a*b*c*d+3*b^2*c^2)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/b^3/d^2-1/24*(7*a*d+3*b*c)*(d*x+
c)^(5/2)*(b*x+a)^(1/2)/b^2/d^2+1/4*x*(d*x+c)^(5/2)*(b*x+a)^(1/2)/b/d+1/64*(-a*d+b*c)*(35*a^2*d^2+10*a*b*c*d+3*
b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^4/d^2

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {92, 81, 52, 65, 223, 212} \[ \int \frac {x^2 (c+d x)^{3/2}}{\sqrt {a+b x}} \, dx=\frac {(b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{5/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d) \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{64 b^4 d^2}+\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{96 b^3 d^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (7 a d+3 b c)}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d} \]

[In]

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

[Out]

((b*c - a*d)*(3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^4*d^2) + ((3*b^2*c^2 + 1
0*a*b*c*d + 35*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*b^3*d^2) - ((3*b*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)
^(5/2))/(24*b^2*d^2) + (x*Sqrt[a + b*x]*(c + d*x)^(5/2))/(4*b*d) + ((b*c - a*d)^2*(3*b^2*c^2 + 10*a*b*c*d + 35
*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(9/2)*d^(5/2))

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*} \text {integral}& = \frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {\int \frac {(c+d x)^{3/2} \left (-a c-\frac {1}{2} (3 b c+7 a d) x\right )}{\sqrt {a+b x}} \, dx}{4 b d} \\ & = -\frac {(3 b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx}{48 b^2 d^2} \\ & = \frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {(3 b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {\left ((b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{64 b^3 d^2} \\ & = \frac {(b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {(3 b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b^4 d^2} \\ & = \frac {(b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {(3 b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 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 )}{64 b^5 d^2} \\ & = \frac {(b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {(3 b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 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 )}{64 b^5 d^2} \\ & = \frac {(b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {(3 b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 (c+d x)^{3/2}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^3 d^3+5 a^2 b d^2 (29 c+14 d x)-a b^2 d \left (15 c^2+92 c d x+56 d^2 x^2\right )+b^3 \left (-9 c^3+6 c^2 d x+72 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b^4 d^2}+\frac {(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{5/2}} \]

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^3*d^3 + 5*a^2*b*d^2*(29*c + 14*d*x) - a*b^2*d*(15*c^2 + 92*c*d*x + 56*d^2
*x^2) + b^3*(-9*c^3 + 6*c^2*d*x + 72*c*d^2*x^2 + 48*d^3*x^3)))/(192*b^4*d^2) + ((b*c - a*d)^2*(3*b^2*c^2 + 10*
a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(9/2)*d^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(573\) vs. \(2(216)=432\).

Time = 0.58 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.26

method	result	size
default	\(\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (96 b^{3} d^{3} x^{3} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-112 a \,b^{2} d^{3} x^{2} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+144 b^{3} c \,d^{2} x^{2} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+105 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} d^{4}-180 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b c \,d^{3}+54 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c^{2} d^{2}+12 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{3} d +9 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{4} c^{4}+140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b \,d^{3} x -184 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c \,d^{2} x +12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{2} d x -210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} d^{3}+290 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b c \,d^{2}-30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c^{2} d -18 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{3}\right )}{384 b^{4} d^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}\)	\(574\)

[In]

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

[Out]

1/384*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(96*b^3*d^3*x^3*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-112*a*b^2*d^3*x^2*(b*d)^
(1/2)*((b*x+a)*(d*x+c))^(1/2)+144*b^3*c*d^2*x^2*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+105*ln(1/2*(2*b*d*x+2*((b*
x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*d^4-180*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b
*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b*c*d^3+54*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/
(b*d)^(1/2))*a^2*b^2*c^2*d^2+12*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*
b^3*c^3*d+9*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^4*c^4+140*((b*x+a)*(
d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b*d^3*x-184*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^2*c*d^2*x+12*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)*b^3*c^2*d*x-210*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*d^3+290*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)*a^2*b*c*d^2-30*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^2*c^2*d-18*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/
2)*b^3*c^3)/b^4/d^2/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.15 \[ \int \frac {x^2 (c+d x)^{3/2}}{\sqrt {a+b x}} \, dx=\left [\frac {3 \, {\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (48 \, b^{4} d^{4} x^{3} - 9 \, b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} + 145 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4} + 8 \, {\left (9 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (3 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b^{5} d^{3}}, -\frac {3 \, {\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (48 \, b^{4} d^{4} x^{3} - 9 \, b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} + 145 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4} + 8 \, {\left (9 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (3 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b^{5} d^{3}}\right ] \]

[In]

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

[Out]

[1/768*(3*(3*b^4*c^4 + 4*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 60*a^3*b*c*d^3 + 35*a^4*d^4)*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*(48*b^4*d^4*x^3 - 9*b^4*c^3*d - 15*a*b^3*c^2*d^2 + 145*a^2*b^2*c*d^3 - 105*a^3*b*d^4 +
8*(9*b^4*c*d^3 - 7*a*b^3*d^4)*x^2 + 2*(3*b^4*c^2*d^2 - 46*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sqrt
(d*x + c))/(b^5*d^3), -1/384*(3*(3*b^4*c^4 + 4*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 60*a^3*b*c*d^3 + 35*a^4*d^4)
*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*(48*b^4*d^4*x^3 - 9*b^4*c^3*d - 15*a*b^3*c^2*d^2 + 145*a^2*b^2*c*d^3 - 105*a^3*b*d^
4 + 8*(9*b^4*c*d^3 - 7*a*b^3*d^4)*x^2 + 2*(3*b^4*c^2*d^2 - 46*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x)*sqrt(b*x + a)*s
qrt(d*x + c))/(b^5*d^3)]

Sympy [F]

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

[In]

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

[Out]

Integral(x**2*(c + d*x)**(3/2)/sqrt(a + b*x), x)

Maxima [F(-2)]

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

[In]

integrate(x^2*(d*x+c)^(3/2)/(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*d-b*c>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (216) = 432\).

Time = 0.33 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.97 \[ \int \frac {x^2 (c+d x)^{3/2}}{\sqrt {a+b x}} \, dx=\frac {\frac {8 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{2}}\right )} c {\left | b \right |}}{b^{2}} + \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} + \frac {b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac {5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{3}}\right )} d {\left | b \right |}}{b^{2}}}{192 \, b} \]

[In]

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

[Out]

1/192*(8*(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))*c*abs(b)/b^2 + (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) + sqr
t(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3))*d*abs(b)/b^2)/b

Mupad [F(-1)]

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

[In]

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

[Out]

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