3.21.86 \(\int (a+b x)^{5/2} (A+B x) \sqrt {d+e x} \, dx\)

Optimal. Leaf size=304 \[ \frac {(b d-a e)^4 (3 a B e-10 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{5/2} e^{9/2}}-\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^3 (3 a B e-10 A b e+7 b B d)}{128 b^2 e^4}+\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^2 (3 a B e-10 A b e+7 b B d)}{192 b^2 e^3}-\frac {(a+b x)^{5/2} \sqrt {d+e x} (b d-a e) (3 a B e-10 A b e+7 b B d)}{240 b^2 e^2}-\frac {(a+b x)^{7/2} \sqrt {d+e x} (3 a B e-10 A b e+7 b B d)}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e} \]

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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 {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^3 (3 a B e-10 A b e+7 b B d)}{128 b^2 e^4}+\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^2 (3 a B e-10 A b e+7 b B d)}{192 b^2 e^3}-\frac {(a+b x)^{5/2} \sqrt {d+e x} (b d-a e) (3 a B e-10 A b e+7 b B d)}{240 b^2 e^2}+\frac {(b d-a e)^4 (3 a B e-10 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{5/2} e^{9/2}}-\frac {(a+b x)^{7/2} \sqrt {d+e x} (3 a B e-10 A b e+7 b B d)}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/2)*(A + B*x)*Sqrt[d + e*x],x]

[Out]

-((b*d - a*e)^3*(7*b*B*d - 10*A*b*e + 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(128*b^2*e^4) + ((b*d - a*e)^2*(7*
b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(192*b^2*e^3) - ((b*d - a*e)*(7*b*B*d - 10*A*b*e +
3*a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(240*b^2*e^2) - ((7*b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(7/2)*Sqrt[d
+ e*x])/(40*b^2*e) + (B*(a + b*x)^(7/2)*(d + e*x)^(3/2))/(5*b*e) + ((b*d - a*e)^4*(7*b*B*d - 10*A*b*e + 3*a*B
*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(128*b^(5/2)*e^(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 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)^{5/2} (A+B x) \sqrt {d+e x} \, dx &=\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac {\left (5 A b e-B \left (\frac {7 b d}{2}+\frac {3 a e}{2}\right )\right ) \int (a+b x)^{5/2} \sqrt {d+e x} \, dx}{5 b e}\\ &=-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}-\frac {((b d-a e) (7 b B d-10 A b e+3 a B e)) \int \frac {(a+b x)^{5/2}}{\sqrt {d+e x}} \, dx}{80 b^2 e}\\ &=-\frac {(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{240 b^2 e^2}-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac {\left ((b d-a e)^2 (7 b B d-10 A b e+3 a B e)\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{96 b^2 e^2}\\ &=\frac {(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{192 b^2 e^3}-\frac {(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{240 b^2 e^2}-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}-\frac {\left ((b d-a e)^3 (7 b B d-10 A b e+3 a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{128 b^2 e^3}\\ &=-\frac {(b d-a e)^3 (7 b B d-10 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^2 e^4}+\frac {(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{192 b^2 e^3}-\frac {(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{240 b^2 e^2}-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac {\left ((b d-a e)^4 (7 b B d-10 A b e+3 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{256 b^2 e^4}\\ &=-\frac {(b d-a e)^3 (7 b B d-10 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^2 e^4}+\frac {(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{192 b^2 e^3}-\frac {(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{240 b^2 e^2}-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac {\left ((b d-a e)^4 (7 b B d-10 A b e+3 a B e)\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 )}{128 b^3 e^4}\\ &=-\frac {(b d-a e)^3 (7 b B d-10 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^2 e^4}+\frac {(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{192 b^2 e^3}-\frac {(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{240 b^2 e^2}-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac {\left ((b d-a e)^4 (7 b B d-10 A b e+3 a B e)\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 )}{128 b^3 e^4}\\ &=-\frac {(b d-a e)^3 (7 b B d-10 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^2 e^4}+\frac {(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{192 b^2 e^3}-\frac {(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{240 b^2 e^2}-\frac {(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{40 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac {(b d-a e)^4 (7 b B d-10 A b e+3 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{5/2} e^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 3.46, size = 359, normalized size = 1.18 \begin {gather*} \frac {(a+b x)^{7/2} (d+e x)^{3/2} \left (\frac {7 \left (-\frac {3 a B e}{2}+5 A b e-\frac {7}{2} b B d\right ) \left (48 b^4 e^4 (a+b x)^4 \sqrt {b d-a e} \sqrt {\frac {b (d+e x)}{b d-a e}}+b (b d-a e) \left (8 b^3 e^3 (a+b x)^3 \sqrt {b d-a e} \sqrt {\frac {b (d+e x)}{b d-a e}}-10 b^3 e^2 (a+b x)^2 (b d-a e)^{3/2} \sqrt {\frac {b (d+e x)}{b d-a e}}+15 b^3 e (a+b x) (b d-a e)^{5/2} \sqrt {\frac {b (d+e x)}{b d-a e}}-15 b^3 \sqrt {e} \sqrt {a+b x} (b d-a e)^3 \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )\right )\right )}{192 b^4 e^4 (a+b x)^4 (b d-a e)^{3/2} \left (\frac {b (d+e x)}{b d-a e}\right )^{3/2}}+7 B\right )}{35 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/2)*(A + B*x)*Sqrt[d + e*x],x]

[Out]

((a + b*x)^(7/2)*(d + e*x)^(3/2)*(7*B + (7*((-7*b*B*d)/2 + 5*A*b*e - (3*a*B*e)/2)*(48*b^4*e^4*Sqrt[b*d - a*e]*
(a + b*x)^4*Sqrt[(b*(d + e*x))/(b*d - a*e)] + b*(b*d - a*e)*(15*b^3*e*(b*d - a*e)^(5/2)*(a + b*x)*Sqrt[(b*(d +
e*x))/(b*d - a*e)] - 10*b^3*e^2*(b*d - a*e)^(3/2)*(a + b*x)^2*Sqrt[(b*(d + e*x))/(b*d - a*e)] + 8*b^3*e^3*Sqr
t[b*d - a*e]*(a + b*x)^3*Sqrt[(b*(d + e*x))/(b*d - a*e)] - 15*b^3*Sqrt[e]*(b*d - a*e)^3*Sqrt[a + b*x]*ArcSinh[
(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]])))/(192*b^4*e^4*(b*d - a*e)^(3/2)*(a + b*x)^4*((b*(d + e*x))/(b*d - a
*e))^(3/2))))/(35*b*e)

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IntegrateAlgebraic [A]  time = 0.73, size = 419, normalized size = 1.38 \begin {gather*} \frac {(b d-a e)^4 (3 a B e-10 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{128 b^{5/2} e^{9/2}}+\frac {\sqrt {d+e x} (b d-a e)^4 \left (\frac {150 A b^5 e (d+e x)^4}{(a+b x)^4}-\frac {700 A b^4 e^2 (d+e x)^3}{(a+b x)^3}+\frac {1280 A b^3 e^3 (d+e x)^2}{(a+b x)^2}-\frac {580 A b^2 e^4 (d+e x)}{a+b x}-\frac {105 b^5 B d (d+e x)^4}{(a+b x)^4}-\frac {45 a b^4 B e (d+e x)^4}{(a+b x)^4}+\frac {490 b^4 B d e (d+e x)^3}{(a+b x)^3}+\frac {210 a b^3 B e^2 (d+e x)^3}{(a+b x)^3}-\frac {896 b^3 B d e^2 (d+e x)^2}{(a+b x)^2}-\frac {384 a b^2 B e^3 (d+e x)^2}{(a+b x)^2}+\frac {790 b^2 B d e^3 (d+e x)}{a+b x}-\frac {210 a b B e^4 (d+e x)}{a+b x}+45 a B e^5-150 A b e^5+105 b B d e^4\right )}{1920 b^2 e^4 \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)^(5/2)*(A + B*x)*Sqrt[d + e*x],x]

[Out]

((b*d - a*e)^4*Sqrt[d + e*x]*(105*b*B*d*e^4 - 150*A*b*e^5 + 45*a*B*e^5 + (790*b^2*B*d*e^3*(d + e*x))/(a + b*x)
- (580*A*b^2*e^4*(d + e*x))/(a + b*x) - (210*a*b*B*e^4*(d + e*x))/(a + b*x) - (896*b^3*B*d*e^2*(d + e*x)^2)/(
a + b*x)^2 + (1280*A*b^3*e^3*(d + e*x)^2)/(a + b*x)^2 - (384*a*b^2*B*e^3*(d + e*x)^2)/(a + b*x)^2 + (490*b^4*B
*d*e*(d + e*x)^3)/(a + b*x)^3 - (700*A*b^4*e^2*(d + e*x)^3)/(a + b*x)^3 + (210*a*b^3*B*e^2*(d + e*x)^3)/(a + b
*x)^3 - (105*b^5*B*d*(d + e*x)^4)/(a + b*x)^4 + (150*A*b^5*e*(d + e*x)^4)/(a + b*x)^4 - (45*a*b^4*B*e*(d + e*x
)^4)/(a + b*x)^4))/(1920*b^2*e^4*Sqrt[a + b*x]*(-e + (b*(d + e*x))/(a + b*x))^5) + ((b*d - a*e)^4*(7*b*B*d - 1
0*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/(128*b^(5/2)*e^(9/2))

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fricas [A]  time = 1.24, size = 1046, normalized size = 3.44

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

[-1/7680*(15*(7*B*b^5*d^5 - 5*(5*B*a*b^4 + 2*A*b^5)*d^4*e + 10*(3*B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 - 10*(B*a^3*b
^2 + 6*A*a^2*b^3)*d^2*e^3 - 5*(B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (3*B*a^5 - 10*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*(384*B*b^5*e^5*x^4 - 105*B*b^5*d^4*e + 10*(34*B*a*b^4 + 15*A*b^5)*d^3*e^2 - 2*(173*B*
a^2*b^3 + 275*A*a*b^4)*d^2*e^3 + 10*(6*B*a^3*b^2 + 73*A*a^2*b^3)*d*e^4 - 15*(3*B*a^4*b - 10*A*a^3*b^2)*e^5 + 4
8*(B*b^5*d*e^4 + (21*B*a*b^4 + 10*A*b^5)*e^5)*x^3 - 8*(7*B*b^5*d^2*e^3 - 2*(11*B*a*b^4 + 5*A*b^5)*d*e^4 - (93*
B*a^2*b^3 + 170*A*a*b^4)*e^5)*x^2 + 2*(35*B*b^5*d^3*e^2 - (111*B*a*b^4 + 50*A*b^5)*d^2*e^3 + (109*B*a^2*b^3 +
180*A*a*b^4)*d*e^4 + 5*(3*B*a^3*b^2 + 118*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^3*e^5), -1/3840*(
15*(7*B*b^5*d^5 - 5*(5*B*a*b^4 + 2*A*b^5)*d^4*e + 10*(3*B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 - 10*(B*a^3*b^2 + 6*A*a
^2*b^3)*d^2*e^3 - 5*(B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (3*B*a^5 - 10*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)*sqrt(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) - 2*(384
*B*b^5*e^5*x^4 - 105*B*b^5*d^4*e + 10*(34*B*a*b^4 + 15*A*b^5)*d^3*e^2 - 2*(173*B*a^2*b^3 + 275*A*a*b^4)*d^2*e^
3 + 10*(6*B*a^3*b^2 + 73*A*a^2*b^3)*d*e^4 - 15*(3*B*a^4*b - 10*A*a^3*b^2)*e^5 + 48*(B*b^5*d*e^4 + (21*B*a*b^4
+ 10*A*b^5)*e^5)*x^3 - 8*(7*B*b^5*d^2*e^3 - 2*(11*B*a*b^4 + 5*A*b^5)*d*e^4 - (93*B*a^2*b^3 + 170*A*a*b^4)*e^5)
*x^2 + 2*(35*B*b^5*d^3*e^2 - (111*B*a*b^4 + 50*A*b^5)*d^2*e^3 + (109*B*a^2*b^3 + 180*A*a*b^4)*d*e^4 + 5*(3*B*a
^3*b^2 + 118*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^3*e^5)]

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giac [B]  time = 2.96, size = 1670, normalized size = 5.49

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

1/1920*(240*(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) + 30*(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) - 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^3*abs(b)/b^2 + 240*(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)/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^1
3*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 + 2
0*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*b*abs(b) + (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) - 1
5*(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^2
3)*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 - 6
3*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*
b*abs(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^3*abs(b)/b^3 + 1440*((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^2*abs(b)/b^2)/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)^(5/2)*(B*x+A)*(e*x+d)^(1/2),x)

[Out]

-1/3840*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-300*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*a^3*b*e^4+150*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))-45*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))-105*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))-720*(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-352*(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+90*(b*e*x^2+a*e*x+b*d*x+a*d
)^(1/2)*(b*e)^(1/2)*B*a^4*e^4+210*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*b^4*d^4+150*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))-436*(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+444*(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-768*(b*e*x
^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*b^4*e^4*x^4-960*(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+375*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))+692*
(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^2-680*(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+200*(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-60*(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-300*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*A*b^4*d^3*e-600*A*a^3*b^2*d*e^4*l
n(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))+900*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))-600*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))+75*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))+150*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))-450*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))-2720*(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^2-160*(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^2-1488*(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+112*(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^2-120*(b*e*x^2
+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*B*a^3*b*d*e^3-140*(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-1460*(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+1100*(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^2-2360*(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-2016*(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^3-96*(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^2/
(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)/e^4/(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)^(5/2)*(B*x+A)*(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(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 )}^{5/2}\,\sqrt {d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^(5/2)*(d + e*x)^(1/2),x)

[Out]

int((A + B*x)*(a + b*x)^(5/2)*(d + e*x)^(1/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(B*x+A)*(e*x+d)**(1/2),x)

[Out]

Timed out

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