\(\int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx\) [768]

Optimal result

Integrand size = 22, antiderivative size = 306 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=-\frac {5 (b c-a d) \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^5 d}-\frac {5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^4 d}-\frac {\left (14 a c+\frac {b c^2}{d}-\frac {63 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac {5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{11/2} d^{3/2}} \]

[Out]

-5/64*(-a*d+b*c)^2*(-63*a^2*d^2+14*a*b*c*d+b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(11
/2)/d^(3/2)-2*a^2*(d*x+c)^(7/2)/b^2/(-a*d+b*c)/(b*x+a)^(1/2)-5/96*(-63*a^2*d^2+14*a*b*c*d+b^2*c^2)*(d*x+c)^(3/
2)*(b*x+a)^(1/2)/b^4/d-1/24*(14*a*c+b*c^2/d-63*a^2*d/b)*(d*x+c)^(5/2)*(b*x+a)^(1/2)/b^2/(-a*d+b*c)+1/4*(d*x+c)
^(7/2)*(b*x+a)^(1/2)/b^2/d-5/64*(-a*d+b*c)*(-63*a^2*d^2+14*a*b*c*d+b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^5/d

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {91, 81, 52, 65, 223, 212} \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=-\frac {5 (b c-a d)^2 \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{11/2} d^{3/2}}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-\frac {63 a^2 d}{b}+14 a c+\frac {b c^2}{d}\right )}{24 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 \sqrt {a+b x} (b c-a d)}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d) \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right )}{64 b^5 d}-\frac {5 \sqrt {a+b x} (c+d x)^{3/2} \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right )}{96 b^4 d}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b^2 d} \]

[In]

Int[(x^2*(c + d*x)^(5/2))/(a + b*x)^(3/2),x]

[Out]

(-5*(b*c - a*d)*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^5*d) - (5*(b^2*c^2 + 14
*a*b*c*d - 63*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*b^4*d) - ((14*a*c + (b*c^2)/d - (63*a^2*d)/b)*Sqrt[a
+ b*x]*(c + d*x)^(5/2))/(24*b^2*(b*c - a*d)) - (2*a^2*(c + d*x)^(7/2))/(b^2*(b*c - a*d)*Sqrt[a + b*x]) + (Sqr
t[a + b*x]*(c + d*x)^(7/2))/(4*b^2*d) - (5*(b*c - a*d)^2*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*ArcTanh[(Sqrt[d]*
Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(11/2)*d^(3/2))

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 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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 {2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {2 \int \frac {(c+d x)^{5/2} \left (-\frac {1}{2} a (b c-7 a d)+\frac {1}{2} b (b c-a d) x\right )}{\sqrt {a+b x}} \, dx}{b^2 (b c-a d)} \\ & = -\frac {2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac {\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \int \frac {(c+d x)^{5/2}}{\sqrt {a+b x}} \, dx}{8 b^2 d (b c-a d)} \\ & = -\frac {\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^3 d (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac {\left (5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right )\right ) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx}{48 b^3 d} \\ & = -\frac {5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^4 d}-\frac {\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^3 d (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac {\left (5 (b c-a d) \left (b^2 c^2+14 a b c d-63 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{64 b^4 d} \\ & = -\frac {5 (b c-a d) \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^5 d}-\frac {5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^4 d}-\frac {\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^3 d (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac {\left (5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b^5 d} \\ & = -\frac {5 (b c-a d) \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^5 d}-\frac {5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^4 d}-\frac {\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^3 d (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac {\left (5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 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^6 d} \\ & = -\frac {5 (b c-a d) \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^5 d}-\frac {5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^4 d}-\frac {\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^3 d (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac {\left (5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 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^6 d} \\ & = -\frac {5 (b c-a d) \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^5 d}-\frac {5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^4 d}-\frac {\left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^3 d (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac {5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{11/2} d^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.75 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {\sqrt {c+d x} \left (-945 a^4 d^3+105 a^3 b d^2 (17 c-3 d x)+a^2 b^2 d \left (-839 c^2+637 c d x+126 d^2 x^2\right )+a b^3 \left (15 c^3-337 c^2 d x-244 c d^2 x^2-72 d^3 x^3\right )+b^4 x \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b^5 d \sqrt {a+b x}}-\frac {5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 b^{11/2} d^{3/2}} \]

[In]

Integrate[(x^2*(c + d*x)^(5/2))/(a + b*x)^(3/2),x]

[Out]

(Sqrt[c + d*x]*(-945*a^4*d^3 + 105*a^3*b*d^2*(17*c - 3*d*x) + a^2*b^2*d*(-839*c^2 + 637*c*d*x + 126*d^2*x^2) +
a*b^3*(15*c^3 - 337*c^2*d*x - 244*c*d^2*x^2 - 72*d^3*x^3) + b^4*x*(15*c^3 + 118*c^2*d*x + 136*c*d^2*x^2 + 48*
d^3*x^3)))/(192*b^5*d*Sqrt[a + b*x]) - (5*(b*c - a*d)^2*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*ArcTanh[(Sqrt[b]*S
qrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(64*b^(11/2)*d^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(960\) vs. \(2(264)=528\).

Time = 1.69 (sec) , antiderivative size = 961, normalized size of antiderivative = 3.14

method	result	size
default	\(\frac {\sqrt {d x +c}\, \left (96 b^{4} d^{3} x^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-144 a \,b^{3} d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+272 b^{4} c \,d^{2} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+945 \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} b \,d^{4} x -2100 \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^{2} c \,d^{3} x +1350 \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^{3} c^{2} d^{2} x -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 \,b^{4} c^{3} d x -15 \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^{5} c^{4} x +252 a^{2} b^{2} d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-488 a \,b^{3} c \,d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+236 b^{4} c^{2} d \,x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+945 \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^{5} d^{4}-2100 \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} b c \,d^{3}+1350 \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^{2} c^{2} d^{2}-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^{2} b^{3} c^{3} d -15 \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^{4} c^{4}-630 a^{3} b \,d^{3} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+1274 a^{2} b^{2} c \,d^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-674 a \,b^{3} c^{2} d x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+30 b^{4} c^{3} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-1890 a^{4} d^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+3570 a^{3} b c \,d^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-1678 a^{2} b^{2} c^{2} d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+30 a \,b^{3} c^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\right )}{384 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {b x +a}\, b^{5} d}\)	\(961\)

[In]

int(x^2*(d*x+c)^(5/2)/(b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/384*(d*x+c)^(1/2)*(96*b^4*d^3*x^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-144*a*b^3*d^3*x^3*((b*x+a)*(d*x+c))^(1
/2)*(b*d)^(1/2)+272*b^4*c*d^2*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+945*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*b*d^4*x-2100*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^2*c*d^3*x+1350*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^3*c^2*d^2*x-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*
b^4*c^3*d*x-15*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^5*c^4*x+252*a^2*b
^2*d^3*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-488*a*b^3*c*d^2*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+236*b^4
*c^2*d*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+945*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^5*d^4-2100*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*
b*c*d^3+1350*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^2*c^2*d^2-180*l
n(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^3*c^3*d-15*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^4*c^4-630*a^3*b*d^3*x*((b*x+a)*(d*x+c))^(1/2)*(b
*d)^(1/2)+1274*a^2*b^2*c*d^2*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-674*a*b^3*c^2*d*x*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)+30*b^4*c^3*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-1890*a^4*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3
570*a^3*b*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-1678*a^2*b^2*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+30*
a*b^3*c^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(b*x+a)^(1/2)/b^5/d

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 790, normalized size of antiderivative = 2.58 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\left [-\frac {15 \, {\left (a b^{4} c^{4} + 12 \, a^{2} b^{3} c^{3} d - 90 \, a^{3} b^{2} c^{2} d^{2} + 140 \, a^{4} b c d^{3} - 63 \, a^{5} d^{4} + {\left (b^{5} c^{4} + 12 \, a b^{4} c^{3} d - 90 \, a^{2} b^{3} c^{2} d^{2} + 140 \, a^{3} b^{2} c d^{3} - 63 \, a^{4} b d^{4}\right )} x\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^{5} d^{4} x^{4} + 15 \, a b^{4} c^{3} d - 839 \, a^{2} b^{3} c^{2} d^{2} + 1785 \, a^{3} b^{2} c d^{3} - 945 \, a^{4} b d^{4} + 8 \, {\left (17 \, b^{5} c d^{3} - 9 \, a b^{4} d^{4}\right )} x^{3} + 2 \, {\left (59 \, b^{5} c^{2} d^{2} - 122 \, a b^{4} c d^{3} + 63 \, a^{2} b^{3} d^{4}\right )} x^{2} + {\left (15 \, b^{5} c^{3} d - 337 \, a b^{4} c^{2} d^{2} + 637 \, a^{2} b^{3} c d^{3} - 315 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, {\left (b^{7} d^{2} x + a b^{6} d^{2}\right )}}, \frac {15 \, {\left (a b^{4} c^{4} + 12 \, a^{2} b^{3} c^{3} d - 90 \, a^{3} b^{2} c^{2} d^{2} + 140 \, a^{4} b c d^{3} - 63 \, a^{5} d^{4} + {\left (b^{5} c^{4} + 12 \, a b^{4} c^{3} d - 90 \, a^{2} b^{3} c^{2} d^{2} + 140 \, a^{3} b^{2} c d^{3} - 63 \, a^{4} b d^{4}\right )} x\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^{5} d^{4} x^{4} + 15 \, a b^{4} c^{3} d - 839 \, a^{2} b^{3} c^{2} d^{2} + 1785 \, a^{3} b^{2} c d^{3} - 945 \, a^{4} b d^{4} + 8 \, {\left (17 \, b^{5} c d^{3} - 9 \, a b^{4} d^{4}\right )} x^{3} + 2 \, {\left (59 \, b^{5} c^{2} d^{2} - 122 \, a b^{4} c d^{3} + 63 \, a^{2} b^{3} d^{4}\right )} x^{2} + {\left (15 \, b^{5} c^{3} d - 337 \, a b^{4} c^{2} d^{2} + 637 \, a^{2} b^{3} c d^{3} - 315 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, {\left (b^{7} d^{2} x + a b^{6} d^{2}\right )}}\right ] \]

[In]

integrate(x^2*(d*x+c)^(5/2)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/768*(15*(a*b^4*c^4 + 12*a^2*b^3*c^3*d - 90*a^3*b^2*c^2*d^2 + 140*a^4*b*c*d^3 - 63*a^5*d^4 + (b^5*c^4 + 12*
a*b^4*c^3*d - 90*a^2*b^3*c^2*d^2 + 140*a^3*b^2*c*d^3 - 63*a^4*b*d^4)*x)*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^5*d^4*x^4 + 15*a*b^4*c^3*d - 839*a^2*b^3*c^2*d^2 + 1785*a^3*b^2*c*d^3 - 945*a^4*b*d^4 + 8*(17*b^5
*c*d^3 - 9*a*b^4*d^4)*x^3 + 2*(59*b^5*c^2*d^2 - 122*a*b^4*c*d^3 + 63*a^2*b^3*d^4)*x^2 + (15*b^5*c^3*d - 337*a*
b^4*c^2*d^2 + 637*a^2*b^3*c*d^3 - 315*a^3*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^7*d^2*x + a*b^6*d^2), 1/
384*(15*(a*b^4*c^4 + 12*a^2*b^3*c^3*d - 90*a^3*b^2*c^2*d^2 + 140*a^4*b*c*d^3 - 63*a^5*d^4 + (b^5*c^4 + 12*a*b^
4*c^3*d - 90*a^2*b^3*c^2*d^2 + 140*a^3*b^2*c*d^3 - 63*a^4*b*d^4)*x)*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^5*d^4*x^4
+ 15*a*b^4*c^3*d - 839*a^2*b^3*c^2*d^2 + 1785*a^3*b^2*c*d^3 - 945*a^4*b*d^4 + 8*(17*b^5*c*d^3 - 9*a*b^4*d^4)*x
^3 + 2*(59*b^5*c^2*d^2 - 122*a*b^4*c*d^3 + 63*a^2*b^3*d^4)*x^2 + (15*b^5*c^3*d - 337*a*b^4*c^2*d^2 + 637*a^2*b
^3*c*d^3 - 315*a^3*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^7*d^2*x + a*b^6*d^2)]

Sympy [F]

\[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\int \frac {x^{2} \left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(x**2*(d*x+c)**(5/2)/(b*x+a)**(3/2),x)

[Out]

Integral(x**2*(c + d*x)**(5/2)/(a + b*x)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^2*(d*x+c)^(5/2)/(b*x+a)^(3/2),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 [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.41 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {1}{192} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{7}} + \frac {17 \, b^{28} c d^{7} {\left | b \right |} - 33 \, a b^{27} d^{8} {\left | b \right |}}{b^{34} d^{6}}\right )} + \frac {59 \, b^{29} c^{2} d^{6} {\left | b \right |} - 326 \, a b^{28} c d^{7} {\left | b \right |} + 315 \, a^{2} b^{27} d^{8} {\left | b \right |}}{b^{34} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{30} c^{3} d^{5} {\left | b \right |} - 191 \, a b^{29} c^{2} d^{6} {\left | b \right |} + 511 \, a^{2} b^{28} c d^{7} {\left | b \right |} - 325 \, a^{3} b^{27} d^{8} {\left | b \right |}\right )}}{b^{34} d^{6}}\right )} \sqrt {b x + a} - \frac {4 \, {\left (a^{2} b^{3} c^{3} d {\left | b \right |} - 3 \, a^{3} b^{2} c^{2} d^{2} {\left | b \right |} + 3 \, a^{4} b c d^{3} {\left | b \right |} - a^{5} d^{4} {\left | b \right |}\right )}}{{\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} \sqrt {b d} b^{5}} + \frac {5 \, {\left (b^{4} c^{4} {\left | b \right |} + 12 \, a b^{3} c^{3} d {\left | b \right |} - 90 \, a^{2} b^{2} c^{2} d^{2} {\left | b \right |} + 140 \, a^{3} b c d^{3} {\left | b \right |} - 63 \, a^{4} d^{4} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{128 \, \sqrt {b d} b^{6} d} \]

[In]

integrate(x^2*(d*x+c)^(5/2)/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

1/192*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)*d^2*abs(b)/b^7 + (17*b^28*c*d
^7*abs(b) - 33*a*b^27*d^8*abs(b))/(b^34*d^6)) + (59*b^29*c^2*d^6*abs(b) - 326*a*b^28*c*d^7*abs(b) + 315*a^2*b^
27*d^8*abs(b))/(b^34*d^6)) + 3*(5*b^30*c^3*d^5*abs(b) - 191*a*b^29*c^2*d^6*abs(b) + 511*a^2*b^28*c*d^7*abs(b)
- 325*a^3*b^27*d^8*abs(b))/(b^34*d^6))*sqrt(b*x + a) - 4*(a^2*b^3*c^3*d*abs(b) - 3*a^3*b^2*c^2*d^2*abs(b) + 3*
a^4*b*c*d^3*abs(b) - a^5*d^4*abs(b))/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^2)*sqrt(b*d)*b^5) + 5/128*(b^4*c^4*abs(b) + 12*a*b^3*c^3*d*abs(b) - 90*a^2*b^2*c^2*d^2*abs(b) + 140*a
^3*b*c*d^3*abs(b) - 63*a^4*d^4*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/
(sqrt(b*d)*b^6*d)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\int \frac {x^2\,{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{3/2}} \,d x \]

[In]

int((x^2*(c + d*x)^(5/2))/(a + b*x)^(3/2),x)

[Out]

int((x^2*(c + d*x)^(5/2))/(a + b*x)^(3/2), x)