3.21.73 \(\int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx\)

Optimal. Leaf size=304 \[ \frac {3 (b d-a e)^4 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{7/2} e^{7/2}}-\frac {3 \sqrt {a+b x} \sqrt {d+e x} (b d-a e)^3 (2 A b e-B (a e+b d))}{128 b^3 e^3}+\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^2 (2 A b e-B (a e+b d))}{64 b^3 e^2}+\frac {(a+b x)^{5/2} \sqrt {d+e x} (b d-a e) (2 A b e-B (a e+b d))}{16 b^3 e}+\frac {(a+b x)^{5/2} (d+e x)^{3/2} (2 A b e-B (a e+b d))}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e} \]

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Rubi [A]  time = 0.24, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {80, 50, 63, 217, 206} \begin {gather*} -\frac {3 \sqrt {a+b x} \sqrt {d+e x} (b d-a e)^3 (2 A b e-B (a e+b d))}{128 b^3 e^3}+\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^2 (2 A b e-B (a e+b d))}{64 b^3 e^2}+\frac {3 (b d-a e)^4 (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 )}{128 b^{7/2} e^{7/2}}+\frac {(a+b x)^{5/2} \sqrt {d+e x} (b d-a e) (2 A b e-B (a e+b d))}{16 b^3 e}+\frac {(a+b x)^{5/2} (d+e x)^{3/2} (2 A b e-B (a e+b d))}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(3/2)*(A + B*x)*(d + e*x)^(3/2),x]

[Out]

(-3*(b*d - a*e)^3*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(128*b^3*e^3) + ((b*d - a*e)^2*(2*A*b
*e - B*(b*d + a*e))*(a + b*x)^(3/2)*Sqrt[d + e*x])/(64*b^3*e^2) + ((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*(a +
b*x)^(5/2)*Sqrt[d + e*x])/(16*b^3*e) + ((2*A*b*e - B*(b*d + a*e))*(a + b*x)^(5/2)*(d + e*x)^(3/2))/(8*b^2*e) +
(B*(a + b*x)^(5/2)*(d + e*x)^(5/2))/(5*b*e) + (3*(b*d - a*e)^4*(2*A*b*e - B*(b*d + a*e))*ArcTanh[(Sqrt[e]*Sqr
t[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(128*b^(7/2)*e^(7/2))

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 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 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 (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx &=\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {\left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right ) \int (a+b x)^{3/2} (d+e x)^{3/2} \, dx}{5 b e}\\ &=\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {\left (3 (b d-a e) \left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right )\right ) \int (a+b x)^{3/2} \sqrt {d+e x} \, dx}{40 b^2 e}\\ &=\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {\left ((b d-a e)^2 \left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{80 b^3 e}\\ &=\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^3 e^2}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}-\frac {\left (3 (b d-a e)^3 \left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{320 b^3 e^2}\\ &=-\frac {3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{128 b^3 e^3}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^3 e^2}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {\left (3 (b d-a e)^4 \left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{640 b^3 e^3}\\ &=-\frac {3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{128 b^3 e^3}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^3 e^2}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {\left (3 (b d-a e)^4 \left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{320 b^4 e^3}\\ &=-\frac {3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{128 b^3 e^3}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^3 e^2}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {\left (3 (b d-a e)^4 \left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{320 b^4 e^3}\\ &=-\frac {3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{128 b^3 e^3}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^3 e^2}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {3 (b d-a e)^4 (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 )}{128 b^{7/2} e^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 3.21, size = 313, normalized size = 1.03 \begin {gather*} \frac {128 b^9 B e^3 (a+b x)^3 (d+e x)^4-5 \sqrt {b d-a e} \left (\frac {b (d+e x)}{b d-a e}\right )^{3/2} (a B e-2 A b e+b B d) \left (8 b^5 e^3 (a+b x)^3 (b d-a e)^{3/2} \sqrt {\frac {b (d+e x)}{b d-a e}} (-a e+3 b d+2 b e x)+2 b^5 e^2 (a+b x)^2 (b d-a e)^{7/2} \sqrt {\frac {b (d+e x)}{b d-a e}}-3 b^5 e (a+b x) (b d-a e)^{9/2} \sqrt {\frac {b (d+e x)}{b d-a e}}+3 b^5 \sqrt {e} \sqrt {a+b x} (b d-a e)^5 \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )\right )}{640 b^{10} e^4 \sqrt {a+b x} (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(3/2)*(A + B*x)*(d + e*x)^(3/2),x]

[Out]

(128*b^9*B*e^3*(a + b*x)^3*(d + e*x)^4 - 5*Sqrt[b*d - a*e]*(b*B*d - 2*A*b*e + a*B*e)*((b*(d + e*x))/(b*d - a*e
))^(3/2)*(-3*b^5*e*(b*d - a*e)^(9/2)*(a + b*x)*Sqrt[(b*(d + e*x))/(b*d - a*e)] + 2*b^5*e^2*(b*d - a*e)^(7/2)*(
a + b*x)^2*Sqrt[(b*(d + e*x))/(b*d - a*e)] + 8*b^5*e^3*(b*d - a*e)^(3/2)*(a + b*x)^3*Sqrt[(b*(d + e*x))/(b*d -
a*e)]*(3*b*d - a*e + 2*b*e*x) + 3*b^5*Sqrt[e]*(b*d - a*e)^5*Sqrt[a + b*x]*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqr
t[b*d - a*e]]))/(640*b^10*e^4*Sqrt[a + b*x]*(d + e*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.75, size = 394, normalized size = 1.30 \begin {gather*} -\frac {3 (b d-a e)^4 (a B e-2 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{128 b^{7/2} e^{7/2}}-\frac {\sqrt {d+e x} (b d-a e)^4 \left (\frac {30 A b^5 e (d+e x)^4}{(a+b x)^4}-\frac {140 A b^4 e^2 (d+e x)^3}{(a+b x)^3}+\frac {140 A b^2 e^4 (d+e x)}{a+b x}-\frac {15 b^5 B d (d+e x)^4}{(a+b x)^4}-\frac {15 a b^4 B e (d+e x)^4}{(a+b x)^4}+\frac {70 b^4 B d e (d+e x)^3}{(a+b x)^3}+\frac {70 a b^3 B e^2 (d+e x)^3}{(a+b x)^3}-\frac {128 b^3 B d e^2 (d+e x)^2}{(a+b x)^2}+\frac {128 a b^2 B e^3 (d+e x)^2}{(a+b x)^2}-\frac {70 b^2 B d e^3 (d+e x)}{a+b x}-\frac {70 a b B e^4 (d+e x)}{a+b x}+15 a B e^5-30 A b e^5+15 b B d e^4\right )}{640 b^3 e^3 \sqrt {a+b x} \left (\frac {b (d+e x)}{a+b x}-e\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(3/2)*(A + B*x)*(d + e*x)^(3/2),x]

[Out]

-1/640*((b*d - a*e)^4*Sqrt[d + e*x]*(15*b*B*d*e^4 - 30*A*b*e^5 + 15*a*B*e^5 - (70*b^2*B*d*e^3*(d + e*x))/(a +
b*x) + (140*A*b^2*e^4*(d + e*x))/(a + b*x) - (70*a*b*B*e^4*(d + e*x))/(a + b*x) - (128*b^3*B*d*e^2*(d + e*x)^2
)/(a + b*x)^2 + (128*a*b^2*B*e^3*(d + e*x)^2)/(a + b*x)^2 + (70*b^4*B*d*e*(d + e*x)^3)/(a + b*x)^3 - (140*A*b^
4*e^2*(d + e*x)^3)/(a + b*x)^3 + (70*a*b^3*B*e^2*(d + e*x)^3)/(a + b*x)^3 - (15*b^5*B*d*(d + e*x)^4)/(a + b*x)
^4 + (30*A*b^5*e*(d + e*x)^4)/(a + b*x)^4 - (15*a*b^4*B*e*(d + e*x)^4)/(a + b*x)^4))/(b^3*e^3*Sqrt[a + b*x]*(-
e + (b*(d + e*x))/(a + b*x))^5) - (3*(b*d - a*e)^4*(b*B*d - 2*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(
Sqrt[e]*Sqrt[a + b*x])])/(128*b^(7/2)*e^(7/2))

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fricas [A]  time = 2.33, size = 1036, normalized size = 3.41 \begin {gather*} \left [\frac {15 \, {\left (B b^{5} d^{5} - {\left (3 \, B a b^{4} + 2 \, A b^{5}\right )} d^{4} e + 2 \, {\left (B a^{2} b^{3} + 4 \, A a b^{4}\right )} d^{3} e^{2} + 2 \, {\left (B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} d^{2} e^{3} - {\left (3 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} d e^{4} + {\left (B a^{5} - 2 \, A a^{4} b\right )} e^{5}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} - 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (128 \, B b^{5} e^{5} x^{4} + 15 \, B b^{5} d^{4} e - 10 \, {\left (4 \, B a b^{4} + 3 \, A b^{5}\right )} d^{3} e^{2} + 2 \, {\left (9 \, B a^{2} b^{3} + 55 \, A a b^{4}\right )} d^{2} e^{3} - 10 \, {\left (4 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} d e^{4} + 15 \, {\left (B a^{4} b - 2 \, A a^{3} b^{2}\right )} e^{5} + 16 \, {\left (11 \, B b^{5} d e^{4} + {\left (11 \, B a b^{4} + 10 \, A b^{5}\right )} e^{5}\right )} x^{3} + 8 \, {\left (B b^{5} d^{2} e^{3} + 2 \, {\left (17 \, B a b^{4} + 15 \, A b^{5}\right )} d e^{4} + {\left (B a^{2} b^{3} + 30 \, A a b^{4}\right )} e^{5}\right )} x^{2} - 2 \, {\left (5 \, B b^{5} d^{3} e^{2} - {\left (13 \, B a b^{4} + 10 \, A b^{5}\right )} d^{2} e^{3} - {\left (13 \, B a^{2} b^{3} + 220 \, A a b^{4}\right )} d e^{4} + 5 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} e^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{2560 \, b^{4} e^{4}}, \frac {15 \, {\left (B b^{5} d^{5} - {\left (3 \, B a b^{4} + 2 \, A b^{5}\right )} d^{4} e + 2 \, {\left (B a^{2} b^{3} + 4 \, A a b^{4}\right )} d^{3} e^{2} + 2 \, {\left (B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} d^{2} e^{3} - {\left (3 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} d e^{4} + {\left (B a^{5} - 2 \, A a^{4} b\right )} e^{5}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (128 \, B b^{5} e^{5} x^{4} + 15 \, B b^{5} d^{4} e - 10 \, {\left (4 \, B a b^{4} + 3 \, A b^{5}\right )} d^{3} e^{2} + 2 \, {\left (9 \, B a^{2} b^{3} + 55 \, A a b^{4}\right )} d^{2} e^{3} - 10 \, {\left (4 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} d e^{4} + 15 \, {\left (B a^{4} b - 2 \, A a^{3} b^{2}\right )} e^{5} + 16 \, {\left (11 \, B b^{5} d e^{4} + {\left (11 \, B a b^{4} + 10 \, A b^{5}\right )} e^{5}\right )} x^{3} + 8 \, {\left (B b^{5} d^{2} e^{3} + 2 \, {\left (17 \, B a b^{4} + 15 \, A b^{5}\right )} d e^{4} + {\left (B a^{2} b^{3} + 30 \, A a b^{4}\right )} e^{5}\right )} x^{2} - 2 \, {\left (5 \, B b^{5} d^{3} e^{2} - {\left (13 \, B a b^{4} + 10 \, A b^{5}\right )} d^{2} e^{3} - {\left (13 \, B a^{2} b^{3} + 220 \, A a b^{4}\right )} d e^{4} + 5 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} e^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{1280 \, b^{4} e^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

[1/2560*(15*(B*b^5*d^5 - (3*B*a*b^4 + 2*A*b^5)*d^4*e + 2*(B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 + 2*(B*a^3*b^2 - 6*A*
a^2*b^3)*d^2*e^3 - (3*B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (B*a^5 - 2*A*a^4*b)*e^5)*sqrt(b*e)*log(8*b^2*e^2*x^2 + b^
2*d^2 + 6*a*b*d*e + a^2*e^2 - 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b
*e^2)*x) + 4*(128*B*b^5*e^5*x^4 + 15*B*b^5*d^4*e - 10*(4*B*a*b^4 + 3*A*b^5)*d^3*e^2 + 2*(9*B*a^2*b^3 + 55*A*a*
b^4)*d^2*e^3 - 10*(4*B*a^3*b^2 - 11*A*a^2*b^3)*d*e^4 + 15*(B*a^4*b - 2*A*a^3*b^2)*e^5 + 16*(11*B*b^5*d*e^4 + (
11*B*a*b^4 + 10*A*b^5)*e^5)*x^3 + 8*(B*b^5*d^2*e^3 + 2*(17*B*a*b^4 + 15*A*b^5)*d*e^4 + (B*a^2*b^3 + 30*A*a*b^4
)*e^5)*x^2 - 2*(5*B*b^5*d^3*e^2 - (13*B*a*b^4 + 10*A*b^5)*d^2*e^3 - (13*B*a^2*b^3 + 220*A*a*b^4)*d*e^4 + 5*(B*
a^3*b^2 - 2*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^4*e^4), 1/1280*(15*(B*b^5*d^5 - (3*B*a*b^4 + 2*
A*b^5)*d^4*e + 2*(B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 + 2*(B*a^3*b^2 - 6*A*a^2*b^3)*d^2*e^3 - (3*B*a^4*b - 8*A*a^3*
b^2)*d*e^4 + (B*a^5 - 2*A*a^4*b)*e^5)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqr
t(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) + 2*(128*B*b^5*e^5*x^4 + 15*B*b^5*d^4*e - 10*(4*B*
a*b^4 + 3*A*b^5)*d^3*e^2 + 2*(9*B*a^2*b^3 + 55*A*a*b^4)*d^2*e^3 - 10*(4*B*a^3*b^2 - 11*A*a^2*b^3)*d*e^4 + 15*(
B*a^4*b - 2*A*a^3*b^2)*e^5 + 16*(11*B*b^5*d*e^4 + (11*B*a*b^4 + 10*A*b^5)*e^5)*x^3 + 8*(B*b^5*d^2*e^3 + 2*(17*
B*a*b^4 + 15*A*b^5)*d*e^4 + (B*a^2*b^3 + 30*A*a*b^4)*e^5)*x^2 - 2*(5*B*b^5*d^3*e^2 - (13*B*a*b^4 + 10*A*b^5)*d
^2*e^3 - (13*B*a^2*b^3 + 220*A*a*b^4)*d*e^4 + 5*(B*a^3*b^2 - 2*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))
/(b^4*e^4)]

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giac [B]  time = 3.86, size = 2489, normalized size = 8.19

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(3/2),x, algorithm="giac")

[Out]

1/1920*(80*(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))*A*d*abs(b) + 10*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)
/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)*e^(-6)/b^14) - (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)*e^(-6
)/b^14) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)*e^(-6)/b^14)*sqrt(b*x +
a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*
x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*B*d*abs(b) - 1920*((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^2*d*abs(b)/b^2 + 160*(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*a*d*abs(b)/b + 10*(sqrt(b^2*d + (b*x +
a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)*e^(-6)/b^14) - (5*b^
13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)*e^(-6)/b^14) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*
b^12*d*e^5 - 93*a^3*b^11*e^6)*e^(-6)/b^14)*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 +
20*a^3*b*d*e^3 - 35*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*
b*e)))/b^(5/2))*A*abs(b)*e + (sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^
4 + (b^20*d*e^7 - 41*a*b^19*e^8)*e^(-8)/b^23) - (7*b^21*d^2*e^6 + 26*a*b^20*d*e^7 - 513*a^2*b^19*e^8)*e^(-8)/b
^23) + 5*(7*b^22*d^3*e^5 + 19*a*b^21*d^2*e^6 + 37*a^2*b^20*d*e^7 - 447*a^3*b^19*e^8)*e^(-8)/b^23)*(b*x + a) -
15*(7*b^23*d^4*e^4 + 12*a*b^22*d^3*e^5 + 18*a^2*b^21*d^2*e^6 + 28*a^3*b^20*d*e^7 - 193*a^4*b^19*e^8)*e^(-8)/b^
23)*sqrt(b*x + a) - 15*(7*b^5*d^5 + 5*a*b^4*d^4*e + 6*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 35*a^4*b*d*e^4 -
63*a^5*e^5)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(7/2))*B
*abs(b)*e + 80*(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*a^2*abs(b)*e/b^2 + 160*(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))*A*a*abs(b)*e/b + 20*(sqrt(b^2*d + (b*x + a)*b*e - a*b*
e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)*e^(-6)/b^14) - (5*b^13*d^2*e^4 +
14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)*e^(-6)/b^14) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 -
93*a^3*b^11*e^6)*e^(-6)/b^14)*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^
3 - 35*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2
))*B*a*abs(b)*e/b + 480*((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^2*d*abs(b)/b^3 + 960*((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*a*d*abs(b)/b^2 + 480*((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)))/sqr
t(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*a^2*abs(b
)*e/b^3)/b

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maple [B]  time = 0.02, size = 1631, normalized size = 5.37

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(3/2),x)

[Out]

1/1280*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-60*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a^3*b*e^4+30*A*b^5*d^4*e
*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))-15*B*a^5*e^5*ln(1/2*(2*b*
e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))-15*B*b^5*d^5*ln(1/2*(2*b*e*x+a*e+b*d+2
*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))+880*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a
*b^3*d*e^3*x+544*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a*b^3*d*e^3*x^2+30*(b*e*x^2+a*e*x+b*d*x+a*d)^(1
/2)*(b*e)^(1/2)*B*a^4*e^4+30*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*b^4*d^4+30*A*a^4*b*e^5*ln(1/2*(2*b*
e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))+52*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*
e)^(1/2)*B*a^2*b^2*d*e^3*x+52*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a*b^3*d^2*e^2*x+256*(b*e*x^2+a*e*x
+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*b^4*e^4*x^4+320*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*b^4*e^4*x^3+45*B
*a*b^4*d^4*e*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))+36*(b*e*x^2+a
*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a^2*b^2*d^2*e^2-80*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a*b^3*d^3
*e+40*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*b^4*d^2*e^2*x-20*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/
2)*B*a^3*b*e^4*x-60*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*b^4*d^3*e-120*A*a^3*b^2*d*e^4*ln(1/2*(2*b*e*
x+a*e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))+180*A*a^2*b^3*d^2*e^3*ln(1/2*(2*b*e*x+a*
e+b*d+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))-120*A*a*b^4*d^3*e^2*ln(1/2*(2*b*e*x+a*e+b*d+
2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))+45*B*a^4*b*d*e^4*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^
2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))-30*B*a^3*b^2*d^2*e^3*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e
*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))-30*B*a^2*b^3*d^3*e^2*ln(1/2*(2*b*e*x+a*e+b*d+2*(b*e*x^2+a*e*x+b*
d*x+a*d)^(1/2)*(b*e)^(1/2))/(b*e)^(1/2))+480*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a*b^3*e^4*x^2+480*(
b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*b^4*d*e^3*x^2+16*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a^
2*b^2*e^4*x^2+16*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*b^4*d^2*e^2*x^2-80*(b*e*x^2+a*e*x+b*d*x+a*d)^(1
/2)*(b*e)^(1/2)*B*a^3*b*d*e^3-20*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*b^4*d^3*e*x+220*(b*e*x^2+a*e*x+
b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a^2*b^2*d*e^3+220*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a*b^3*d^2*e^2+4
0*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a^2*b^2*e^4*x+352*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*
B*a*b^3*e^4*x^3+352*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*b^4*d*e^3*x^3)/b^3/e^3/(b*e*x^2+a*e*x+b*d*x+
a*d)^(1/2)/(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)^(3/2)*(B*x+A)*(e*x+d)^(3/2),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*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^(3/2)*(d + e*x)^(3/2),x)

[Out]

int((A + B*x)*(a + b*x)^(3/2)*(d + e*x)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(B*x+A)*(e*x+d)**(3/2),x)

[Out]

Integral((A + B*x)*(a + b*x)**(3/2)*(d + e*x)**(3/2), x)

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