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

Optimal. Leaf size=304 \[ \frac {(b d-a e)^4 (7 a B e-10 A b e+3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{9/2} e^{5/2}}-\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^3 (7 a B e-10 A b e+3 b B d)}{128 b^4 e^2}-\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^2 (7 a B e-10 A b e+3 b B d)}{64 b^4 e}-\frac {(a+b x)^{3/2} (d+e x)^{3/2} (b d-a e) (7 a B e-10 A b e+3 b B d)}{48 b^3 e}-\frac {(a+b x)^{3/2} (d+e x)^{5/2} (7 a B e-10 A b e+3 b B d)}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e} \]

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Rubi [A]  time = 0.27, 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 (7 a B e-10 A b e+3 b B d)}{128 b^4 e^2}+\frac {(b d-a e)^4 (7 a B e-10 A b e+3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{9/2} e^{5/2}}-\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^2 (7 a B e-10 A b e+3 b B d)}{64 b^4 e}-\frac {(a+b x)^{3/2} (d+e x)^{3/2} (b d-a e) (7 a B e-10 A b e+3 b B d)}{48 b^3 e}-\frac {(a+b x)^{3/2} (d+e x)^{5/2} (7 a B e-10 A b e+3 b B d)}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*d - a*e)^3*(3*b*B*d - 10*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(128*b^4*e^2) - ((b*d - a*e)^2*(3*
b*B*d - 10*A*b*e + 7*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(64*b^4*e) - ((b*d - a*e)*(3*b*B*d - 10*A*b*e + 7*a
*B*e)*(a + b*x)^(3/2)*(d + e*x)^(3/2))/(48*b^3*e) - ((3*b*B*d - 10*A*b*e + 7*a*B*e)*(a + b*x)^(3/2)*(d + e*x)^
(5/2))/(40*b^2*e) + (B*(a + b*x)^(3/2)*(d + e*x)^(7/2))/(5*b*e) + ((b*d - a*e)^4*(3*b*B*d - 10*A*b*e + 7*a*B*e
)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(128*b^(9/2)*e^(5/2))

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 \sqrt {a+b x} (A+B x) (d+e x)^{5/2} \, dx &=\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}+\frac {\left (5 A b e-B \left (\frac {3 b d}{2}+\frac {7 a e}{2}\right )\right ) \int \sqrt {a+b x} (d+e x)^{5/2} \, dx}{5 b e}\\ &=-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}-\frac {((b d-a e) (3 b B d-10 A b e+7 a B e)) \int \sqrt {a+b x} (d+e x)^{3/2} \, dx}{16 b^2 e}\\ &=-\frac {(b d-a e) (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{48 b^3 e}-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}-\frac {\left ((b d-a e)^2 (3 b B d-10 A b e+7 a B e)\right ) \int \sqrt {a+b x} \sqrt {d+e x} \, dx}{32 b^3 e}\\ &=-\frac {(b d-a e)^2 (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^4 e}-\frac {(b d-a e) (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{48 b^3 e}-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}-\frac {\left ((b d-a e)^3 (3 b B d-10 A b e+7 a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{128 b^4 e}\\ &=-\frac {(b d-a e)^3 (3 b B d-10 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^4 e^2}-\frac {(b d-a e)^2 (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^4 e}-\frac {(b d-a e) (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{48 b^3 e}-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}+\frac {\left ((b d-a e)^4 (3 b B d-10 A b e+7 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{256 b^4 e^2}\\ &=-\frac {(b d-a e)^3 (3 b B d-10 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^4 e^2}-\frac {(b d-a e)^2 (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^4 e}-\frac {(b d-a e) (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{48 b^3 e}-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}+\frac {\left ((b d-a e)^4 (3 b B d-10 A b e+7 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^5 e^2}\\ &=-\frac {(b d-a e)^3 (3 b B d-10 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^4 e^2}-\frac {(b d-a e)^2 (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^4 e}-\frac {(b d-a e) (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{48 b^3 e}-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}+\frac {\left ((b d-a e)^4 (3 b B d-10 A b e+7 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^5 e^2}\\ &=-\frac {(b d-a e)^3 (3 b B d-10 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{128 b^4 e^2}-\frac {(b d-a e)^2 (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^4 e}-\frac {(b d-a e) (3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{48 b^3 e}-\frac {(3 b B d-10 A b e+7 a B e) (a+b x)^{3/2} (d+e x)^{5/2}}{40 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{7/2}}{5 b e}+\frac {(b d-a e)^4 (3 b B d-10 A b e+7 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{9/2} e^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 3.37, size = 301, normalized size = 0.99 \begin {gather*} \frac {\sqrt {d+e x} \left (\frac {\left (-\frac {7 a B e}{2}+5 A b e-\frac {3}{2} b B d\right ) \left (2 b^5 e^2 (a+b x)^2 (b d-a e)^{3/2} \sqrt {\frac {b (d+e x)}{b d-a e}} \left (15 a^2 e^2-10 a b e (5 d+2 e x)+b^2 \left (59 d^2+68 d e x+24 e^2 x^2\right )\right )+15 b^5 e (a+b x) (b d-a e)^{9/2} \sqrt {\frac {b (d+e x)}{b d-a e}}-15 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 )}{(b d-a e)^{3/2} \sqrt {\frac {b (d+e x)}{b d-a e}}}+192 b^8 B e^2 (a+b x)^2 (d+e x)^3\right )}{960 b^9 e^3 \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(192*b^8*B*e^2*(a + b*x)^2*(d + e*x)^3 + (((-3*b*B*d)/2 + 5*A*b*e - (7*a*B*e)/2)*(15*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)^(3/2)*(a + b*x)^2*Sqrt[(b*(d +
e*x))/(b*d - a*e)]*(15*a^2*e^2 - 10*a*b*e*(5*d + 2*e*x) + b^2*(59*d^2 + 68*d*e*x + 24*e^2*x^2)) - 15*b^5*Sqrt
[e]*(b*d - a*e)^5*Sqrt[a + b*x]*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]]))/((b*d - a*e)^(3/2)*Sqrt[(b*
(d + e*x))/(b*d - a*e)])))/(960*b^9*e^3*Sqrt[a + b*x])

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IntegrateAlgebraic [A]  time = 0.66, size = 418, normalized size = 1.38 \begin {gather*} \frac {(b d-a e)^4 (7 a B e-10 A b e+3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{9/2} e^{5/2}}+\frac {\sqrt {a+b x} (b d-a e)^4 \left (\frac {580 A b^4 e^2 (a+b x)}{d+e x}-\frac {1280 A b^3 e^3 (a+b x)^2}{(d+e x)^2}+\frac {700 A b^2 e^4 (a+b x)^3}{(d+e x)^3}-\frac {150 A b e^5 (a+b x)^4}{(d+e x)^4}+\frac {210 b^4 B d e (a+b x)}{d+e x}-105 a b^4 B e-\frac {790 a b^3 B e^2 (a+b x)}{d+e x}+\frac {384 b^3 B d e^2 (a+b x)^2}{(d+e x)^2}+\frac {896 a b^2 B e^3 (a+b x)^2}{(d+e x)^2}-\frac {210 b^2 B d e^3 (a+b x)^3}{(d+e x)^3}+\frac {105 a B e^5 (a+b x)^4}{(d+e x)^4}-\frac {490 a b B e^4 (a+b x)^3}{(d+e x)^3}+\frac {45 b B d e^4 (a+b x)^4}{(d+e x)^4}+150 A b^5 e-45 b^5 B d\right )}{1920 b^4 e^2 \sqrt {d+e x} \left (b-\frac {e (a+b x)}{d+e x}\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.59, size = 1044, normalized size = 3.43

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/7680*(15*(3*B*b^5*d^5 - 5*(B*a*b^4 + 2*A*b^5)*d^4*e - 10*(B*a^2*b^3 - 4*A*a*b^4)*d^3*e^2 + 30*(B*a^3*b^2 -
2*A*a^2*b^3)*d^2*e^3 - 5*(5*B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (7*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 - 45*B*b^5*d^4*e + 30*(2*B*a*b^4 + 5*A*b^5)*d^3*e^2 - 2*(173*B*a^2*b
^3 - 365*A*a*b^4)*d^2*e^3 + 10*(34*B*a^3*b^2 - 55*A*a^2*b^3)*d*e^4 - 15*(7*B*a^4*b - 10*A*a^3*b^2)*e^5 + 48*(2
1*B*b^5*d*e^4 + (B*a*b^4 + 10*A*b^5)*e^5)*x^3 + 8*(93*B*b^5*d^2*e^3 + 2*(11*B*a*b^4 + 85*A*b^5)*d*e^4 - (7*B*a
^2*b^3 - 10*A*a*b^4)*e^5)*x^2 + 2*(15*B*b^5*d^3*e^2 + (109*B*a*b^4 + 590*A*b^5)*d^2*e^3 - 3*(37*B*a^2*b^3 - 60
*A*a*b^4)*d*e^4 + 5*(7*B*a^3*b^2 - 10*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*e^3), -1/3840*(15*(
3*B*b^5*d^5 - 5*(B*a*b^4 + 2*A*b^5)*d^4*e - 10*(B*a^2*b^3 - 4*A*a*b^4)*d^3*e^2 + 30*(B*a^3*b^2 - 2*A*a^2*b^3)*
d^2*e^3 - 5*(5*B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (7*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 - 45*B*b^5*d^4*e + 30*(2*B*a*b^4 + 5*A*b^5)*d^3*e^2 - 2*(173*B*a^2*b^3 - 365*A*a*b^4)*d^2*e^3 + 10*(3
4*B*a^3*b^2 - 55*A*a^2*b^3)*d*e^4 - 15*(7*B*a^4*b - 10*A*a^3*b^2)*e^5 + 48*(21*B*b^5*d*e^4 + (B*a*b^4 + 10*A*b
^5)*e^5)*x^3 + 8*(93*B*b^5*d^2*e^3 + 2*(11*B*a*b^4 + 85*A*b^5)*d*e^4 - (7*B*a^2*b^3 - 10*A*a*b^4)*e^5)*x^2 + 2
*(15*B*b^5*d^3*e^2 + (109*B*a*b^4 + 590*A*b^5)*d^2*e^3 - 3*(37*B*a^2*b^3 - 60*A*a*b^4)*d*e^4 + 5*(7*B*a^3*b^2
- 10*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*e^3)]

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giac [B]  time = 3.84, size = 2506, normalized size = 8.24

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1920*(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*d^2*abs(b)/b^2 - 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*d^2*ab
s(b)/b - 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 - 1
3*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)*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*d*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*d*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*d^2*abs(b)/b^3 - 480*((b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*e^(-3/2)*log(ab
s(-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*d^2*abs(b)/b^2 - 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*a*abs(b)*e^2/
b^2 - 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) + s
qrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*B*a*abs(b)*e^2/b^2 - 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*a
bs(b)*e^2/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^2
2*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^2/b -
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)*s
qrt(b*x + a))*A*a*d*abs(b)*e/b^3)/b

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maple [B]  time = 0.03, 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)*(e*x+d)^(5/2)*(b*x+a)^(1/2),x)

[Out]

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

[Out]

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

[Out]

Timed out

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