3.23.11 \(\int (a+b x)^{3/2} (A+B x) (d+e x)^{5/2} \, dx\) [2211]

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

[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 = 0.20, 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 = {81, 52, 65, 223, 212} \begin {gather*} -\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} \end {gather*}

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 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int (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 ) \text {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 ) \text {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 = 1.15, size = 432, normalized size = 1.21 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {d+e x} \left (-105 a^5 B e^5+5 a^4 b e^4 (83 B d+36 A e+14 B e x)-2 a^3 b^2 e^3 \left (60 A e (7 d+e x)+B \left (273 d^2+136 d e x+28 e^2 x^2\right )\right )+6 a^2 b^3 e^2 \left (4 A e \left (64 d^2+23 d e x+4 e^2 x^2\right )+B \left (25 d^3+58 d^2 e x+36 d e^2 x^2+8 e^3 x^3\right )\right )+a b^4 e \left (24 A e \left (35 d^3+233 d^2 e x+256 d e^2 x^2+88 e^3 x^3\right )+B \left (-245 d^4+160 d^3 e x+3384 d^2 e^2 x^2+4448 d e^3 x^3+1664 e^4 x^4\right )\right )+b^5 \left (12 A e \left (-15 d^4+10 d^3 e x+248 d^2 e^2 x^2+336 d e^3 x^3+128 e^4 x^4\right )+5 B \left (15 d^5-10 d^4 e x+8 d^3 e^2 x^2+432 d^2 e^3 x^3+640 d e^4 x^4+256 e^5 x^5\right )\right )\right )}{7680 b^4 e^3}+\frac {(b d-a e)^5 (-5 b B d+12 A b e-7 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{512 b^{9/2} e^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-105*a^5*B*e^5 + 5*a^4*b*e^4*(83*B*d + 36*A*e + 14*B*e*x) - 2*a^3*b^2*e^3*(60*A*
e*(7*d + e*x) + B*(273*d^2 + 136*d*e*x + 28*e^2*x^2)) + 6*a^2*b^3*e^2*(4*A*e*(64*d^2 + 23*d*e*x + 4*e^2*x^2) +
B*(25*d^3 + 58*d^2*e*x + 36*d*e^2*x^2 + 8*e^3*x^3)) + a*b^4*e*(24*A*e*(35*d^3 + 233*d^2*e*x + 256*d*e^2*x^2 +
88*e^3*x^3) + B*(-245*d^4 + 160*d^3*e*x + 3384*d^2*e^2*x^2 + 4448*d*e^3*x^3 + 1664*e^4*x^4)) + b^5*(12*A*e*(-
15*d^4 + 10*d^3*e*x + 248*d^2*e^2*x^2 + 336*d*e^3*x^3 + 128*e^4*x^4) + 5*B*(15*d^5 - 10*d^4*e*x + 8*d^3*e^2*x^
2 + 432*d^2*e^3*x^3 + 640*d*e^4*x^4 + 256*e^5*x^5))))/(7680*b^4*e^3) + ((b*d - a*e)^5*(-5*b*B*d + 12*A*b*e - 7
*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/(512*b^(9/2)*e^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1847\) vs. \(2(308)=616\).
time = 0.09, size = 1848, normalized size = 5.16

method	result	size
default	\(\text {Expression too large to display}\)	\(1848\)

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,method=_RETURNVERBOSE)

[Out]

-1/15360*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-2560*B*b^5*e^5*x^5*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-3072*A*b^5*e^5*x
^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-900*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b
*e)^(1/2))*a^4*b^2*d*e^5+1800*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^
3*b^3*d^2*e^4+300*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^3*d^3*e^
3-1800*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^4*d^3*e^3+900*A*ln(
1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^5*d^4*e^2+450*B*ln(1/2*(2*b*e*x+2
*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^5*b*d*e^5-675*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d
))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*b^2*d^2*e^4-360*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^4*b*e^5
+360*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^5*d^4*e+225*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/
2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^4*d^4*e^2-270*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/
(b*e)^(1/2))*a*b^5*d^5*e-105*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^6
*e^6+75*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^6*d^6-300*B*(b*e)^(1/2
)*((b*x+a)*(e*x+d))^(1/2)*a^2*b^3*d^3*e^2-432*B*a^2*b^3*d*e^4*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-6768*B*a
*b^4*d^2*e^3*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-830*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^4*b*d*e^4+109
2*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^3*b^2*d^2*e^3-3328*B*a*b^4*e^5*x^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/
2)-6400*B*b^5*d*e^4*x^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-4224*A*a*b^4*e^5*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)
^(1/2)-8064*A*b^5*d*e^4*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-96*B*a^2*b^3*e^5*x^3*((b*x+a)*(e*x+d))^(1/2)*(
b*e)^(1/2)-4320*B*b^5*d^2*e^3*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-192*A*a^2*b^3*e^5*x^2*((b*x+a)*(e*x+d))^
(1/2)*(b*e)^(1/2)-5952*A*b^5*d^2*e^3*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+112*B*a^3*b^2*e^5*x^2*((b*x+a)*(e
*x+d))^(1/2)*(b*e)^(1/2)-80*B*b^5*d^3*e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-3072*A*a^2*b^3*d^2*e^3*((b*x
+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-8896*B*a*b^4*d*e^4*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-12288*A*a*b^4*d*e^4*
x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-11184*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^4*d^2*e^3*x-696*B*(b*e
)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b^3*d^2*e^3*x-320*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^4*d^3*e^2*x+54
4*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^3*b^2*d*e^4*x-1104*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^2*b^3*d*e
^4*x-240*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^5*d^3*e^2*x+240*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^3*b^2
*e^5*x+1680*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^3*b^2*d*e^4-1680*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b
^4*d^3*e^2+490*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*b^4*d^4*e-140*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^4
*b*e^5*x+100*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b^5*d^4*e*x+180*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)
*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^5*b*e^6-180*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b
*d)/(b*e)^(1/2))*b^6*d^5*e+210*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a^5*e^5-150*B*(b*e)^(1/2)*((b*x+a)*(e*x+d
))^(1/2)*b^5*d^5)/b^4/e^3/((b*x+a)*(e*x+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)^(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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 655 vs. \(2 (324) = 648\).
time = 1.35, size = 1320, normalized size = 3.69 \begin {gather*} \left [\frac {{\left (15 \, {\left (5 \, B b^{6} d^{6} - 6 \, {\left (3 \, B a b^{5} + 2 \, A b^{6}\right )} d^{5} e + 15 \, {\left (B a^{2} b^{4} + 4 \, A a b^{5}\right )} d^{4} e^{2} + 20 \, {\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} d^{3} e^{3} - 15 \, {\left (3 \, B a^{4} b^{2} - 8 \, A a^{3} b^{3}\right )} d^{2} e^{4} + 30 \, {\left (B a^{5} b - 2 \, A a^{4} b^{2}\right )} d e^{5} - {\left (7 \, B a^{6} - 12 \, A a^{5} b\right )} e^{6}\right )} \sqrt {b} e^{\frac {1}{2}} \log \left (b^{2} d^{2} - 4 \, {\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {b} e^{\frac {1}{2}} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right ) + 4 \, {\left (75 \, B b^{6} d^{5} e + {\left (1280 \, B b^{6} x^{5} - 105 \, B a^{5} b + 180 \, A a^{4} b^{2} + 128 \, {\left (13 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 48 \, {\left (B a^{2} b^{4} + 44 \, A a b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} e^{6} + {\left (3200 \, B b^{6} d x^{4} + 32 \, {\left (139 \, B a b^{5} + 126 \, A b^{6}\right )} d x^{3} + 24 \, {\left (9 \, B a^{2} b^{4} + 256 \, A a b^{5}\right )} d x^{2} - 8 \, {\left (34 \, B a^{3} b^{3} - 69 \, A a^{2} b^{4}\right )} d x + 5 \, {\left (83 \, B a^{4} b^{2} - 168 \, A a^{3} b^{3}\right )} d\right )} e^{5} + 6 \, {\left (360 \, B b^{6} d^{2} x^{3} + 4 \, {\left (141 \, B a b^{5} + 124 \, A b^{6}\right )} d^{2} x^{2} + 2 \, {\left (29 \, B a^{2} b^{4} + 466 \, A a b^{5}\right )} d^{2} x - {\left (91 \, B a^{3} b^{3} - 256 \, A a^{2} b^{4}\right )} d^{2}\right )} e^{4} + 10 \, {\left (4 \, B b^{6} d^{3} x^{2} + 4 \, {\left (4 \, B a b^{5} + 3 \, A b^{6}\right )} d^{3} x + 3 \, {\left (5 \, B a^{2} b^{4} + 28 \, A a b^{5}\right )} d^{3}\right )} e^{3} - 5 \, {\left (10 \, B b^{6} d^{4} x + {\left (49 \, B a b^{5} + 36 \, A b^{6}\right )} d^{4}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-4\right )}}{30720 \, b^{5}}, \frac {{\left (15 \, {\left (5 \, B b^{6} d^{6} - 6 \, {\left (3 \, B a b^{5} + 2 \, A b^{6}\right )} d^{5} e + 15 \, {\left (B a^{2} b^{4} + 4 \, A a b^{5}\right )} d^{4} e^{2} + 20 \, {\left (B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} d^{3} e^{3} - 15 \, {\left (3 \, B a^{4} b^{2} - 8 \, A a^{3} b^{3}\right )} d^{2} e^{4} + 30 \, {\left (B a^{5} b - 2 \, A a^{4} b^{2}\right )} d e^{5} - {\left (7 \, B a^{6} - 12 \, A a^{5} b\right )} e^{6}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {-b e} \sqrt {x e + d}}{2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} e^{2} + {\left (b^{2} d x + a b d\right )} e\right )}}\right ) + 2 \, {\left (75 \, B b^{6} d^{5} e + {\left (1280 \, B b^{6} x^{5} - 105 \, B a^{5} b + 180 \, A a^{4} b^{2} + 128 \, {\left (13 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 48 \, {\left (B a^{2} b^{4} + 44 \, A a b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} e^{6} + {\left (3200 \, B b^{6} d x^{4} + 32 \, {\left (139 \, B a b^{5} + 126 \, A b^{6}\right )} d x^{3} + 24 \, {\left (9 \, B a^{2} b^{4} + 256 \, A a b^{5}\right )} d x^{2} - 8 \, {\left (34 \, B a^{3} b^{3} - 69 \, A a^{2} b^{4}\right )} d x + 5 \, {\left (83 \, B a^{4} b^{2} - 168 \, A a^{3} b^{3}\right )} d\right )} e^{5} + 6 \, {\left (360 \, B b^{6} d^{2} x^{3} + 4 \, {\left (141 \, B a b^{5} + 124 \, A b^{6}\right )} d^{2} x^{2} + 2 \, {\left (29 \, B a^{2} b^{4} + 466 \, A a b^{5}\right )} d^{2} x - {\left (91 \, B a^{3} b^{3} - 256 \, A a^{2} b^{4}\right )} d^{2}\right )} e^{4} + 10 \, {\left (4 \, B b^{6} d^{3} x^{2} + 4 \, {\left (4 \, B a b^{5} + 3 \, A b^{6}\right )} d^{3} x + 3 \, {\left (5 \, B a^{2} b^{4} + 28 \, A a b^{5}\right )} d^{3}\right )} e^{3} - 5 \, {\left (10 \, B b^{6} d^{4} x + {\left (49 \, B a b^{5} + 36 \, A b^{6}\right )} d^{4}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-4\right )}}{15360 \, b^{5}}\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)^(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^(1/2)*log(b^2*d^2 - 4*(b*d + (2*b*x + a)*e)*sqrt(b*x + a)*sqrt(x*e + d)*sqrt(b)*
e^(1/2) + (8*b^2*x^2 + 8*a*b*x + a^2)*e^2 + 2*(4*b^2*d*x + 3*a*b*d)*e) + 4*(75*B*b^6*d^5*e + (1280*B*b^6*x^5 -
105*B*a^5*b + 180*A*a^4*b^2 + 128*(13*B*a*b^5 + 12*A*b^6)*x^4 + 48*(B*a^2*b^4 + 44*A*a*b^5)*x^3 - 8*(7*B*a^3*
b^3 - 12*A*a^2*b^4)*x^2 + 10*(7*B*a^4*b^2 - 12*A*a^3*b^3)*x)*e^6 + (3200*B*b^6*d*x^4 + 32*(139*B*a*b^5 + 126*A
*b^6)*d*x^3 + 24*(9*B*a^2*b^4 + 256*A*a*b^5)*d*x^2 - 8*(34*B*a^3*b^3 - 69*A*a^2*b^4)*d*x + 5*(83*B*a^4*b^2 - 1
68*A*a^3*b^3)*d)*e^5 + 6*(360*B*b^6*d^2*x^3 + 4*(141*B*a*b^5 + 124*A*b^6)*d^2*x^2 + 2*(29*B*a^2*b^4 + 466*A*a*
b^5)*d^2*x - (91*B*a^3*b^3 - 256*A*a^2*b^4)*d^2)*e^4 + 10*(4*B*b^6*d^3*x^2 + 4*(4*B*a*b^5 + 3*A*b^6)*d^3*x + 3
*(5*B*a^2*b^4 + 28*A*a*b^5)*d^3)*e^3 - 5*(10*B*b^6*d^4*x + (49*B*a*b^5 + 36*A*b^6)*d^4)*e^2)*sqrt(b*x + a)*sqr
t(x*e + d))*e^(-4)/b^5, 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*(b*d + (2*b*x + a)*e)*sqrt(b*x + a)*sqrt(-b*e
)*sqrt(x*e + d)/((b^2*x^2 + a*b*x)*e^2 + (b^2*d*x + a*b*d)*e)) + 2*(75*B*b^6*d^5*e + (1280*B*b^6*x^5 - 105*B*a
^5*b + 180*A*a^4*b^2 + 128*(13*B*a*b^5 + 12*A*b^6)*x^4 + 48*(B*a^2*b^4 + 44*A*a*b^5)*x^3 - 8*(7*B*a^3*b^3 - 12
*A*a^2*b^4)*x^2 + 10*(7*B*a^4*b^2 - 12*A*a^3*b^3)*x)*e^6 + (3200*B*b^6*d*x^4 + 32*(139*B*a*b^5 + 126*A*b^6)*d*
x^3 + 24*(9*B*a^2*b^4 + 256*A*a*b^5)*d*x^2 - 8*(34*B*a^3*b^3 - 69*A*a^2*b^4)*d*x + 5*(83*B*a^4*b^2 - 168*A*a^3
*b^3)*d)*e^5 + 6*(360*B*b^6*d^2*x^3 + 4*(141*B*a*b^5 + 124*A*b^6)*d^2*x^2 + 2*(29*B*a^2*b^4 + 466*A*a*b^5)*d^2
*x - (91*B*a^3*b^3 - 256*A*a^2*b^4)*d^2)*e^4 + 10*(4*B*b^6*d^3*x^2 + 4*(4*B*a*b^5 + 3*A*b^6)*d^3*x + 3*(5*B*a^
2*b^4 + 28*A*a*b^5)*d^3)*e^3 - 5*(10*B*b^6*d^4*x + (49*B*a*b^5 + 36*A*b^6)*d^4)*e^2)*sqrt(b*x + a)*sqrt(x*e +
d))*e^(-4)/b^5]

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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 {5}{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)**(5/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4427 vs. \(2 (324) = 648\).
time = 2.19, size = 4427, normalized size = 12.37 \begin {gather*} \text {Too large to display} \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)^(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...

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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 )}^{5/2} \,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)^(5/2),x)

[Out]

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

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