3.7.91 \(\int \frac {x^2 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx\) [691]

Optimal. Leaf size=319 \[ \frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^4 (b c-a d)}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}-\frac {5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{11/2}} \]

[Out]

2/3*c^2*(b*x+a)^(7/2)/d^2/(-a*d+b*c)/(d*x+c)^(3/2)-5/8*(-a*d+b*c)*(a^2*d^2-14*a*b*c*d+21*b^2*c^2)*arctanh(d^(1
/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/d^(11/2)/b^(1/2)-4/3*c*(-3*a*d+5*b*c)*(b*x+a)^(7/2)/d^2/(-a*d+b*c)^2/
(d*x+c)^(1/2)-5/12*(a^2*d^2-14*a*b*c*d+21*b^2*c^2)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/d^4/(-a*d+b*c)+1/3*(a^2*d^2-14*
a*b*c*d+21*b^2*c^2)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/d^3/(-a*d+b*c)^2+5/8*(a^2*d^2-14*a*b*c*d+21*b^2*c^2)*(b*x+a)^(
1/2)*(d*x+c)^(1/2)/d^5

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Rubi [A]
time = 0.23, antiderivative size = 319, 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 = {91, 79, 52, 65, 223, 212} \begin {gather*} -\frac {5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{11/2}}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{8 d^5}-\frac {5 (a+b x)^{3/2} \sqrt {c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{12 d^4 (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{3 d^3 (b c-a d)^2}+\frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {4 c (a+b x)^{7/2} (5 b c-3 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c^2*(a + b*x)^(7/2))/(3*d^2*(b*c - a*d)*(c + d*x)^(3/2)) - (4*c*(5*b*c - 3*a*d)*(a + b*x)^(7/2))/(3*d^2*(b*
c - a*d)^2*Sqrt[c + d*x]) + (5*(21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*d^5) - (5*(
21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*d^4*(b*c - a*d)) + ((21*b^2*c^2 - 14*a*b
*c*d + a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(3*d^3*(b*c - a*d)^2) - (5*(b*c - a*d)*(21*b^2*c^2 - 14*a*b*c*d
+ a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*Sqrt[b]*d^(11/2))

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 79

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

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*} \int \frac {x^2 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx &=\frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {2 \int \frac {(a+b x)^{5/2} \left (\frac {1}{2} c (7 b c-3 a d)-\frac {3}{2} d (b c-a d) x\right )}{(c+d x)^{3/2}} \, dx}{3 d^2 (b c-a d)}\\ &=\frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{d^2 (b c-a d)^2}\\ &=\frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}-\frac {\left (5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{6 d^3 (b c-a d)}\\ &=\frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^4 (b c-a d)}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}+\frac {\left (5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 d^4}\\ &=\frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^4 (b c-a d)}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}-\frac {\left (5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 d^5}\\ &=\frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^4 (b c-a d)}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}-\frac {\left (5 (b c-a d) \left (21 b^2 c^2-14 a b c d+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 )}{8 b d^5}\\ &=\frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^4 (b c-a d)}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}-\frac {\left (5 (b c-a d) \left (21 b^2 c^2-14 a b c d+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 )}{8 b d^5}\\ &=\frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^4 (b c-a d)}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}-\frac {5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 3.70, size = 214, normalized size = 0.67 \begin {gather*} \frac {\frac {d \sqrt {a+b x} \left (a^2 d^2 \left (113 c^2+162 c d x+33 d^2 x^2\right )-2 a b d \left (210 c^3+287 c^2 d x+48 c d^2 x^2-13 d^3 x^3\right )+b^2 \left (315 c^4+420 c^3 d x+63 c^2 d^2 x^2-18 c d^3 x^3+8 d^4 x^4\right )\right )}{(c+d x)^{3/2}}+\frac {15 \left (21 b^3 c^3-35 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) \log \left (\sqrt {a+b x}-\sqrt {\frac {b}{d}} \sqrt {c+d x}\right )}{\sqrt {\frac {b}{d}}}}{24 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d*Sqrt[a + b*x]*(a^2*d^2*(113*c^2 + 162*c*d*x + 33*d^2*x^2) - 2*a*b*d*(210*c^3 + 287*c^2*d*x + 48*c*d^2*x^2
- 13*d^3*x^3) + b^2*(315*c^4 + 420*c^3*d*x + 63*c^2*d^2*x^2 - 18*c*d^3*x^3 + 8*d^4*x^4)))/(c + d*x)^(3/2) + (1
5*(21*b^3*c^3 - 35*a*b^2*c^2*d + 15*a^2*b*c*d^2 - a^3*d^3)*Log[Sqrt[a + b*x] - Sqrt[b/d]*Sqrt[c + d*x]])/Sqrt[
b/d])/(24*d^6)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1001\) vs. \(2(275)=550\).
time = 0.08, size = 1002, normalized size = 3.14

method	result	size
default	\(\frac {\sqrt {b x +a}\, \left (16 b^{2} d^{4} x^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+15 \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} d^{5} x^{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^{2} b c \,d^{4} x^{2}+525 \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^{2} c^{2} d^{3} x^{2}-315 \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^{3} c^{3} d^{2} x^{2}+52 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,d^{4} x^{3}-36 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c \,d^{3} x^{3}+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} c \,d^{4} x -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 \,c^{2} d^{3} x +1050 \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^{2} c^{3} d^{2} x -630 \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^{3} c^{4} d x +66 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{4} x^{2}-192 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b c \,d^{3} x^{2}+126 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{2} d^{2} x^{2}+15 \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} c^{2} d^{3}-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^{2} b \,c^{3} d^{2}+525 \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^{2} c^{4} d -315 \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^{3} c^{5}+324 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c \,d^{3} x -1148 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,c^{2} d^{2} x +840 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{3} d x +226 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c^{2} d^{2}-840 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,c^{3} d +630 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{4}\right )}{48 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, \left (d x +c \right )^{\frac {3}{2}} d^{5}}\)	\(1002\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(b*x+a)^(1/2)*(16*b^2*d^4*x^4*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+15*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*d^5*x^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^2*b*c*d^4*x^2+525*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^2*c^2*d^3*x^2-315*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^3*
c^3*d^2*x^2+52*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b*d^4*x^3-36*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^2*c*d^
3*x^3+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*c*d^4*x-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*c^2*d^3*x+1050*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^2*c^3*d^2*x-630*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^3*c^4*d*x+66*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*d^4*x^2-192*(b*
d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b*c*d^3*x^2+126*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^2*c^2*d^2*x^2+15*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*c^2*d^3-225*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^2*b*c^3*d^2+525*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^2*c^4*d-315*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^3*c^5+324*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*c*d^3*x-1148*(b*d)^(1/2)*((d*x+c)*(b*x
+a))^(1/2)*a*b*c^2*d^2*x+840*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^2*c^3*d*x+226*(b*d)^(1/2)*((d*x+c)*(b*x+a))
^(1/2)*a^2*c^2*d^2-840*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b*c^3*d+630*(b*d)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b
^2*c^4)/((d*x+c)*(b*x+a))^(1/2)/(b*d)^(1/2)/(d*x+c)^(3/2)/d^5

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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)^(5/2)/(d*x+c)^(5/2),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 = 1.70, size = 810, normalized size = 2.54 \begin {gather*} \left [-\frac {15 \, {\left (21 \, b^{3} c^{5} - 35 \, a b^{2} c^{4} d + 15 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} + {\left (21 \, b^{3} c^{3} d^{2} - 35 \, a b^{2} c^{2} d^{3} + 15 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{2} + 2 \, {\left (21 \, b^{3} c^{4} d - 35 \, a b^{2} c^{3} d^{2} + 15 \, a^{2} b c^{2} d^{3} - a^{3} c 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 (8 \, b^{3} d^{5} x^{4} + 315 \, b^{3} c^{4} d - 420 \, a b^{2} c^{3} d^{2} + 113 \, a^{2} b c^{2} d^{3} - 2 \, {\left (9 \, b^{3} c d^{4} - 13 \, a b^{2} d^{5}\right )} x^{3} + 3 \, {\left (21 \, b^{3} c^{2} d^{3} - 32 \, a b^{2} c d^{4} + 11 \, a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (210 \, b^{3} c^{3} d^{2} - 287 \, a b^{2} c^{2} d^{3} + 81 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (b d^{8} x^{2} + 2 \, b c d^{7} x + b c^{2} d^{6}\right )}}, \frac {15 \, {\left (21 \, b^{3} c^{5} - 35 \, a b^{2} c^{4} d + 15 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} + {\left (21 \, b^{3} c^{3} d^{2} - 35 \, a b^{2} c^{2} d^{3} + 15 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{2} + 2 \, {\left (21 \, b^{3} c^{4} d - 35 \, a b^{2} c^{3} d^{2} + 15 \, a^{2} b c^{2} d^{3} - a^{3} c 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 (8 \, b^{3} d^{5} x^{4} + 315 \, b^{3} c^{4} d - 420 \, a b^{2} c^{3} d^{2} + 113 \, a^{2} b c^{2} d^{3} - 2 \, {\left (9 \, b^{3} c d^{4} - 13 \, a b^{2} d^{5}\right )} x^{3} + 3 \, {\left (21 \, b^{3} c^{2} d^{3} - 32 \, a b^{2} c d^{4} + 11 \, a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (210 \, b^{3} c^{3} d^{2} - 287 \, a b^{2} c^{2} d^{3} + 81 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b d^{8} x^{2} + 2 \, b c d^{7} x + b c^{2} d^{6}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/96*(15*(21*b^3*c^5 - 35*a*b^2*c^4*d + 15*a^2*b*c^3*d^2 - a^3*c^2*d^3 + (21*b^3*c^3*d^2 - 35*a*b^2*c^2*d^3
+ 15*a^2*b*c*d^4 - a^3*d^5)*x^2 + 2*(21*b^3*c^4*d - 35*a*b^2*c^3*d^2 + 15*a^2*b*c^2*d^3 - a^3*c*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*(8*b^3*d^5*x^4 + 315*b^3*c^4*d - 420*a*b^2*c^3*d^2 + 113*a^2*b*c^2*d^3
- 2*(9*b^3*c*d^4 - 13*a*b^2*d^5)*x^3 + 3*(21*b^3*c^2*d^3 - 32*a*b^2*c*d^4 + 11*a^2*b*d^5)*x^2 + 2*(210*b^3*c^3
*d^2 - 287*a*b^2*c^2*d^3 + 81*a^2*b*c*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d^8*x^2 + 2*b*c*d^7*x + b*c^2*d^
6), 1/48*(15*(21*b^3*c^5 - 35*a*b^2*c^4*d + 15*a^2*b*c^3*d^2 - a^3*c^2*d^3 + (21*b^3*c^3*d^2 - 35*a*b^2*c^2*d^
3 + 15*a^2*b*c*d^4 - a^3*d^5)*x^2 + 2*(21*b^3*c^4*d - 35*a*b^2*c^3*d^2 + 15*a^2*b*c^2*d^3 - a^3*c*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*(8*b^3*d^5*x^4 + 315*b^3*c^4*d - 420*a*b^2*c^3*d^2 + 113*a^2*b*c^2*d^3 - 2*(9*b^3*c*d^4
- 13*a*b^2*d^5)*x^3 + 3*(21*b^3*c^2*d^3 - 32*a*b^2*c*d^4 + 11*a^2*b*d^5)*x^2 + 2*(210*b^3*c^3*d^2 - 287*a*b^2*
c^2*d^3 + 81*a^2*b*c*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d^8*x^2 + 2*b*c*d^7*x + b*c^2*d^6)]

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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)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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Giac [A]
time = 1.64, size = 514, normalized size = 1.61 \begin {gather*} \frac {{\left ({\left ({\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b^{6} c d^{8} - a b^{5} d^{9}\right )} {\left (b x + a\right )}}{b^{4} c d^{9} {\left | b \right |} - a b^{3} d^{10} {\left | b \right |}} - \frac {3 \, {\left (3 \, b^{7} c^{2} d^{7} - 2 \, a b^{6} c d^{8} - a^{2} b^{5} d^{9}\right )}}{b^{4} c d^{9} {\left | b \right |} - a b^{3} d^{10} {\left | b \right |}}\right )} + \frac {3 \, {\left (21 \, b^{8} c^{3} d^{6} - 35 \, a b^{7} c^{2} d^{7} + 15 \, a^{2} b^{6} c d^{8} - a^{3} b^{5} d^{9}\right )}}{b^{4} c d^{9} {\left | b \right |} - a b^{3} d^{10} {\left | b \right |}}\right )} {\left (b x + a\right )} + \frac {20 \, {\left (21 \, b^{9} c^{4} d^{5} - 56 \, a b^{8} c^{3} d^{6} + 50 \, a^{2} b^{7} c^{2} d^{7} - 16 \, a^{3} b^{6} c d^{8} + a^{4} b^{5} d^{9}\right )}}{b^{4} c d^{9} {\left | b \right |} - a b^{3} d^{10} {\left | b \right |}}\right )} {\left (b x + a\right )} + \frac {15 \, {\left (21 \, b^{10} c^{5} d^{4} - 77 \, a b^{9} c^{4} d^{5} + 106 \, a^{2} b^{8} c^{3} d^{6} - 66 \, a^{3} b^{7} c^{2} d^{7} + 17 \, a^{4} b^{6} c d^{8} - a^{5} b^{5} d^{9}\right )}}{b^{4} c d^{9} {\left | b \right |} - a b^{3} d^{10} {\left | b \right |}}\right )} \sqrt {b x + a}}{24 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {5 \, {\left (21 \, b^{4} c^{3} - 35 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - a^{3} b 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 )}{8 \, \sqrt {b d} d^{5} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(((2*(b*x + a)*(4*(b^6*c*d^8 - a*b^5*d^9)*(b*x + a)/(b^4*c*d^9*abs(b) - a*b^3*d^10*abs(b)) - 3*(3*b^7*c^2
*d^7 - 2*a*b^6*c*d^8 - a^2*b^5*d^9)/(b^4*c*d^9*abs(b) - a*b^3*d^10*abs(b))) + 3*(21*b^8*c^3*d^6 - 35*a*b^7*c^2
*d^7 + 15*a^2*b^6*c*d^8 - a^3*b^5*d^9)/(b^4*c*d^9*abs(b) - a*b^3*d^10*abs(b)))*(b*x + a) + 20*(21*b^9*c^4*d^5
- 56*a*b^8*c^3*d^6 + 50*a^2*b^7*c^2*d^7 - 16*a^3*b^6*c*d^8 + a^4*b^5*d^9)/(b^4*c*d^9*abs(b) - a*b^3*d^10*abs(b
)))*(b*x + a) + 15*(21*b^10*c^5*d^4 - 77*a*b^9*c^4*d^5 + 106*a^2*b^8*c^3*d^6 - 66*a^3*b^7*c^2*d^7 + 17*a^4*b^6
*c*d^8 - a^5*b^5*d^9)/(b^4*c*d^9*abs(b) - a*b^3*d^10*abs(b)))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3
/2) + 5/8*(21*b^4*c^3 - 35*a*b^3*c^2*d + 15*a^2*b^2*c*d^2 - a^3*b*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)*d^5*abs(b))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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