\(\int x^2 (a+b x)^{5/2} \sqrt {c+d x} \, dx\) [647]

Optimal result

Integrand size = 22, antiderivative size = 376 \[ \int x^2 (a+b x)^{5/2} \sqrt {c+d x} \, dx=\frac {(b c-a d)^3 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^5}-\frac {(b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^4}+\frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}-\frac {(b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{11/2}} \]

[Out]

-1/60*(5*a*d+9*b*c)*(b*x+a)^(7/2)*(d*x+c)^(3/2)/b^2/d^2+1/6*x*(b*x+a)^(7/2)*(d*x+c)^(3/2)/b/d-1/512*(-a*d+b*c)
^4*(5*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))/b^(7/2)/d^(11/2)-1/7
68*(-a*d+b*c)^2*(5*a^2*d^2+14*a*b*c*d+21*b^2*c^2)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^3/d^4+1/960*(-a*d+b*c)*(5*a^2*
d^2+14*a*b*c*d+21*b^2*c^2)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/b^3/d^3+1/160*(5*a^2*d^2+14*a*b*c*d+21*b^2*c^2)*(b*x+a)
^(7/2)*(d*x+c)^(1/2)/b^3/d^2+1/512*(-a*d+b*c)^3*(5*a^2*d^2+14*a*b*c*d+21*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/
b^3/d^5

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {92, 81, 52, 65, 223, 212} \[ \int x^2 (a+b x)^{5/2} \sqrt {c+d x} \, dx=-\frac {\left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{11/2}}+\frac {(a+b x)^{7/2} \sqrt {c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right )}{160 b^3 d^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^3}{512 b^3 d^5}-\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^2}{768 b^3 d^4}+\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)}{960 b^3 d^3}-\frac {(a+b x)^{7/2} (c+d x)^{3/2} (5 a d+9 b c)}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d} \]

[In]

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

[Out]

((b*c - a*d)^3*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b^3*d^5) - ((b*c - a*d)
^2*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(768*b^3*d^4) + ((b*c - a*d)*(21*b^2*c
^2 + 14*a*b*c*d + 5*a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(960*b^3*d^3) + ((21*b^2*c^2 + 14*a*b*c*d + 5*a^2*
d^2)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(160*b^3*d^2) - ((9*b*c + 5*a*d)*(a + b*x)^(7/2)*(c + d*x)^(3/2))/(60*b^2*
d^2) + (x*(a + b*x)^(7/2)*(c + d*x)^(3/2))/(6*b*d) - ((b*c - a*d)^4*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*ArcT
anh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(7/2)*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 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 (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}+\frac {\int (a+b x)^{5/2} \sqrt {c+d x} \left (-a c-\frac {1}{2} (9 b c+5 a d) x\right ) \, dx}{6 b d} \\ & = -\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \int (a+b x)^{5/2} \sqrt {c+d x} \, dx}{40 b^2 d^2} \\ & = \frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}+\frac {\left ((b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{320 b^3 d^2} \\ & = \frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}-\frac {\left ((b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{384 b^3 d^3} \\ & = -\frac {(b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^4}+\frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}+\frac {\left ((b c-a d)^3 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{512 b^3 d^4} \\ & = \frac {(b c-a d)^3 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^5}-\frac {(b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^4}+\frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{1024 b^3 d^5} \\ & = \frac {(b c-a d)^3 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^5}-\frac {(b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^4}+\frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 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 )}{512 b^4 d^5} \\ & = \frac {(b c-a d)^3 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^5}-\frac {(b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^4}+\frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}-\frac {\left ((b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 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 )}{512 b^4 d^5} \\ & = \frac {(b c-a d)^3 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^5}-\frac {(b c-a d)^2 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^4}+\frac {(b c-a d) \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^3}+\frac {\left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d^2}-\frac {(9 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d}-\frac {(b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.81 \[ \int x^2 (a+b x)^{5/2} \sqrt {c+d x} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (75 a^5 d^5-5 a^4 b d^4 (13 c+10 d x)+10 a^3 b^2 d^3 \left (-9 c^2+4 c d x+4 d^2 x^2\right )+2 a^2 b^3 d^2 \left (419 c^3-262 c^2 d x+204 c d^2 x^2+1080 d^3 x^3\right )+a b^4 d \left (-945 c^4+616 c^3 d x-488 c^2 d^2 x^2+416 c d^3 x^3+3200 d^4 x^4\right )+b^5 \left (315 c^5-210 c^4 d x+168 c^3 d^2 x^2-144 c^2 d^3 x^3+128 c d^4 x^4+1280 d^5 x^5\right )\right )}{7680 b^3 d^5}-\frac {(b c-a d)^4 \left (21 b^2 c^2+14 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{512 b^{7/2} d^{11/2}} \]

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(75*a^5*d^5 - 5*a^4*b*d^4*(13*c + 10*d*x) + 10*a^3*b^2*d^3*(-9*c^2 + 4*c*d*x + 4*
d^2*x^2) + 2*a^2*b^3*d^2*(419*c^3 - 262*c^2*d*x + 204*c*d^2*x^2 + 1080*d^3*x^3) + a*b^4*d*(-945*c^4 + 616*c^3*
d*x - 488*c^2*d^2*x^2 + 416*c*d^3*x^3 + 3200*d^4*x^4) + b^5*(315*c^5 - 210*c^4*d*x + 168*c^3*d^2*x^2 - 144*c^2
*d^3*x^3 + 128*c*d^4*x^4 + 1280*d^5*x^5)))/(7680*b^3*d^5) - ((b*c - a*d)^4*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^
2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(512*b^(7/2)*d^(11/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1036\) vs. \(2(326)=652\).

Time = 0.56 (sec) , antiderivative size = 1037, normalized size of antiderivative = 2.76

method	result	size
default	\(\text {Expression too large to display}\)	\(1037\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 892, normalized size of antiderivative = 2.37 \[ \int x^2 (a+b x)^{5/2} \sqrt {c+d x} \, dx=\left [\frac {15 \, {\left (21 \, b^{6} c^{6} - 70 \, a b^{5} c^{5} d + 75 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 5 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + 5 \, 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} + 315 \, b^{6} c^{5} d - 945 \, a b^{5} c^{4} d^{2} + 838 \, a^{2} b^{4} c^{3} d^{3} - 90 \, a^{3} b^{3} c^{2} d^{4} - 65 \, a^{4} b^{2} c d^{5} + 75 \, a^{5} b d^{6} + 128 \, {\left (b^{6} c d^{5} + 25 \, a b^{5} d^{6}\right )} x^{4} - 16 \, {\left (9 \, b^{6} c^{2} d^{4} - 26 \, a b^{5} c d^{5} - 135 \, a^{2} b^{4} d^{6}\right )} x^{3} + 8 \, {\left (21 \, b^{6} c^{3} d^{3} - 61 \, a b^{5} c^{2} d^{4} + 51 \, a^{2} b^{4} c d^{5} + 5 \, a^{3} b^{3} d^{6}\right )} x^{2} - 2 \, {\left (105 \, b^{6} c^{4} d^{2} - 308 \, a b^{5} c^{3} d^{3} + 262 \, a^{2} b^{4} c^{2} d^{4} - 20 \, a^{3} b^{3} c d^{5} + 25 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30720 \, b^{4} d^{6}}, \frac {15 \, {\left (21 \, b^{6} c^{6} - 70 \, a b^{5} c^{5} d + 75 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 5 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + 5 \, 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} + 315 \, b^{6} c^{5} d - 945 \, a b^{5} c^{4} d^{2} + 838 \, a^{2} b^{4} c^{3} d^{3} - 90 \, a^{3} b^{3} c^{2} d^{4} - 65 \, a^{4} b^{2} c d^{5} + 75 \, a^{5} b d^{6} + 128 \, {\left (b^{6} c d^{5} + 25 \, a b^{5} d^{6}\right )} x^{4} - 16 \, {\left (9 \, b^{6} c^{2} d^{4} - 26 \, a b^{5} c d^{5} - 135 \, a^{2} b^{4} d^{6}\right )} x^{3} + 8 \, {\left (21 \, b^{6} c^{3} d^{3} - 61 \, a b^{5} c^{2} d^{4} + 51 \, a^{2} b^{4} c d^{5} + 5 \, a^{3} b^{3} d^{6}\right )} x^{2} - 2 \, {\left (105 \, b^{6} c^{4} d^{2} - 308 \, a b^{5} c^{3} d^{3} + 262 \, a^{2} b^{4} c^{2} d^{4} - 20 \, a^{3} b^{3} c d^{5} + 25 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15360 \, b^{4} d^{6}}\right ] \]

[In]

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

[Out]

[1/30720*(15*(21*b^6*c^6 - 70*a*b^5*c^5*d + 75*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 6*a^
5*b*c*d^5 + 5*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 + 315*b^6*c^5*d - 945*a*
b^5*c^4*d^2 + 838*a^2*b^4*c^3*d^3 - 90*a^3*b^3*c^2*d^4 - 65*a^4*b^2*c*d^5 + 75*a^5*b*d^6 + 128*(b^6*c*d^5 + 25
*a*b^5*d^6)*x^4 - 16*(9*b^6*c^2*d^4 - 26*a*b^5*c*d^5 - 135*a^2*b^4*d^6)*x^3 + 8*(21*b^6*c^3*d^3 - 61*a*b^5*c^2
*d^4 + 51*a^2*b^4*c*d^5 + 5*a^3*b^3*d^6)*x^2 - 2*(105*b^6*c^4*d^2 - 308*a*b^5*c^3*d^3 + 262*a^2*b^4*c^2*d^4 -
20*a^3*b^3*c*d^5 + 25*a^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d^6), 1/15360*(15*(21*b^6*c^6 - 70*a*b
^5*c^5*d + 75*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + 5*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 + 315*b^6*c^5*d - 945*a*b^5*c^4*d^2 + 838*a^2*b^4*c^3*d^3 - 90*a^3*b^3*c^2*d
^4 - 65*a^4*b^2*c*d^5 + 75*a^5*b*d^6 + 128*(b^6*c*d^5 + 25*a*b^5*d^6)*x^4 - 16*(9*b^6*c^2*d^4 - 26*a*b^5*c*d^5
- 135*a^2*b^4*d^6)*x^3 + 8*(21*b^6*c^3*d^3 - 61*a*b^5*c^2*d^4 + 51*a^2*b^4*c*d^5 + 5*a^3*b^3*d^6)*x^2 - 2*(10
5*b^6*c^4*d^2 - 308*a*b^5*c^3*d^3 + 262*a^2*b^4*c^2*d^4 - 20*a^3*b^3*c*d^5 + 25*a^4*b^2*d^6)*x)*sqrt(b*x + a)*
sqrt(d*x + c))/(b^4*d^6)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x^2*(b*x+a)^(5/2)*(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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1361 vs. \(2 (326) = 652\).

Time = 0.45 (sec) , antiderivative size = 1361, normalized size of antiderivative = 3.62 \[ \int x^2 (a+b x)^{5/2} \sqrt {c+d x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/7680*(12*(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(-s
qrt(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*abs(b) + 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^3*abs(b)
/b^2 + 120*(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^2*abs(b)/b + (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^3
2*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 - 23
1*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))*b*abs
(b))/b

Mupad [F(-1)]

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

[In]

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

[Out]

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