3.2211 \(\int (a+b x)^{3/2} (A+B x) (d+e x)^{5/2} \, dx\)

Optimal. Leaf size=358 \[ -\frac {(b d-a e)^5 (7 a B e-12 A b e+5 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{512 b^{9/2} e^{7/2}}+\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^4 (7 a B e-12 A b e+5 b B d)}{512 b^4 e^3}-\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^3 (7 a B e-12 A b e+5 b B d)}{768 b^4 e^2}-\frac {(a+b x)^{5/2} \sqrt {d+e x} (b d-a e)^2 (7 a B e-12 A b e+5 b B d)}{192 b^4 e}-\frac {(a+b x)^{5/2} (d+e x)^{3/2} (b d-a e) (7 a B e-12 A b e+5 b B d)}{96 b^3 e}-\frac {(a+b x)^{5/2} (d+e x)^{5/2} (7 a B e-12 A b e+5 b B d)}{60 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e} \]

[Out]

-1/96*(-a*e+b*d)*(-12*A*b*e+7*B*a*e+5*B*b*d)*(b*x+a)^(5/2)*(e*x+d)^(3/2)/b^3/e-1/60*(-12*A*b*e+7*B*a*e+5*B*b*d
)*(b*x+a)^(5/2)*(e*x+d)^(5/2)/b^2/e+1/6*B*(b*x+a)^(5/2)*(e*x+d)^(7/2)/b/e-1/512*(-a*e+b*d)^5*(-12*A*b*e+7*B*a*
e+5*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(9/2)/e^(7/2)-1/768*(-a*e+b*d)^3*(-12*A*b*e+
7*B*a*e+5*B*b*d)*(b*x+a)^(3/2)*(e*x+d)^(1/2)/b^4/e^2-1/192*(-a*e+b*d)^2*(-12*A*b*e+7*B*a*e+5*B*b*d)*(b*x+a)^(5
/2)*(e*x+d)^(1/2)/b^4/e+1/512*(-a*e+b*d)^4*(-12*A*b*e+7*B*a*e+5*B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^4/e^3

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Rubi [A]  time = 0.32, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^4 (7 a B e-12 A b e+5 b B d)}{512 b^4 e^3}-\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^3 (7 a B e-12 A b e+5 b B d)}{768 b^4 e^2}-\frac {(b d-a e)^5 (7 a B e-12 A b e+5 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{512 b^{9/2} e^{7/2}}-\frac {(a+b x)^{5/2} \sqrt {d+e x} (b d-a e)^2 (7 a B e-12 A b e+5 b B d)}{192 b^4 e}-\frac {(a+b x)^{5/2} (d+e x)^{3/2} (b d-a e) (7 a B e-12 A b e+5 b B d)}{96 b^3 e}-\frac {(a+b x)^{5/2} (d+e x)^{5/2} (7 a B e-12 A b e+5 b B d)}{60 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(3/2)*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

((b*d - a*e)^4*(5*b*B*d - 12*A*b*e + 7*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(512*b^4*e^3) - ((b*d - a*e)^3*(5*b
*B*d - 12*A*b*e + 7*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(768*b^4*e^2) - ((b*d - a*e)^2*(5*b*B*d - 12*A*b*e +
7*a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(192*b^4*e) - ((b*d - a*e)*(5*b*B*d - 12*A*b*e + 7*a*B*e)*(a + b*x)^(
5/2)*(d + e*x)^(3/2))/(96*b^3*e) - ((5*b*B*d - 12*A*b*e + 7*a*B*e)*(a + b*x)^(5/2)*(d + e*x)^(5/2))/(60*b^2*e)
+ (B*(a + b*x)^(5/2)*(d + e*x)^(7/2))/(6*b*e) - ((b*d - a*e)^5*(5*b*B*d - 12*A*b*e + 7*a*B*e)*ArcTanh[(Sqrt[e
]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(512*b^(9/2)*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)^{5/2} \, dx &=\frac {B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e}+\frac {\left (6 A b e-B \left (\frac {5 b d}{2}+\frac {7 a e}{2}\right )\right ) \int (a+b x)^{3/2} (d+e x)^{5/2} \, dx}{6 b e}\\ &=-\frac {(5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{5/2}}{60 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e}-\frac {((b d-a e) (5 b B d-12 A b e+7 a B e)) \int (a+b x)^{3/2} (d+e x)^{3/2} \, dx}{24 b^2 e}\\ &=-\frac {(b d-a e) (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{3/2}}{96 b^3 e}-\frac {(5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{5/2}}{60 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e}-\frac {\left ((b d-a e)^2 (5 b B d-12 A b e+7 a B e)\right ) \int (a+b x)^{3/2} \sqrt {d+e x} \, dx}{64 b^3 e}\\ &=-\frac {(b d-a e)^2 (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{192 b^4 e}-\frac {(b d-a e) (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{3/2}}{96 b^3 e}-\frac {(5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{5/2}}{60 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e}-\frac {\left ((b d-a e)^3 (5 b B d-12 A b e+7 a B e)\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{384 b^4 e}\\ &=-\frac {(b d-a e)^3 (5 b B d-12 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{768 b^4 e^2}-\frac {(b d-a e)^2 (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{192 b^4 e}-\frac {(b d-a e) (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{3/2}}{96 b^3 e}-\frac {(5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{5/2}}{60 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e}+\frac {\left ((b d-a e)^4 (5 b B d-12 A b e+7 a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{512 b^4 e^2}\\ &=\frac {(b d-a e)^4 (5 b B d-12 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{512 b^4 e^3}-\frac {(b d-a e)^3 (5 b B d-12 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{768 b^4 e^2}-\frac {(b d-a e)^2 (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{192 b^4 e}-\frac {(b d-a e) (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{3/2}}{96 b^3 e}-\frac {(5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{5/2}}{60 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e}-\frac {\left ((b d-a e)^5 (5 b B d-12 A b e+7 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{1024 b^4 e^3}\\ &=\frac {(b d-a e)^4 (5 b B d-12 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{512 b^4 e^3}-\frac {(b d-a e)^3 (5 b B d-12 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{768 b^4 e^2}-\frac {(b d-a e)^2 (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{192 b^4 e}-\frac {(b d-a e) (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{3/2}}{96 b^3 e}-\frac {(5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{5/2}}{60 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e}-\frac {\left ((b d-a e)^5 (5 b B d-12 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 )}{512 b^5 e^3}\\ &=\frac {(b d-a e)^4 (5 b B d-12 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{512 b^4 e^3}-\frac {(b d-a e)^3 (5 b B d-12 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{768 b^4 e^2}-\frac {(b d-a e)^2 (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{192 b^4 e}-\frac {(b d-a e) (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{3/2}}{96 b^3 e}-\frac {(5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{5/2}}{60 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e}-\frac {\left ((b d-a e)^5 (5 b B d-12 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 )}{512 b^5 e^3}\\ &=\frac {(b d-a e)^4 (5 b B d-12 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{512 b^4 e^3}-\frac {(b d-a e)^3 (5 b B d-12 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{768 b^4 e^2}-\frac {(b d-a e)^2 (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{192 b^4 e}-\frac {(b d-a e) (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{3/2}}{96 b^3 e}-\frac {(5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{5/2}}{60 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e}-\frac {(b d-a e)^5 (5 b B d-12 A b e+7 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{512 b^{9/2} e^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 4.29, size = 357, normalized size = 1.00 \[ \frac {(a+b x)^{5/2} \sqrt {d+e x} \left (\frac {\left (-\frac {7 a B e}{2}+6 A b e-\frac {5}{2} b B d\right ) \left (8 b^6 e^3 (a+b x)^3 (b d-a e)^{3/2} \sqrt {\frac {b (d+e x)}{b d-a e}} \left (5 a^2 e^2-10 a b e (2 d+e x)+b^2 \left (31 d^2+42 d e x+16 e^2 x^2\right )\right )+5 b^6 \sqrt {e} \left (2 e^{3/2} (a+b x)^2 (b d-a e)^{9/2} \sqrt {\frac {b (d+e x)}{b d-a e}}-3 \sqrt {e} (a+b x) (b d-a e)^{11/2} \sqrt {\frac {b (d+e x)}{b d-a e}}+3 \sqrt {a+b x} (b d-a e)^6 \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )\right )\right )}{128 b^6 e^3 (a+b x)^3 (b d-a e)^{3/2} \sqrt {\frac {b (d+e x)}{b d-a e}}}+5 b^3 B (d+e x)^3\right )}{30 b^4 e} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(3/2)*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

((a + b*x)^(5/2)*Sqrt[d + e*x]*(5*b^3*B*(d + e*x)^3 + (((-5*b*B*d)/2 + 6*A*b*e - (7*a*B*e)/2)*(8*b^6*e^3*(b*d
- a*e)^(3/2)*(a + b*x)^3*Sqrt[(b*(d + e*x))/(b*d - a*e)]*(5*a^2*e^2 - 10*a*b*e*(2*d + e*x) + b^2*(31*d^2 + 42*
d*e*x + 16*e^2*x^2)) + 5*b^6*Sqrt[e]*(-3*Sqrt[e]*(b*d - a*e)^(11/2)*(a + b*x)*Sqrt[(b*(d + e*x))/(b*d - a*e)]
+ 2*e^(3/2)*(b*d - a*e)^(9/2)*(a + b*x)^2*Sqrt[(b*(d + e*x))/(b*d - a*e)] + 3*(b*d - a*e)^6*Sqrt[a + b*x]*ArcS
inh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]])))/(128*b^6*e^3*(b*d - a*e)^(3/2)*(a + b*x)^3*Sqrt[(b*(d + e*x))/
(b*d - a*e)])))/(30*b^4*e)

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fricas [B]  time = 1.21, size = 1384, normalized size = 3.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

[1/30720*(15*(5*B*b^6*d^6 - 6*(3*B*a*b^5 + 2*A*b^6)*d^5*e + 15*(B*a^2*b^4 + 4*A*a*b^5)*d^4*e^2 + 20*(B*a^3*b^3
- 6*A*a^2*b^4)*d^3*e^3 - 15*(3*B*a^4*b^2 - 8*A*a^3*b^3)*d^2*e^4 + 30*(B*a^5*b - 2*A*a^4*b^2)*d*e^5 - (7*B*a^6
- 12*A*a^5*b)*e^6)*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*(1280*B*b^6*e^6*x^5 + 75*B*b^6*d^5*e - 5*(49*
B*a*b^5 + 36*A*b^6)*d^4*e^2 + 30*(5*B*a^2*b^4 + 28*A*a*b^5)*d^3*e^3 - 6*(91*B*a^3*b^3 - 256*A*a^2*b^4)*d^2*e^4
+ 5*(83*B*a^4*b^2 - 168*A*a^3*b^3)*d*e^5 - 15*(7*B*a^5*b - 12*A*a^4*b^2)*e^6 + 128*(25*B*b^6*d*e^5 + (13*B*a*
b^5 + 12*A*b^6)*e^6)*x^4 + 16*(135*B*b^6*d^2*e^4 + 2*(139*B*a*b^5 + 126*A*b^6)*d*e^5 + 3*(B*a^2*b^4 + 44*A*a*b
^5)*e^6)*x^3 + 8*(5*B*b^6*d^3*e^3 + 3*(141*B*a*b^5 + 124*A*b^6)*d^2*e^4 + 3*(9*B*a^2*b^4 + 256*A*a*b^5)*d*e^5
- (7*B*a^3*b^3 - 12*A*a^2*b^4)*e^6)*x^2 - 2*(25*B*b^6*d^4*e^2 - 20*(4*B*a*b^5 + 3*A*b^6)*d^3*e^3 - 6*(29*B*a^2
*b^4 + 466*A*a*b^5)*d^2*e^4 + 4*(34*B*a^3*b^3 - 69*A*a^2*b^4)*d*e^5 - 5*(7*B*a^4*b^2 - 12*A*a^3*b^3)*e^6)*x)*s
qrt(b*x + a)*sqrt(e*x + d))/(b^5*e^4), 1/15360*(15*(5*B*b^6*d^6 - 6*(3*B*a*b^5 + 2*A*b^6)*d^5*e + 15*(B*a^2*b^
4 + 4*A*a*b^5)*d^4*e^2 + 20*(B*a^3*b^3 - 6*A*a^2*b^4)*d^3*e^3 - 15*(3*B*a^4*b^2 - 8*A*a^3*b^3)*d^2*e^4 + 30*(B
*a^5*b - 2*A*a^4*b^2)*d*e^5 - (7*B*a^6 - 12*A*a^5*b)*e^6)*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*(1280*B*b^6*e^6*x^5 + 75*B
*b^6*d^5*e - 5*(49*B*a*b^5 + 36*A*b^6)*d^4*e^2 + 30*(5*B*a^2*b^4 + 28*A*a*b^5)*d^3*e^3 - 6*(91*B*a^3*b^3 - 256
*A*a^2*b^4)*d^2*e^4 + 5*(83*B*a^4*b^2 - 168*A*a^3*b^3)*d*e^5 - 15*(7*B*a^5*b - 12*A*a^4*b^2)*e^6 + 128*(25*B*b
^6*d*e^5 + (13*B*a*b^5 + 12*A*b^6)*e^6)*x^4 + 16*(135*B*b^6*d^2*e^4 + 2*(139*B*a*b^5 + 126*A*b^6)*d*e^5 + 3*(B
*a^2*b^4 + 44*A*a*b^5)*e^6)*x^3 + 8*(5*B*b^6*d^3*e^3 + 3*(141*B*a*b^5 + 124*A*b^6)*d^2*e^4 + 3*(9*B*a^2*b^4 +
256*A*a*b^5)*d*e^5 - (7*B*a^3*b^3 - 12*A*a^2*b^4)*e^6)*x^2 - 2*(25*B*b^6*d^4*e^2 - 20*(4*B*a*b^5 + 3*A*b^6)*d^
3*e^3 - 6*(29*B*a^2*b^4 + 466*A*a*b^5)*d^2*e^4 + 4*(34*B*a^3*b^3 - 69*A*a^2*b^4)*d*e^5 - 5*(7*B*a^4*b^2 - 12*A
*a^3*b^3)*e^6)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^5*e^4)]

________________________________________________________________________________________

giac [B]  time = 5.73, size = 4427, normalized size = 12.37 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(5/2),x, algorithm="giac")

[Out]

1/7680*(320*(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^2*abs(b) + 40*(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^2*abs(b) - 7680*((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^2*abs(b)/b^2 + 640*(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(-sqr
t(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^2*abs(b)/b + 80*(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*d*abs(b)*e + 8*(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*d*abs(b)*e + 640*(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*d*abs(b)*e/b^2 + 1280*(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*d*abs(b)*e/b + 160*(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*d*abs(b)*e/b + 1920*((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^2*abs(b)/b^3 + 3840*((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^2*abs(b)/
b^2 + 4*(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) - 1
5*(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))*A*abs(b)*e^2 + (sqrt(b
^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*(10*(b*x + a)/b^5 + (b^30*d*e^9 - 61*a*b^29*e^10
)*e^(-10)/b^34) - 3*(3*b^31*d^2*e^8 + 14*a*b^30*d*e^9 - 417*a^2*b^29*e^10)*e^(-10)/b^34) + (21*b^32*d^3*e^7 +
77*a*b^31*d^2*e^8 + 183*a^2*b^30*d*e^9 - 3481*a^3*b^29*e^10)*e^(-10)/b^34)*(b*x + a) - 5*(21*b^33*d^4*e^6 + 56
*a*b^32*d^3*e^7 + 106*a^2*b^31*d^2*e^8 + 176*a^3*b^30*d*e^9 - 2279*a^4*b^29*e^10)*e^(-10)/b^34)*(b*x + a) + 15
*(21*b^34*d^5*e^5 + 35*a*b^33*d^4*e^6 + 50*a^2*b^32*d^3*e^7 + 70*a^3*b^31*d^2*e^8 + 105*a^4*b^30*d*e^9 - 793*a
^5*b^29*e^10)*e^(-10)/b^34)*sqrt(b*x + a) + 15*(21*b^6*d^6 + 14*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 + 20*a^3*b^3*
d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 - 231*a^6*e^6)*e^(-11/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2)
+ sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(9/2))*B*abs(b)*e^2 + 320*(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(-sq
rt(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^2*abs(b)*e^2/b^2 + 40*(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^2*abs(b)*e^2/b^2 + 80*(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 - 16
3*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*abs(b)*e^2/b
+ 8*(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*a*abs(b)*e^2/b + 3840*((
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*d*abs(b)*e/b^3)/b

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maple [B]  time = 0.03, size = 2198, normalized size = 6.14 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(5/2),x)

[Out]

-1/15360*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(1800*A*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))*a^3*b^3*d^2*e^4+180*A*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))*a^5*b*e^6-105*B*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))*a^6*e^6+75*B*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))*b^6*d^6-
180*A*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))*b^6*d^5*e+210*B*(b*e
*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*a^5*e^5-150*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*b^5*d^5-4320
*B*x^3*b^5*d^2*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-192*A*x^2*a^2*b^3*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)
^(1/2)*(b*e)^(1/2)-5952*A*x^2*b^5*d^2*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^3*b^2*e^5*(b
*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+240*A*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*x*a^3*b^2*e^5-240*
A*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*x*b^5*d^3*e^2-830*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*
a^4*b*d*e^4+1092*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*a^3*b^2*d^2*e^3-300*B*(b*e*x^2+a*e*x+b*d*x+a*d)
^(1/2)*(b*e)^(1/2)*a^2*b^3*d^3*e^2+490*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*a*b^4*d^4*e-900*A*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))*a^4*b^2*d*e^5+450*B*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))*a^5*b*d*e^5-675*B*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))*a^4*b^2*d^2*e^4+300*B*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))*a^3*b^3*d^3*e^3+225*B*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))*a^2*b^4*d^4*e^2-1800*A*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))*a^2*b^4*d^3*e^3+900*A*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))*a*b^5*d^4*e^2-270*B*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))*a*b^5*d^5*e-360*A*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*a^4*b*
e^5+360*A*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*b^5*d^4*e-2560*B*x^5*b^5*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(
1/2)*(b*e)^(1/2)-3072*A*x^4*b^5*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+100*B*(b*e*x^2+a*e*x+b*d*x+a*d
)^(1/2)*(b*e)^(1/2)*x*b^5*d^4*e+1680*A*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*a^3*b^2*d*e^4-1680*A*(b*e*x
^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*a*b^4*d^3*e^2-4224*A*x^3*a*b^4*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)
^(1/2)-8064*A*x^3*b^5*d*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-96*B*x^3*a^2*b^3*e^5*(b*e*x^2+a*e*x+b*
d*x+a*d)^(1/2)*(b*e)^(1/2)-432*B*x^2*a^2*b^3*d*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-6768*B*x^2*a*b^
4*d^2*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-8896*B*x^3*a*b^4*d*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(
b*e)^(1/2)-12288*A*x^2*a*b^4*d*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-696*B*(b*e*x^2+a*e*x+b*d*x+a*d)
^(1/2)*(b*e)^(1/2)*x*a^2*b^3*d^2*e^3-320*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*x*a*b^4*d^3*e^2-11184*A
*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*x*a*b^4*d^2*e^3+544*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)
*x*a^3*b^2*d*e^4-1104*A*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*x*a^2*b^3*d*e^4-80*B*x^2*b^5*d^3*e^2*(b*e*
x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-3072*A*a^2*b^3*d^2*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-3328
*B*x^4*a*b^4*e^5*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-6400*B*x^4*b^5*d*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1
/2)*(b*e)^(1/2)-140*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*x*a^4*b*e^5)/b^4/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 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(5/2),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 \[ \int \left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^(3/2)*(d + e*x)^(5/2),x)

[Out]

int((A + B*x)*(a + b*x)^(3/2)*(d + e*x)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(B*x+A)*(e*x+d)**(5/2),x)

[Out]

Timed out

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