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

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

________________________________________________________________________________________

Rubi [A]  time = 0.25, antiderivative size = 302, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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 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 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^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 = 1.15, size = 282, normalized size = 0.93 \begin {gather*} \frac {15 (b c-a d)^{5/2} \left (7 a^3 d^3+9 a^2 b c d^2+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 (105 a^4 d^4-10 a^3 b d^3 (4 c+7 d x)+2 a^2 b^2 d^2 \left (-17 c^2+11 c d x+28 d^2 x^2\right )-2 a b^3 d \left (20 c^3-11 c^2 d x+8 c d^2 x^2+24 d^3 x^3\right )+b^4 \left (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\right )\right )}{1920 b^5 d^{9/2} \sqrt {c+d x}} \end {gather*}

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])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.94, size = 583, normalized size = 1.93 \begin {gather*} \frac {(b c-a d)^2 \left (7 a^3 d^3+9 a^2 b c d^2+9 a b^2 c^2 d+7 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{128 b^{9/2} d^{9/2}}-\frac {\sqrt {c+d x} (b c-a d)^2 \left (\frac {105 a^3 b^4 d^3 (c+d x)^4}{(a+b x)^4}+\frac {790 a^3 b^3 d^4 (c+d x)^3}{(a+b x)^3}-\frac {896 a^3 b^2 d^5 (c+d x)^2}{(a+b x)^2}+\frac {490 a^3 b d^6 (c+d x)}{a+b x}-105 a^3 d^7+\frac {135 a^2 b^5 c d^2 (c+d x)^4}{(a+b x)^4}-\frac {630 a^2 b^4 c d^3 (c+d x)^3}{(a+b x)^3}-\frac {1152 a^2 b^3 c d^4 (c+d x)^2}{(a+b x)^2}+\frac {630 a^2 b^2 c d^5 (c+d x)}{a+b x}-135 a^2 b c d^6+\frac {105 b^7 c^3 (c+d x)^4}{(a+b x)^4}-\frac {490 b^6 c^3 d (c+d x)^3}{(a+b x)^3}+\frac {135 a b^6 c^2 d (c+d x)^4}{(a+b x)^4}+\frac {896 b^5 c^3 d^2 (c+d x)^2}{(a+b x)^2}-\frac {630 a b^5 c^2 d^2 (c+d x)^3}{(a+b x)^3}-\frac {790 b^4 c^3 d^3 (c+d x)}{a+b x}+\frac {1152 a b^4 c^2 d^3 (c+d x)^2}{(a+b x)^2}+\frac {630 a b^3 c^2 d^4 (c+d x)}{a+b x}-135 a b^2 c^2 d^5-105 b^3 c^3 d^4\right )}{1920 b^4 d^4 \sqrt {a+b x} \left (\frac {b (c+d x)}{a+b x}-d\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1920*((b*c - a*d)^2*Sqrt[c + d*x]*(-105*b^3*c^3*d^4 - 135*a*b^2*c^2*d^5 - 135*a^2*b*c*d^6 - 105*a^3*d^7 - (
790*b^4*c^3*d^3*(c + d*x))/(a + b*x) + (630*a*b^3*c^2*d^4*(c + d*x))/(a + b*x) + (630*a^2*b^2*c*d^5*(c + d*x))
/(a + b*x) + (490*a^3*b*d^6*(c + d*x))/(a + b*x) + (896*b^5*c^3*d^2*(c + d*x)^2)/(a + b*x)^2 + (1152*a*b^4*c^2
*d^3*(c + d*x)^2)/(a + b*x)^2 - (1152*a^2*b^3*c*d^4*(c + d*x)^2)/(a + b*x)^2 - (896*a^3*b^2*d^5*(c + d*x)^2)/(
a + b*x)^2 - (490*b^6*c^3*d*(c + d*x)^3)/(a + b*x)^3 - (630*a*b^5*c^2*d^2*(c + d*x)^3)/(a + b*x)^3 - (630*a^2*
b^4*c*d^3*(c + d*x)^3)/(a + b*x)^3 + (790*a^3*b^3*d^4*(c + d*x)^3)/(a + b*x)^3 + (105*b^7*c^3*(c + d*x)^4)/(a
+ b*x)^4 + (135*a*b^6*c^2*d*(c + d*x)^4)/(a + b*x)^4 + (135*a^2*b^5*c*d^2*(c + d*x)^4)/(a + b*x)^4 + (105*a^3*
b^4*d^3*(c + d*x)^4)/(a + b*x)^4))/(b^4*d^4*Sqrt[a + b*x]*(-d + (b*(c + d*x))/(a + b*x))^5) + ((b*c - a*d)^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)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])
])/(128*b^(9/2)*d^(9/2))

________________________________________________________________________________________

fricas [A]  time = 1.13, size = 702, normalized size = 2.32 \begin {gather*} \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 {gather*}

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)]

________________________________________________________________________________________

giac [B]  time = 1.89, size = 671, normalized size = 2.22 \begin {gather*} \frac {\frac {10 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} + \frac {b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac {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}}\right )} + \frac {3 \, {\left (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}\right )}}{b^{14} d^{6}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (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}\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^{3}}\right )} a {\left | b \right |}}{b^{2}} + \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^{4}} + \frac {b^{20} c d^{7} - 41 \, a b^{19} d^{8}}{b^{23} d^{8}}\right )} - \frac {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}}\right )} + \frac {5 \, {\left (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}\right )}}{b^{23} d^{8}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (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}\right )}}{b^{23} d^{8}}\right )} \sqrt {b x + a} - \frac {15 \, {\left (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}\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^{3} d^{4}}\right )} {\left | b \right |}}{b}}{1920 \, b} \end {gather*}

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*(10*(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*abs(b)/b^2 + (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 - 19
3*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))*abs(b)/b)/b

________________________________________________________________________________________

maple [B]  time = 0.02, size = 942, normalized size = 3.12 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (768 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} d^{4} x^{4}+105 a^{5} 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 )-75 a^{4} b c \,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 )-30 a^{3} b^{2} c^{2} 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 )-30 a^{2} b^{3} c^{3} 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 )-75 a \,b^{4} c^{4} 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 )+96 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a \,b^{3} d^{4} x^{3}+105 b^{5} c^{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 )+96 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} c \,d^{3} x^{3}-112 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a^{2} b^{2} d^{4} x^{2}+32 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a \,b^{3} c \,d^{3} x^{2}-112 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} c^{2} d^{2} x^{2}+140 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,d^{4} x -44 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c \,d^{3} x -44 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{2} d^{2} x +140 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{3} d x -210 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} d^{4}+80 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b c \,d^{3}+68 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{2} d^{2}+80 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{3} d -210 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4}\right )}{3840 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} d^{4}} \end {gather*}

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)

________________________________________________________________________________________

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^3*(b*x+a)^(1/2)*(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 details)Is a*d-b*c zero or nonzero?

________________________________________________________________________________________

mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \sqrt {a + b x} \sqrt {c + d x}\, dx \end {gather*}

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)

________________________________________________________________________________________