3.6.41 \(\int x^2 \sqrt {a+b x} (c+d x)^{5/2} \, dx\)

Optimal. Leaf size=376 \[ -\frac {\left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{11/2} d^{7/2}}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^2}{256 b^5 d^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^3}{512 b^5 d^3}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)}{192 b^4 d^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right )}{160 b^3 d^2}-\frac {(a+b x)^{3/2} (c+d x)^{7/2} (9 a d+5 b c)}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d} \]

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Rubi [A]  time = 0.37, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {90, 80, 50, 63, 217, 206} \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^3}{512 b^5 d^3}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^2}{256 b^5 d^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)}{192 b^4 d^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right )}{160 b^3 d^2}-\frac {\left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{11/2} d^{7/2}}-\frac {(a+b x)^{3/2} (c+d x)^{7/2} (9 a d+5 b c)}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)^3*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b^5*d^3) + ((b*c - a*d)
^2*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(256*b^5*d^2) + ((b*c - a*d)*(5*b^2*c^
2 + 14*a*b*c*d + 21*a^2*d^2)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(192*b^4*d^2) + ((5*b^2*c^2 + 14*a*b*c*d + 21*a^
2*d^2)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(160*b^3*d^2) - ((5*b*c + 9*a*d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(60*
b^2*d^2) + (x*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(6*b*d) - ((b*c - a*d)^4*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*
ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(11/2)*d^(7/2))

Rule 50

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 63

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 80

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 90

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

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 \sqrt {a+b x} (c+d x)^{5/2} \, dx &=\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\int \sqrt {a+b x} (c+d x)^{5/2} \left (-a c-\frac {1}{2} (5 b c+9 a d) x\right ) \, dx}{6 b d}\\ &=-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \int \sqrt {a+b x} (c+d x)^{5/2} \, dx}{40 b^2 d^2}\\ &=\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\left ((b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \int \sqrt {a+b x} (c+d x)^{3/2} \, dx}{64 b^3 d^2}\\ &=\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\left ((b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \int \sqrt {a+b x} \sqrt {c+d x} \, dx}{128 b^4 d^2}\\ &=\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}+\frac {\left ((b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{512 b^5 d^2}\\ &=\frac {(b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{1024 b^5 d^3}\\ &=\frac {(b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{512 b^6 d^3}\\ &=\frac {(b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{512 b^6 d^3}\\ &=\frac {(b c-a d)^3 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{256 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{5/2}}{160 b^3 d^2}-\frac {(5 b c+9 a d) (a+b x)^{3/2} (c+d x)^{7/2}}{60 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d}-\frac {(b c-a d)^4 \left (5 b^2 c^2+14 a b c d+21 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{11/2} d^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 2.01, size = 320, normalized size = 0.85 \begin {gather*} \frac {(a+b x)^{3/2} (c+d x)^{7/2} \left (\frac {3 \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \left (2 b^5 d^2 (a+b x)^2 (b c-a d)^{3/2} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (15 a^2 d^2-10 a b d (5 c+2 d x)+b^2 \left (59 c^2+68 c d x+24 d^2 x^2\right )\right )+15 b^5 d (a+b x) (b c-a d)^{9/2} \sqrt {\frac {b (c+d x)}{b c-a d}}-15 b^5 \sqrt {d} \sqrt {a+b x} (b c-a d)^5 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )}{128 b^9 d^2 (a+b x)^2 (c+d x)^4 \sqrt {b c-a d}}-3 (9 a d+5 b c)+30 b d x\right )}{180 b^2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(3/2)*(c + d*x)^(7/2)*(-3*(5*b*c + 9*a*d) + 30*b*d*x + (3*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*Sqr
t[(b*(c + d*x))/(b*c - a*d)]*(15*b^5*d*(b*c - a*d)^(9/2)*(a + b*x)*Sqrt[(b*(c + d*x))/(b*c - a*d)] + 2*b^5*d^2
*(b*c - a*d)^(3/2)*(a + b*x)^2*Sqrt[(b*(c + d*x))/(b*c - a*d)]*(15*a^2*d^2 - 10*a*b*d*(5*c + 2*d*x) + b^2*(59*
c^2 + 68*c*d*x + 24*d^2*x^2)) - 15*b^5*Sqrt[d]*(b*c - a*d)^5*Sqrt[a + b*x]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqr
t[b*c - a*d]]))/(128*b^9*d^2*Sqrt[b*c - a*d]*(a + b*x)^2*(c + d*x)^4)))/(180*b^2*d^2)

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IntegrateAlgebraic [A]  time = 0.89, size = 518, normalized size = 1.38 \begin {gather*} \frac {\sqrt {a+b x} (b c-a d)^4 \left (315 a^2 b^5 d^2+\frac {3335 a^2 b^4 d^3 (a+b x)}{c+d x}-\frac {5058 a^2 b^3 d^4 (a+b x)^2}{(c+d x)^2}+\frac {4158 a^2 b^2 d^5 (a+b x)^3}{(c+d x)^3}+\frac {315 a^2 d^7 (a+b x)^5}{(c+d x)^5}-\frac {1785 a^2 b d^6 (a+b x)^4}{(c+d x)^4}-\frac {425 b^6 c^2 d (a+b x)}{c+d x}+210 a b^6 c d+\frac {990 b^5 c^2 d^2 (a+b x)^2}{(c+d x)^2}-\frac {1190 a b^5 c d^2 (a+b x)}{c+d x}+\frac {990 b^4 c^2 d^3 (a+b x)^3}{(c+d x)^3}-\frac {3372 a b^4 c d^3 (a+b x)^2}{(c+d x)^2}-\frac {425 b^3 c^2 d^4 (a+b x)^4}{(c+d x)^4}+\frac {2772 a b^3 c d^4 (a+b x)^3}{(c+d x)^3}+\frac {75 b^2 c^2 d^5 (a+b x)^5}{(c+d x)^5}-\frac {1190 a b^2 c d^5 (a+b x)^4}{(c+d x)^4}+\frac {210 a b c d^6 (a+b x)^5}{(c+d x)^5}+75 b^7 c^2\right )}{7680 b^5 d^3 \sqrt {c+d x} \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {(b c-a d)^4 \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{11/2} d^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)^4*Sqrt[a + b*x]*(75*b^7*c^2 + 210*a*b^6*c*d + 315*a^2*b^5*d^2 + (75*b^2*c^2*d^5*(a + b*x)^5)/(c +
d*x)^5 + (210*a*b*c*d^6*(a + b*x)^5)/(c + d*x)^5 + (315*a^2*d^7*(a + b*x)^5)/(c + d*x)^5 - (425*b^3*c^2*d^4*(
a + b*x)^4)/(c + d*x)^4 - (1190*a*b^2*c*d^5*(a + b*x)^4)/(c + d*x)^4 - (1785*a^2*b*d^6*(a + b*x)^4)/(c + d*x)^
4 + (990*b^4*c^2*d^3*(a + b*x)^3)/(c + d*x)^3 + (2772*a*b^3*c*d^4*(a + b*x)^3)/(c + d*x)^3 + (4158*a^2*b^2*d^5
*(a + b*x)^3)/(c + d*x)^3 + (990*b^5*c^2*d^2*(a + b*x)^2)/(c + d*x)^2 - (3372*a*b^4*c*d^3*(a + b*x)^2)/(c + d*
x)^2 - (5058*a^2*b^3*d^4*(a + b*x)^2)/(c + d*x)^2 - (425*b^6*c^2*d*(a + b*x))/(c + d*x) - (1190*a*b^5*c*d^2*(a
+ b*x))/(c + d*x) + (3335*a^2*b^4*d^3*(a + b*x))/(c + d*x)))/(7680*b^5*d^3*Sqrt[c + d*x]*(b - (d*(a + b*x))/(
c + d*x))^6) - ((b*c - a*d)^4*(5*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*S
qrt[c + d*x])])/(512*b^(11/2)*d^(7/2))

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fricas [A]  time = 1.77, size = 892, normalized size = 2.37 \begin {gather*} \left [\frac {15 \, {\left (5 \, b^{6} c^{6} - 6 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 75 \, a^{4} b^{2} c^{2} d^{4} - 70 \, a^{5} b c d^{5} + 21 \, a^{6} d^{6}\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 (1280 \, b^{6} d^{6} x^{5} + 75 \, b^{6} c^{5} d - 65 \, a b^{5} c^{4} d^{2} - 90 \, a^{2} b^{4} c^{3} d^{3} + 838 \, a^{3} b^{3} c^{2} d^{4} - 945 \, a^{4} b^{2} c d^{5} + 315 \, a^{5} b d^{6} + 128 \, {\left (25 \, b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (135 \, b^{6} c^{2} d^{4} + 26 \, a b^{5} c d^{5} - 9 \, a^{2} b^{4} d^{6}\right )} x^{3} + 8 \, {\left (5 \, b^{6} c^{3} d^{3} + 51 \, a b^{5} c^{2} d^{4} - 61 \, a^{2} b^{4} c d^{5} + 21 \, a^{3} b^{3} d^{6}\right )} x^{2} - 2 \, {\left (25 \, b^{6} c^{4} d^{2} - 20 \, a b^{5} c^{3} d^{3} + 262 \, a^{2} b^{4} c^{2} d^{4} - 308 \, a^{3} b^{3} c d^{5} + 105 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30720 \, b^{6} d^{4}}, \frac {15 \, {\left (5 \, b^{6} c^{6} - 6 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 75 \, a^{4} b^{2} c^{2} d^{4} - 70 \, a^{5} b c d^{5} + 21 \, a^{6} d^{6}\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 (1280 \, b^{6} d^{6} x^{5} + 75 \, b^{6} c^{5} d - 65 \, a b^{5} c^{4} d^{2} - 90 \, a^{2} b^{4} c^{3} d^{3} + 838 \, a^{3} b^{3} c^{2} d^{4} - 945 \, a^{4} b^{2} c d^{5} + 315 \, a^{5} b d^{6} + 128 \, {\left (25 \, b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (135 \, b^{6} c^{2} d^{4} + 26 \, a b^{5} c d^{5} - 9 \, a^{2} b^{4} d^{6}\right )} x^{3} + 8 \, {\left (5 \, b^{6} c^{3} d^{3} + 51 \, a b^{5} c^{2} d^{4} - 61 \, a^{2} b^{4} c d^{5} + 21 \, a^{3} b^{3} d^{6}\right )} x^{2} - 2 \, {\left (25 \, b^{6} c^{4} d^{2} - 20 \, a b^{5} c^{3} d^{3} + 262 \, a^{2} b^{4} c^{2} d^{4} - 308 \, a^{3} b^{3} c d^{5} + 105 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15360 \, b^{6} d^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/30720*(15*(5*b^6*c^6 - 6*a*b^5*c^5*d - 5*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 75*a^4*b^2*c^2*d^4 - 70*a^5
*b*c*d^5 + 21*a^6*d^6)*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)*s
qrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(1280*b^6*d^6*x^5 + 75*b^6*c^5*d - 65*a*b^
5*c^4*d^2 - 90*a^2*b^4*c^3*d^3 + 838*a^3*b^3*c^2*d^4 - 945*a^4*b^2*c*d^5 + 315*a^5*b*d^6 + 128*(25*b^6*c*d^5 +
a*b^5*d^6)*x^4 + 16*(135*b^6*c^2*d^4 + 26*a*b^5*c*d^5 - 9*a^2*b^4*d^6)*x^3 + 8*(5*b^6*c^3*d^3 + 51*a*b^5*c^2*
d^4 - 61*a^2*b^4*c*d^5 + 21*a^3*b^3*d^6)*x^2 - 2*(25*b^6*c^4*d^2 - 20*a*b^5*c^3*d^3 + 262*a^2*b^4*c^2*d^4 - 30
8*a^3*b^3*c*d^5 + 105*a^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*d^4), 1/15360*(15*(5*b^6*c^6 - 6*a*b^5
*c^5*d - 5*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 75*a^4*b^2*c^2*d^4 - 70*a^5*b*c*d^5 + 21*a^6*d^6)*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*(1280*b^6*d^6*x^5 + 75*b^6*c^5*d - 65*a*b^5*c^4*d^2 - 90*a^2*b^4*c^3*d^3 + 838*a^3*b^3*c^2*d^4
- 945*a^4*b^2*c*d^5 + 315*a^5*b*d^6 + 128*(25*b^6*c*d^5 + a*b^5*d^6)*x^4 + 16*(135*b^6*c^2*d^4 + 26*a*b^5*c*d
^5 - 9*a^2*b^4*d^6)*x^3 + 8*(5*b^6*c^3*d^3 + 51*a*b^5*c^2*d^4 - 61*a^2*b^4*c*d^5 + 21*a^3*b^3*d^6)*x^2 - 2*(25
*b^6*c^4*d^2 - 20*a*b^5*c^3*d^3 + 262*a^2*b^4*c^2*d^4 - 308*a^3*b^3*c*d^5 + 105*a^4*b^2*d^6)*x)*sqrt(b*x + a)*
sqrt(d*x + c))/(b^6*d^4)]

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giac [B]  time = 3.05, size = 2043, normalized size = 5.43

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7680*(320*(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*c^2*abs(b)/b^2 + 40*(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))*c^2*abs(b)/b + 80*(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*c*d*abs(b)/b^2 + 8*(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))*sqr
t(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))*c*d*abs(b)/
b + 4*(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))*a*d^2*abs(b)/b^2 + (sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*(10*(b*x + a)/b^5 + (b^30*c*d^9 - 61*a*b^29*d^10)/(b^
34*d^10)) - 3*(3*b^31*c^2*d^8 + 14*a*b^30*c*d^9 - 417*a^2*b^29*d^10)/(b^34*d^10)) + (21*b^32*c^3*d^7 + 77*a*b^
31*c^2*d^8 + 183*a^2*b^30*c*d^9 - 3481*a^3*b^29*d^10)/(b^34*d^10))*(b*x + a) - 5*(21*b^33*c^4*d^6 + 56*a*b^32*
c^3*d^7 + 106*a^2*b^31*c^2*d^8 + 176*a^3*b^30*c*d^9 - 2279*a^4*b^29*d^10)/(b^34*d^10))*(b*x + a) + 15*(21*b^34
*c^5*d^5 + 35*a*b^33*c^4*d^6 + 50*a^2*b^32*c^3*d^7 + 70*a^3*b^31*c^2*d^8 + 105*a^4*b^30*c*d^9 - 793*a^5*b^29*d
^10)/(b^34*d^10))*sqrt(b*x + a) + 15*(21*b^6*c^6 + 14*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 + 20*a^3*b^3*c^3*d^3 +
35*a^4*b^2*c^2*d^4 + 126*a^5*b*c*d^5 - 231*a^6*d^6)*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^4*d^5))*d^2*abs(b)/b)/b

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maple [B]  time = 0.02, size = 1240, normalized size = 3.30 \begin {gather*} -\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (-2560 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} d^{5} x^{5}-256 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} d^{5} x^{4}-6400 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c \,d^{4} x^{4}+315 a^{6} d^{6} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-1050 a^{5} b c \,d^{5} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+1125 a^{4} b^{2} c^{2} d^{4} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-300 a^{3} b^{3} c^{3} d^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-75 a^{2} b^{4} c^{4} d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+288 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} d^{5} x^{3}-90 a \,b^{5} c^{5} d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-832 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c \,d^{4} x^{3}+75 b^{6} c^{6} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-4320 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{2} d^{3} x^{3}-336 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} d^{5} x^{2}+976 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c \,d^{4} x^{2}-816 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{2} d^{3} x^{2}-80 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{3} d^{2} x^{2}+420 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b \,d^{5} x -1232 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c \,d^{4} x +1048 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c^{2} d^{3} x -80 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{3} d^{2} x +100 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{4} d x -630 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{5} d^{5}+1890 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} b c \,d^{4}-1676 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b^{2} c^{2} d^{3}+180 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{3} c^{3} d^{2}+130 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{4} c^{4} d -150 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{5} c^{5}\right )}{15360 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{5} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15360*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(-256*x^4*a*b^4*d^5*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)-6400*x^4*
b^5*c*d^4*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+288*x^3*a^2*b^3*d^5*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c
)^(1/2)-4320*x^3*b^5*c^2*d^3*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)-336*x^2*a^3*b^2*d^5*(b*d)^(1/2)*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)-80*x^2*b^5*c^3*d^2*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+420*(b*d)^(1/2)*(b*d
*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*b*d^5+100*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*b^5*c^4*d+1890*(b*d)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*b*c*d^4-1676*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b^2*c^
2*d^3+180*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^3*c^3*d^2+130*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c
)^(1/2)*a*b^4*c^4*d-1050*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))*a
^5*b*c*d^5+1125*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))*a^4*b^2*c^
2*d^4-300*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))*a^3*b^3*c^3*d^3-
75*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))*a^2*b^4*c^4*d^2-90*ln(1
/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))*a*b^5*c^5*d-2560*x^5*b^5*d^5*(
b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)-630*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*d^5-150*(b*d)^(
1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^5*c^5+315*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*
d)^(1/2))/(b*d)^(1/2))*a^6*d^6+75*ln(1/2*(2*b*d*x+a*d+b*c+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d)
^(1/2))*b^6*c^6-832*x^3*a*b^4*c*d^4*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+976*x^2*a^2*b^3*c*d^4*(b*d)^(1
/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)-816*x^2*a*b^4*c^2*d^3*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)-1232*(b*
d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*b^2*c*d^4+1048*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^
2*b^3*c^2*d^3-80*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a*b^4*c^3*d^2)/d^3/(b*d*x^2+a*d*x+b*c*x+a*c)^(1
/2)/b^5/(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*(d*x+c)^(5/2)*(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 details)Is a*d-b*c zero or nonzero?

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

[Out]

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

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sympy [F(-1)]  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*(d*x+c)**(5/2)*(b*x+a)**(1/2),x)

[Out]

Timed out

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