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

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.367761, antiderivative size = 376, normalized size of antiderivative = 1., 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} \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^3}{512 b^5 d^3}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^2}{256 b^5 d^2}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)}{192 b^4 d^2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2} \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right )}{160 b^3 d^2}-\frac{\left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{11/2} d^{7/2}}-\frac{(a+b x)^{3/2} (c+d x)^{7/2} (9 a d+5 b c)}{60 b^2 d^2}+\frac{x (a+b x)^{3/2} (c+d x)^{7/2}}{6 b d} \]

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 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 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 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 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]

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])

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*}

Mathematica [A]  time = 1.92316, size = 320, normalized size = 0.85 \[ \frac{(a+b x)^{3/2} (c+d x)^{7/2} \left (\frac{3 \left (21 a^2 d^2+14 a b c d+5 b^2 c^2\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \left (2 b^5 d^2 (a+b x)^2 (b c-a d)^{3/2} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (15 a^2 d^2-10 a b d (5 c+2 d x)+b^2 \left (59 c^2+68 c d x+24 d^2 x^2\right )\right )+15 b^5 d (a+b x) (b c-a d)^{9/2} \sqrt{\frac{b (c+d x)}{b c-a d}}-15 b^5 \sqrt{d} \sqrt{a+b x} (b c-a d)^5 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )\right )}{128 b^9 d^2 (a+b x)^2 (c+d x)^4 \sqrt{b c-a d}}-3 (9 a d+5 b c)+30 b d x\right )}{180 b^2 d^2} \]

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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Maple [B]  time = 0.019, size = 1240, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

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)*(-4320*x^3*b^5*c^2*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-832*x^
3*a*b^4*c*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(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+976*x^2*a^2*b^3*c*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-816*x^2*a*b
^4*c^2*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-2560*x^5*b^5*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^
(1/2)-336*x^2*a^3*b^2*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-80*x^2*b^5*c^3*d^2*(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)*(b*d)^(1/2)-256*x^4*a*b^4*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-6400*x^4*b^5*c*d^4*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+288*x^3*a^2*b^3*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(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+315*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*
d)^(1/2))*a^6*d^6+75*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^6*c
^6-630*(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^5-1050*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b*c*d^5+1125
*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*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*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^3*c^3*d^3-75*ln(1/2*(2*b*
d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^4*c^4*d^2-90*ln(1/2*(2*b*d*x+2*(
b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^5*c^5*d)/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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [A]  time = 3.24065, size = 1970, normalized size = 5.24 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [B]  time = 1.50583, size = 1546, normalized size = 4.11 \begin{align*} \text{result too large to display} \end{align*}

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*(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^2 + (b^7*c*d^5 - 17*a
*b^6*d^6)/(b^8*d^6)) - (5*b^8*c^2*d^4 + 6*a*b^7*c*d^5 - 59*a^2*b^6*d^6)/(b^8*d^6)) + 3*(5*b^9*c^3*d^3 + a*b^8*
c^2*d^4 - a^2*b^7*c*d^5 - 5*a^3*b^6*d^6)/(b^8*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2*c
^2*d^2 - 4*a^3*b*c*d^3 + 5*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*d^3))*c^2*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^3 + (b^13*c*d^7 - 31*a*b^12*d^8)/(b^15*d^8)) - (7*b^14*c^2*d^6 + 16*a*b^13*c*d^7 - 263*a^2*b^12*d^8)/(
b^15*d^8)) + 5*(7*b^15*c^3*d^5 + 9*a*b^14*c^2*d^6 + 9*a^2*b^13*c*d^7 - 121*a^3*b^12*d^8)/(b^15*d^8))*(b*x + a)
- 15*(7*b^16*c^4*d^4 + 2*a*b^15*c^3*d^5 - 2*a^3*b^13*c*d^7 - 7*a^4*b^12*d^8)/(b^15*d^8))*sqrt(b*x + a) - 15*(
7*b^5*c^5 - 5*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 7*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^2*d^4))*c*d*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^4 + (b^21*c*d^9 - 49*a*b^20*d^10)/(b^24
*d^10)) - 3*(3*b^22*c^2*d^8 + 10*a*b^21*c*d^9 - 253*a^2*b^20*d^10)/(b^24*d^10)) + (21*b^23*c^3*d^7 + 49*a*b^22
*c^2*d^8 + 79*a^2*b^21*c*d^9 - 1429*a^3*b^20*d^10)/(b^24*d^10))*(b*x + a) - 5*(21*b^24*c^4*d^6 + 28*a*b^23*c^3
*d^7 + 30*a^2*b^22*c^2*d^8 + 28*a^3*b^21*c*d^9 - 491*a^4*b^20*d^10)/(b^24*d^10))*(b*x + a) + 15*(21*b^25*c^5*d
^5 + 7*a*b^24*c^4*d^6 + 2*a^2*b^23*c^3*d^7 - 2*a^3*b^22*c^2*d^8 - 7*a^4*b^21*c*d^9 - 21*a^5*b^20*d^10)/(b^24*d
^10))*sqrt(b*x + a) + 15*(21*b^6*c^6 - 14*a*b^5*c^5*d - 5*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 - 5*a^4*b^2*c^2*
d^4 - 14*a^5*b*c*d^5 + 21*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)))/(s
qrt(b*d)*b^3*d^5))*d^2*abs(b)/b^2)/b