3.23.2 \(\int \sqrt {a+b x} (A+B x) \sqrt {d+e x} \, dx\) [2202]

Optimal. Leaf size=196 \[ \frac {(b d-a e) (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{8 b^2 e^2}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{4 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{3/2}}{3 b e}-\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{5/2} e^{5/2}} \]

[Out]

1/3*B*(b*x+a)^(3/2)*(e*x+d)^(3/2)/b/e-1/8*(-a*e+b*d)^2*(2*A*b*e-B*(a*e+b*d))*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(
1/2)/(e*x+d)^(1/2))/b^(5/2)/e^(5/2)+1/4*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(3/2)*(e*x+d)^(1/2)/b^2/e+1/8*(-a*e+b*d)
*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^2/e^2

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Rubi [A]
time = 0.09, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {81, 52, 65, 223, 212} \begin {gather*} -\frac {(b d-a e)^2 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{5/2} e^{5/2}}+\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e) (2 A b e-B (a e+b d))}{8 b^2 e^2}+\frac {(a+b x)^{3/2} \sqrt {d+e x} (2 A b e-B (a e+b d))}{4 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{3/2}}{3 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x]*(A + B*x)*Sqrt[d + e*x],x]

[Out]

((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^2*e^2) + ((2*A*b*e - B*(b*d + a*e))*(
a + b*x)^(3/2)*Sqrt[d + e*x])/(4*b^2*e) + (B*(a + b*x)^(3/2)*(d + e*x)^(3/2))/(3*b*e) - ((b*d - a*e)^2*(2*A*b*
e - B*(b*d + a*e))*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(5/2)*e^(5/2))

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 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*} \int \sqrt {a+b x} (A+B x) \sqrt {d+e x} \, dx &=\frac {B (a+b x)^{3/2} (d+e x)^{3/2}}{3 b e}+\frac {\left (3 A b e-B \left (\frac {3 b d}{2}+\frac {3 a e}{2}\right )\right ) \int \sqrt {a+b x} \sqrt {d+e x} \, dx}{3 b e}\\ &=\frac {(2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{4 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{3/2}}{3 b e}+\frac {\left ((b d-a e) \left (3 A b e-B \left (\frac {3 b d}{2}+\frac {3 a e}{2}\right )\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{12 b^2 e}\\ &=\frac {(b d-a e) (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{8 b^2 e^2}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{4 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{3/2}}{3 b e}-\frac {\left ((b d-a e)^2 \left (3 A b e-B \left (\frac {3 b d}{2}+\frac {3 a e}{2}\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{24 b^2 e^2}\\ &=\frac {(b d-a e) (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{8 b^2 e^2}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{4 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{3/2}}{3 b e}-\frac {\left ((b d-a e)^2 \left (3 A b e-B \left (\frac {3 b d}{2}+\frac {3 a e}{2}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{12 b^3 e^2}\\ &=\frac {(b d-a e) (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{8 b^2 e^2}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{4 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{3/2}}{3 b e}-\frac {\left ((b d-a e)^2 \left (3 A b e-B \left (\frac {3 b d}{2}+\frac {3 a e}{2}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{12 b^3 e^2}\\ &=\frac {(b d-a e) (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{8 b^2 e^2}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{4 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{3/2}}{3 b e}-\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{5/2} e^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.47, size = 160, normalized size = 0.82 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {d+e x} \left (-3 a^2 B e^2+2 a b e (3 A e+B (d+e x))+b^2 \left (6 A e (d+2 e x)+B \left (-3 d^2+2 d e x+8 e^2 x^2\right )\right )\right )}{24 b^2 e^2}+\frac {(b d-a e)^2 (b B d-2 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{8 b^{5/2} e^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x]*(A + B*x)*Sqrt[d + e*x],x]

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-3*a^2*B*e^2 + 2*a*b*e*(3*A*e + B*(d + e*x)) + b^2*(6*A*e*(d + 2*e*x) + B*(-3*d^
2 + 2*d*e*x + 8*e^2*x^2))))/(24*b^2*e^2) + ((b*d - a*e)^2*(b*B*d - 2*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d +
e*x])/(Sqrt[e]*Sqrt[a + b*x])])/(8*b^(5/2)*e^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(635\) vs. \(2(164)=328\).
time = 0.09, size = 636, normalized size = 3.24

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {e x +d}\, \left (-16 B \,b^{2} e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+6 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b \,e^{3}-12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} d \,e^{2}+6 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d^{2} e -24 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{2} e^{2} x -3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} e^{3}+3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b d \,e^{2}+3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} d^{2} e -3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d^{3}-4 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a b \,e^{2} x -4 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{2} d e x -12 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a b \,e^{2}-12 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{2} d e +6 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a^{2} e^{2}-4 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a b d e +6 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{2} d^{2}\right )}{48 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{2} e^{2} \sqrt {b e}}\)	\(636\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b*x+a)^(1/2)*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/48*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-16*B*b^2*e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+6*A*ln(1/2*(2*b*e*x+2
*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b*e^3-12*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^
(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^2*d*e^2+6*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+
a*e+b*d)/(b*e)^(1/2))*b^3*d^2*e-24*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^2*e^2*x-3*B*ln(1/2*(2*b*e*x+2*((b*x
+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*e^3+3*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*
e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b*d*e^2+3*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/
(b*e)^(1/2))*a*b^2*d^2*e-3*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*d
^3-4*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b*e^2*x-4*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^2*d*e*x-12*A*(b
*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b*e^2-12*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^2*d*e+6*B*(b*e)^(1/2)*((b
*x+a)*(e*x+d))^(1/2)*a^2*e^2-4*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b*d*e+6*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))
^(1/2)*b^2*d^2)/((b*x+a)*(e*x+d))^(1/2)/b^2/e^2/(b*e)^(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((B*x+A)*(b*x+a)^(1/2)*(e*x+d)^(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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 1.16, size = 512, normalized size = 2.61 \begin {gather*} \left [-\frac {{\left (3 \, {\left (B b^{3} d^{3} - {\left (B a b^{2} + 2 \, A b^{3}\right )} d^{2} e - {\left (B a^{2} b - 4 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3}\right )} \sqrt {b} e^{\frac {1}{2}} \log \left (b^{2} d^{2} - 4 \, {\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {b} e^{\frac {1}{2}} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right ) + 4 \, {\left (3 \, B b^{3} d^{2} e - {\left (8 \, B b^{3} x^{2} - 3 \, B a^{2} b + 6 \, A a b^{2} + 2 \, {\left (B a b^{2} + 6 \, A b^{3}\right )} x\right )} e^{3} - 2 \, {\left (B b^{3} d x + {\left (B a b^{2} + 3 \, A b^{3}\right )} d\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-3\right )}}{96 \, b^{3}}, -\frac {{\left (3 \, {\left (B b^{3} d^{3} - {\left (B a b^{2} + 2 \, A b^{3}\right )} d^{2} e - {\left (B a^{2} b - 4 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {-b e} \sqrt {x e + d}}{2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} e^{2} + {\left (b^{2} d x + a b d\right )} e\right )}}\right ) + 2 \, {\left (3 \, B b^{3} d^{2} e - {\left (8 \, B b^{3} x^{2} - 3 \, B a^{2} b + 6 \, A a b^{2} + 2 \, {\left (B a b^{2} + 6 \, A b^{3}\right )} x\right )} e^{3} - 2 \, {\left (B b^{3} d x + {\left (B a b^{2} + 3 \, A b^{3}\right )} d\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-3\right )}}{48 \, b^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

[-1/96*(3*(B*b^3*d^3 - (B*a*b^2 + 2*A*b^3)*d^2*e - (B*a^2*b - 4*A*a*b^2)*d*e^2 + (B*a^3 - 2*A*a^2*b)*e^3)*sqrt
(b)*e^(1/2)*log(b^2*d^2 - 4*(b*d + (2*b*x + a)*e)*sqrt(b*x + a)*sqrt(x*e + d)*sqrt(b)*e^(1/2) + (8*b^2*x^2 + 8
*a*b*x + a^2)*e^2 + 2*(4*b^2*d*x + 3*a*b*d)*e) + 4*(3*B*b^3*d^2*e - (8*B*b^3*x^2 - 3*B*a^2*b + 6*A*a*b^2 + 2*(
B*a*b^2 + 6*A*b^3)*x)*e^3 - 2*(B*b^3*d*x + (B*a*b^2 + 3*A*b^3)*d)*e^2)*sqrt(b*x + a)*sqrt(x*e + d))*e^(-3)/b^3
, -1/48*(3*(B*b^3*d^3 - (B*a*b^2 + 2*A*b^3)*d^2*e - (B*a^2*b - 4*A*a*b^2)*d*e^2 + (B*a^3 - 2*A*a^2*b)*e^3)*sqr
t(-b*e)*arctan(1/2*(b*d + (2*b*x + a)*e)*sqrt(b*x + a)*sqrt(-b*e)*sqrt(x*e + d)/((b^2*x^2 + a*b*x)*e^2 + (b^2*
d*x + a*b*d)*e)) + 2*(3*B*b^3*d^2*e - (8*B*b^3*x^2 - 3*B*a^2*b + 6*A*a*b^2 + 2*(B*a*b^2 + 6*A*b^3)*x)*e^3 - 2*
(B*b^3*d*x + (B*a*b^2 + 3*A*b^3)*d)*e^2)*sqrt(b*x + a)*sqrt(x*e + d))*e^(-3)/b^3]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A + B x\right ) \sqrt {a + b x} \sqrt {d + e x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)**(1/2)*(e*x+d)**(1/2),x)

[Out]

Integral((A + B*x)*sqrt(a + b*x)*sqrt(d + e*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (171) = 342\).
time = 0.97, size = 571, normalized size = 2.91 \begin {gather*} -\frac {\frac {24 \, {\left (\frac {{\left (b^{2} d - a b e\right )} e^{\left (-\frac {1}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a}\right )} A a {\left | b \right |}}{b^{2}} - \frac {{\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {{\left (b^{6} d e^{3} - 13 \, a b^{5} e^{4}\right )} e^{\left (-4\right )}}{b^{7}}\right )} - \frac {3 \, {\left (b^{7} d^{2} e^{2} + 2 \, a b^{6} d e^{3} - 11 \, a^{2} b^{5} e^{4}\right )} e^{\left (-4\right )}}{b^{7}}\right )} - \frac {3 \, {\left (b^{3} d^{3} + a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac {3}{2}}}\right )} B {\left | b \right |}}{b} - \frac {6 \, {\left (\frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + {\left (b d e - 5 \, a e^{2}\right )} e^{\left (-2\right )} + 2 \, a\right )} \sqrt {b x + a}\right )} B a {\left | b \right |}}{b^{3}} - \frac {6 \, {\left (\frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + {\left (b d e - 5 \, a e^{2}\right )} e^{\left (-2\right )} + 2 \, a\right )} \sqrt {b x + a}\right )} A {\left | b \right |}}{b^{2}}}{24 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-1/24*(24*((b^2*d - a*b*e)*e^(-1/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*
e)))/sqrt(b) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a))*A*a*abs(b)/b^2 - (sqrt(b^2*d + (b*x + a)*b*e
- a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a*b^5*e^4)*e^(-4)/b^7) - 3*(b^7*d^2*e^
2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)*e^(-4)/b^7) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*e^(-5/
2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*B*abs(b)/b - 6*((b^
3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*e^(-3/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*
e - a*b*e)))/sqrt(b) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + (b*d*e - 5*a*e^2)*e^(-2) + 2*a)*sqrt(b*x +
a))*B*a*abs(b)/b^3 - 6*((b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*e^(-3/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2)
+ sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + (b*d*e - 5*a*e
^2)*e^(-2) + 2*a)*sqrt(b*x + a))*A*abs(b)/b^2)/b

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Mupad [B]
time = 79.38, size = 1207, normalized size = 6.16 \begin {gather*} A\,\left (\frac {x}{2}+\frac {a\,e+b\,d}{4\,b\,e}\right )\,\sqrt {a+b\,x}\,\sqrt {d+e\,x}-\frac {\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (\frac {B\,a^3\,b^3\,e^3}{4}-\frac {B\,a^2\,b^4\,d\,e^2}{4}-\frac {B\,a\,b^5\,d^2\,e}{4}+\frac {B\,b^6\,d^3}{4}\right )}{e^8\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {17\,B\,a^3\,b^2\,e^3}{12}+\frac {101\,B\,a^2\,b^3\,d\,e^2}{4}+\frac {101\,B\,a\,b^4\,d^2\,e}{4}+\frac {17\,B\,b^5\,d^3}{12}\right )}{e^7\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (\frac {19\,B\,a^3\,e^3}{2}+\frac {269\,B\,a^2\,b\,d\,e^2}{2}+\frac {269\,B\,a\,b^2\,d^2\,e}{2}+\frac {19\,B\,b^3\,d^3}{2}\right )}{e^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^7}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (\frac {19\,B\,a^3\,b\,e^3}{2}+\frac {269\,B\,a^2\,b^2\,d\,e^2}{2}+\frac {269\,B\,a\,b^3\,d^2\,e}{2}+\frac {19\,B\,b^4\,d^3}{2}\right )}{e^6\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^5}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{11}\,\left (\frac {B\,a^3\,e^3}{4}-\frac {B\,a^2\,b\,d\,e^2}{4}-\frac {B\,a\,b^2\,d^2\,e}{4}+\frac {B\,b^3\,d^3}{4}\right )}{b^2\,e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{11}}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^9\,\left (\frac {17\,B\,a^3\,e^3}{12}+\frac {101\,B\,a^2\,b\,d\,e^2}{4}+\frac {101\,B\,a\,b^2\,d^2\,e}{4}+\frac {17\,B\,b^3\,d^3}{12}\right )}{b\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^9}+\frac {\sqrt {a}\,\sqrt {d}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (32\,B\,a^2\,b^2\,e^2+96\,B\,a\,b^3\,d\,e+32\,B\,b^4\,d^2\right )}{e^6\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}+\frac {8\,B\,a^{3/2}\,d^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{10}}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{10}}+\frac {\sqrt {a}\,\sqrt {d}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (64\,B\,a^2\,b\,e^2+\frac {656\,B\,a\,b^2\,d\,e}{3}+64\,B\,b^3\,d^2\right )}{e^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}+\frac {\sqrt {a}\,\sqrt {d}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8\,\left (32\,B\,a^2\,e^2+96\,B\,a\,b\,d\,e+32\,B\,b^2\,d^2\right )}{e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^8}+\frac {8\,B\,a^{3/2}\,b^4\,d^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e^6\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{12}}{{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{12}}+\frac {b^6}{e^6}-\frac {6\,b^5\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}+\frac {15\,b^4\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}-\frac {20\,b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}+\frac {15\,b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^8}-\frac {6\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{10}}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{10}}}-\frac {A\,\ln \left (a\,e+b\,d+2\,b\,e\,x+2\,\sqrt {b}\,\sqrt {e}\,\sqrt {a+b\,x}\,\sqrt {d+e\,x}\right )\,{\left (a\,e-b\,d\right )}^2}{8\,b^{3/2}\,e^{3/2}}+\frac {B\,\mathrm {atanh}\left (\frac {B\,\sqrt {e}\,\left (a\,e+b\,d\right )\,{\left (a\,e-b\,d\right )}^2\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )\,\left (B\,a^3\,e^3-B\,a^2\,b\,d\,e^2-B\,a\,b^2\,d^2\,e+B\,b^3\,d^3\right )}\right )\,\left (a\,e+b\,d\right )\,{\left (a\,e-b\,d\right )}^2}{4\,b^{5/2}\,e^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^(1/2)*(d + e*x)^(1/2),x)

[Out]

A*(x/2 + (a*e + b*d)/(4*b*e))*(a + b*x)^(1/2)*(d + e*x)^(1/2) - ((((a + b*x)^(1/2) - a^(1/2))*((B*b^6*d^3)/4 +
(B*a^3*b^3*e^3)/4 - (B*a^2*b^4*d*e^2)/4 - (B*a*b^5*d^2*e)/4))/(e^8*((d + e*x)^(1/2) - d^(1/2))) - (((a + b*x)
^(1/2) - a^(1/2))^3*((17*B*b^5*d^3)/12 + (17*B*a^3*b^2*e^3)/12 + (101*B*a^2*b^3*d*e^2)/4 + (101*B*a*b^4*d^2*e)
/4))/(e^7*((d + e*x)^(1/2) - d^(1/2))^3) - (((a + b*x)^(1/2) - a^(1/2))^7*((19*B*a^3*e^3)/2 + (19*B*b^3*d^3)/2
+ (269*B*a*b^2*d^2*e)/2 + (269*B*a^2*b*d*e^2)/2))/(e^5*((d + e*x)^(1/2) - d^(1/2))^7) - (((a + b*x)^(1/2) - a
^(1/2))^5*((19*B*b^4*d^3)/2 + (19*B*a^3*b*e^3)/2 + (269*B*a^2*b^2*d*e^2)/2 + (269*B*a*b^3*d^2*e)/2))/(e^6*((d
+ e*x)^(1/2) - d^(1/2))^5) + (((a + b*x)^(1/2) - a^(1/2))^11*((B*a^3*e^3)/4 + (B*b^3*d^3)/4 - (B*a*b^2*d^2*e)/
4 - (B*a^2*b*d*e^2)/4))/(b^2*e^3*((d + e*x)^(1/2) - d^(1/2))^11) - (((a + b*x)^(1/2) - a^(1/2))^9*((17*B*a^3*e
^3)/12 + (17*B*b^3*d^3)/12 + (101*B*a*b^2*d^2*e)/4 + (101*B*a^2*b*d*e^2)/4))/(b*e^4*((d + e*x)^(1/2) - d^(1/2)
)^9) + (a^(1/2)*d^(1/2)*((a + b*x)^(1/2) - a^(1/2))^4*(32*B*b^4*d^2 + 32*B*a^2*b^2*e^2 + 96*B*a*b^3*d*e))/(e^6
*((d + e*x)^(1/2) - d^(1/2))^4) + (8*B*a^(3/2)*d^(3/2)*((a + b*x)^(1/2) - a^(1/2))^10)/(e^2*((d + e*x)^(1/2) -
d^(1/2))^10) + (a^(1/2)*d^(1/2)*((a + b*x)^(1/2) - a^(1/2))^6*(64*B*b^3*d^2 + 64*B*a^2*b*e^2 + (656*B*a*b^2*d
*e)/3))/(e^5*((d + e*x)^(1/2) - d^(1/2))^6) + (a^(1/2)*d^(1/2)*((a + b*x)^(1/2) - a^(1/2))^8*(32*B*a^2*e^2 + 3
2*B*b^2*d^2 + 96*B*a*b*d*e))/(e^4*((d + e*x)^(1/2) - d^(1/2))^8) + (8*B*a^(3/2)*b^4*d^(3/2)*((a + b*x)^(1/2) -
a^(1/2))^2)/(e^6*((d + e*x)^(1/2) - d^(1/2))^2))/(((a + b*x)^(1/2) - a^(1/2))^12/((d + e*x)^(1/2) - d^(1/2))^
12 + b^6/e^6 - (6*b^5*((a + b*x)^(1/2) - a^(1/2))^2)/(e^5*((d + e*x)^(1/2) - d^(1/2))^2) + (15*b^4*((a + b*x)^
(1/2) - a^(1/2))^4)/(e^4*((d + e*x)^(1/2) - d^(1/2))^4) - (20*b^3*((a + b*x)^(1/2) - a^(1/2))^6)/(e^3*((d + e*
x)^(1/2) - d^(1/2))^6) + (15*b^2*((a + b*x)^(1/2) - a^(1/2))^8)/(e^2*((d + e*x)^(1/2) - d^(1/2))^8) - (6*b*((a
+ b*x)^(1/2) - a^(1/2))^10)/(e*((d + e*x)^(1/2) - d^(1/2))^10)) - (A*log(a*e + b*d + 2*b*e*x + 2*b^(1/2)*e^(1
/2)*(a + b*x)^(1/2)*(d + e*x)^(1/2))*(a*e - b*d)^2)/(8*b^(3/2)*e^(3/2)) + (B*atanh((B*e^(1/2)*(a*e + b*d)*(a*e
- b*d)^2*((a + b*x)^(1/2) - a^(1/2)))/(b^(1/2)*((d + e*x)^(1/2) - d^(1/2))*(B*a^3*e^3 + B*b^3*d^3 - B*a*b^2*d
^2*e - B*a^2*b*d*e^2)))*(a*e + b*d)*(a*e - b*d)^2)/(4*b^(5/2)*e^(5/2))

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