3.547 \(\int x^3 \sqrt{a+b x} \sqrt{c+d x} \, dx\)

Optimal. Leaf size=302 \[ \frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (35 a^2 d^2-42 b d x (a d+b c)+38 a b c d+35 b^2 c^2\right )}{240 b^3 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right )}{64 b^4 d^3}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-2 a^3 b c d^3-7 a^4 d^4+2 a b^3 c^3 d+7 b^4 c^4\right )}{128 b^4 d^4}+\frac{(a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{9/2} d^{9/2}}+\frac{x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d} \]

[Out]

-((7*b^4*c^4 + 2*a*b^3*c^3*d - 2*a^3*b*c*d^3 - 7*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^4*d^4) - ((b*c +
a*d)*(7*b^2*c^2 + 2*a*b*c*d + 7*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(64*b^4*d^3) + (x^2*(a + b*x)^(3/2)*(
c + d*x)^(3/2))/(5*b*d) + ((a + b*x)^(3/2)*(c + d*x)^(3/2)*(35*b^2*c^2 + 38*a*b*c*d + 35*a^2*d^2 - 42*b*d*(b*c
+ a*d)*x))/(240*b^3*d^3) + ((b*c - a*d)^2*(b*c + a*d)*(7*b^2*c^2 + 2*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[d]*Sq
rt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(9/2)*d^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.250899, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {100, 147, 50, 63, 217, 206} \[ \frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (35 a^2 d^2-42 b d x (a d+b c)+38 a b c d+35 b^2 c^2\right )}{240 b^3 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right )}{64 b^4 d^3}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-2 a^3 b c d^3-7 a^4 d^4+2 a b^3 c^3 d+7 b^4 c^4\right )}{128 b^4 d^4}+\frac{(a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{9/2} d^{9/2}}+\frac{x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[a + b*x]*Sqrt[c + d*x],x]

[Out]

-((7*b^4*c^4 + 2*a*b^3*c^3*d - 2*a^3*b*c*d^3 - 7*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^4*d^4) - ((b*c +
a*d)*(7*b^2*c^2 + 2*a*b*c*d + 7*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(64*b^4*d^3) + (x^2*(a + b*x)^(3/2)*(
c + d*x)^(3/2))/(5*b*d) + ((a + b*x)^(3/2)*(c + d*x)^(3/2)*(35*b^2*c^2 + 38*a*b*c*d + 35*a^2*d^2 - 42*b*d*(b*c
+ a*d)*x))/(240*b^3*d^3) + ((b*c - a*d)^2*(b*c + a*d)*(7*b^2*c^2 + 2*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[d]*Sq
rt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(9/2)*d^(9/2))

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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^3 \sqrt{a+b x} \sqrt{c+d x} \, dx &=\frac{x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac{\int x \sqrt{a+b x} \sqrt{c+d x} \left (-2 a c-\frac{7}{2} (b c+a d) x\right ) \, dx}{5 b d}\\ &=\frac{x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (35 b^2 c^2+38 a b c d+35 a^2 d^2-42 b d (b c+a d) x\right )}{240 b^3 d^3}-\frac{\left ((b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right )\right ) \int \sqrt{a+b x} \sqrt{c+d x} \, dx}{32 b^3 d^3}\\ &=-\frac{(b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^4 d^3}+\frac{x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (35 b^2 c^2+38 a b c d+35 a^2 d^2-42 b d (b c+a d) x\right )}{240 b^3 d^3}-\frac{\left ((b c-a d) (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right )\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{128 b^4 d^3}\\ &=-\frac{(b c-a d) (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^4 d^4}-\frac{(b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^4 d^3}+\frac{x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (35 b^2 c^2+38 a b c d+35 a^2 d^2-42 b d (b c+a d) x\right )}{240 b^3 d^3}+\frac{\left ((b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{256 b^4 d^4}\\ &=-\frac{(b c-a d) (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^4 d^4}-\frac{(b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^4 d^3}+\frac{x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (35 b^2 c^2+38 a b c d+35 a^2 d^2-42 b d (b c+a d) x\right )}{240 b^3 d^3}+\frac{\left ((b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 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 )}{128 b^5 d^4}\\ &=-\frac{(b c-a d) (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^4 d^4}-\frac{(b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^4 d^3}+\frac{x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (35 b^2 c^2+38 a b c d+35 a^2 d^2-42 b d (b c+a d) x\right )}{240 b^3 d^3}+\frac{\left ((b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 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 )}{128 b^5 d^4}\\ &=-\frac{(b c-a d) (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^4 d^4}-\frac{(b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt{c+d x}}{64 b^4 d^3}+\frac{x^2 (a+b x)^{3/2} (c+d x)^{3/2}}{5 b d}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} \left (35 b^2 c^2+38 a b c d+35 a^2 d^2-42 b d (b c+a d) x\right )}{240 b^3 d^3}+\frac{(b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{9/2} d^{9/2}}\\ \end{align*}

Mathematica [A]  time = 1.12642, size = 282, normalized size = 0.93 \[ \frac{15 (b c-a d)^{5/2} \left (9 a^2 b c d^2+7 a^3 d^3+9 a b^2 c^2 d+7 b^3 c^3\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )-b \sqrt{d} \sqrt{a+b x} (c+d x) \left (2 a^2 b^2 d^2 \left (-17 c^2+11 c d x+28 d^2 x^2\right )-10 a^3 b d^3 (4 c+7 d x)+105 a^4 d^4-2 a b^3 d \left (-11 c^2 d x+20 c^3+8 c d^2 x^2+24 d^3 x^3\right )+b^4 \left (56 c^2 d^2 x^2-70 c^3 d x+105 c^4-48 c d^3 x^3-384 d^4 x^4\right )\right )}{1920 b^5 d^{9/2} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[a + b*x]*Sqrt[c + d*x],x]

[Out]

(-(b*Sqrt[d]*Sqrt[a + b*x]*(c + d*x)*(105*a^4*d^4 - 10*a^3*b*d^3*(4*c + 7*d*x) + 2*a^2*b^2*d^2*(-17*c^2 + 11*c
*d*x + 28*d^2*x^2) - 2*a*b^3*d*(20*c^3 - 11*c^2*d*x + 8*c*d^2*x^2 + 24*d^3*x^3) + b^4*(105*c^4 - 70*c^3*d*x +
56*c^2*d^2*x^2 - 48*c*d^3*x^3 - 384*d^4*x^4))) + 15*(b*c - a*d)^(5/2)*(7*b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d
^2 + 7*a^3*d^3)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(1920*b^5*d^
(9/2)*Sqrt[c + d*x])

________________________________________________________________________________________

Maple [B]  time = 0.019, size = 942, normalized size = 3.1 \begin{align*}{\frac{1}{3840\,{b}^{4}{d}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 768\,{x}^{4}{b}^{4}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+96\,{x}^{3}a{b}^{3}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+96\,{x}^{3}{b}^{4}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-112\,{x}^{2}{a}^{2}{b}^{2}{d}^{4}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+32\,{x}^{2}a{b}^{3}c{d}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-112\,{x}^{2}{b}^{4}{c}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+105\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{5}{d}^{5}-75\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}bc{d}^{4}-30\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{b}^{2}{c}^{2}{d}^{3}-30\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{3}{c}^{3}{d}^{2}-75\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{4}{c}^{4}d+105\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{5}{c}^{5}+140\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{d}^{4}-44\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}{b}^{2}c{d}^{3}-44\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}xa{b}^{3}{c}^{2}{d}^{2}+140\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}x{b}^{4}{c}^{3}d-210\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{d}^{4}+80\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}bc{d}^{3}+68\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+80\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}a{b}^{3}{c}^{3}d-210\,\sqrt{bd}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{4}{c}^{4} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^(1/2)*(d*x+c)^(1/2),x)

[Out]

1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(768*x^4*b^4*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+96*x^3*a*b^3*d
^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+96*x^3*b^4*c*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-11
2*x^2*a^2*b^2*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+32*x^2*a*b^3*c*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/
2)*(b*d)^(1/2)-112*x^2*b^4*c^2*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+105*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*d^5-75*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^4*b*c*d^4-30*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^2*c^2*d^3-30*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^3*c^3*d^2-75*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*b^4*c^4*d+105*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^5*c^5+140*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*b*d^4-44*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)*x*a^2*b^2*c*d^3-44*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a*b^3*c^2*d^2+140*(b*d)^(1/2)*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*b^4*c^3*d-210*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*d^4+80*(b*d)^(1/2
)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c*d^3+68*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^2*c^2*d^2+8
0*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^3*d-210*(b*d)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^4*
c^4)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/b^4/d^4/(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^3*(b*x+a)^(1/2)*(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.53082, size = 1542, normalized size = 5.11 \begin{align*} \left [\frac{15 \,{\left (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}\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 (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 40 \, a b^{4} c^{3} d^{2} + 34 \, a^{2} b^{3} c^{2} d^{3} + 40 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \,{\left (b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} - 8 \,{\left (7 \, b^{5} c^{2} d^{3} - 2 \, a b^{4} c d^{4} + 7 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \,{\left (35 \, b^{5} c^{3} d^{2} - 11 \, a b^{4} c^{2} d^{3} - 11 \, a^{2} b^{3} c d^{4} + 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, b^{5} d^{5}}, -\frac{15 \,{\left (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}\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 (384 \, b^{5} d^{5} x^{4} - 105 \, b^{5} c^{4} d + 40 \, a b^{4} c^{3} d^{2} + 34 \, a^{2} b^{3} c^{2} d^{3} + 40 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \,{\left (b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} - 8 \,{\left (7 \, b^{5} c^{2} d^{3} - 2 \, a b^{4} c d^{4} + 7 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \,{\left (35 \, b^{5} c^{3} d^{2} - 11 \, a b^{4} c^{2} d^{3} - 11 \, a^{2} b^{3} c d^{4} + 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, b^{5} d^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(1/2)*(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

[1/7680*(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)*sq
rt(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)*sq
rt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(384*b^5*d^5*x^4 - 105*b^5*c^4*d + 40*a*b^4*c^3*d^2 + 34*a^2*b^3*c^
2*d^3 + 40*a^3*b^2*c*d^4 - 105*a^4*b*d^5 + 48*(b^5*c*d^4 + a*b^4*d^5)*x^3 - 8*(7*b^5*c^2*d^3 - 2*a*b^4*c*d^4 +
7*a^2*b^3*d^5)*x^2 + 2*(35*b^5*c^3*d^2 - 11*a*b^4*c^2*d^3 - 11*a^2*b^3*c*d^4 + 35*a^3*b^2*d^5)*x)*sqrt(b*x +
a)*sqrt(d*x + c))/(b^5*d^5), -1/3840*(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)*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*(384*b^5*d^5*x^4 - 105*b^5*c^4*d + 40*a*b^4*c^3*d^2 + 34*
a^2*b^3*c^2*d^3 + 40*a^3*b^2*c*d^4 - 105*a^4*b*d^5 + 48*(b^5*c*d^4 + a*b^4*d^5)*x^3 - 8*(7*b^5*c^2*d^3 - 2*a*b
^4*c*d^4 + 7*a^2*b^3*d^5)*x^2 + 2*(35*b^5*c^3*d^2 - 11*a*b^4*c^2*d^3 - 11*a^2*b^3*c*d^4 + 35*a^3*b^2*d^5)*x)*s
qrt(b*x + a)*sqrt(d*x + c))/(b^5*d^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{a + b x} \sqrt{c + d x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**(1/2)*(d*x+c)**(1/2),x)

[Out]

Integral(x**3*sqrt(a + b*x)*sqrt(c + d*x), x)

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Giac [A]  time = 3.15392, size = 491, normalized size = 1.63 \begin{align*} \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (4 \,{\left (b x + a\right )}{\left (6 \,{\left (b x + a\right )}{\left (\frac{8 \,{\left (b x + a\right )}}{b^{3}} + \frac{b^{13} c d^{7} - 31 \, a b^{12} d^{8}}{b^{15} d^{8}}\right )} - \frac{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}}\right )} + \frac{5 \,{\left (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}\right )}}{b^{15} d^{8}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (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}\right )}}{b^{15} d^{8}}\right )} \sqrt{b x + a} - \frac{15 \,{\left (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}\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 )}{\sqrt{b d} b^{2} d^{4}}\right )}{\left | b \right |}}{1920 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(1/2)*(d*x+c)^(1/2),x, algorithm="giac")

[Out]

1/1920*(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))*abs(b)/b^3