3.2224 \(\int (a+b x)^{5/2} (A+B x) (d+e x)^{3/2} \, dx\)

Optimal. Leaf size=358 \[ -\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^4 (5 a B e-12 A b e+7 b B d)}{512 b^3 e^4}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^3 (5 a B e-12 A b e+7 b B d)}{768 b^3 e^3}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e)^2 (5 a B e-12 A b e+7 b B d)}{960 b^3 e^2}+\frac{(b d-a e)^5 (5 a B e-12 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{512 b^{7/2} e^{9/2}}-\frac{(a+b x)^{7/2} \sqrt{d+e x} (b d-a e) (5 a B e-12 A b e+7 b B d)}{160 b^3 e}-\frac{(a+b x)^{7/2} (d+e x)^{3/2} (5 a B e-12 A b e+7 b B d)}{60 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e} \]

[Out]

-((b*d - a*e)^4*(7*b*B*d - 12*A*b*e + 5*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(512*b^3*e^4) + ((b*d - a*e)^3*(7*
b*B*d - 12*A*b*e + 5*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(768*b^3*e^3) - ((b*d - a*e)^2*(7*b*B*d - 12*A*b*e
+ 5*a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(960*b^3*e^2) - ((b*d - a*e)*(7*b*B*d - 12*A*b*e + 5*a*B*e)*(a + b*x
)^(7/2)*Sqrt[d + e*x])/(160*b^3*e) - ((7*b*B*d - 12*A*b*e + 5*a*B*e)*(a + b*x)^(7/2)*(d + e*x)^(3/2))/(60*b^2*
e) + (B*(a + b*x)^(7/2)*(d + e*x)^(5/2))/(6*b*e) + ((b*d - a*e)^5*(7*b*B*d - 12*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt
[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(512*b^(7/2)*e^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.295275, antiderivative size = 358, normalized size of antiderivative = 1., 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} \[ -\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^4 (5 a B e-12 A b e+7 b B d)}{512 b^3 e^4}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^3 (5 a B e-12 A b e+7 b B d)}{768 b^3 e^3}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e)^2 (5 a B e-12 A b e+7 b B d)}{960 b^3 e^2}+\frac{(b d-a e)^5 (5 a B e-12 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{512 b^{7/2} e^{9/2}}-\frac{(a+b x)^{7/2} \sqrt{d+e x} (b d-a e) (5 a B e-12 A b e+7 b B d)}{160 b^3 e}-\frac{(a+b x)^{7/2} (d+e x)^{3/2} (5 a B e-12 A b e+7 b B d)}{60 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*d - a*e)^4*(7*b*B*d - 12*A*b*e + 5*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(512*b^3*e^4) + ((b*d - a*e)^3*(7*
b*B*d - 12*A*b*e + 5*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(768*b^3*e^3) - ((b*d - a*e)^2*(7*b*B*d - 12*A*b*e
+ 5*a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(960*b^3*e^2) - ((b*d - a*e)*(7*b*B*d - 12*A*b*e + 5*a*B*e)*(a + b*x
)^(7/2)*Sqrt[d + e*x])/(160*b^3*e) - ((7*b*B*d - 12*A*b*e + 5*a*B*e)*(a + b*x)^(7/2)*(d + e*x)^(3/2))/(60*b^2*
e) + (B*(a + b*x)^(7/2)*(d + e*x)^(5/2))/(6*b*e) + ((b*d - a*e)^5*(7*b*B*d - 12*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt
[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(512*b^(7/2)*e^(9/2))

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 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 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]

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])

Rubi steps

\begin{align*} \int (a+b x)^{5/2} (A+B x) (d+e x)^{3/2} \, dx &=\frac{B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}+\frac{\left (6 A b e-B \left (\frac{7 b d}{2}+\frac{5 a e}{2}\right )\right ) \int (a+b x)^{5/2} (d+e x)^{3/2} \, dx}{6 b e}\\ &=-\frac{(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac{((b d-a e) (7 b B d-12 A b e+5 a B e)) \int (a+b x)^{5/2} \sqrt{d+e x} \, dx}{40 b^2 e}\\ &=-\frac{(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{160 b^3 e}-\frac{(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac{\left ((b d-a e)^2 (7 b B d-12 A b e+5 a B e)\right ) \int \frac{(a+b x)^{5/2}}{\sqrt{d+e x}} \, dx}{320 b^3 e}\\ &=-\frac{(b d-a e)^2 (7 b B d-12 A b e+5 a B e) (a+b x)^{5/2} \sqrt{d+e x}}{960 b^3 e^2}-\frac{(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{160 b^3 e}-\frac{(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}+\frac{\left ((b d-a e)^3 (7 b B d-12 A b e+5 a B e)\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{d+e x}} \, dx}{384 b^3 e^2}\\ &=\frac{(b d-a e)^3 (7 b B d-12 A b e+5 a B e) (a+b x)^{3/2} \sqrt{d+e x}}{768 b^3 e^3}-\frac{(b d-a e)^2 (7 b B d-12 A b e+5 a B e) (a+b x)^{5/2} \sqrt{d+e x}}{960 b^3 e^2}-\frac{(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{160 b^3 e}-\frac{(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac{\left ((b d-a e)^4 (7 b B d-12 A b e+5 a B e)\right ) \int \frac{\sqrt{a+b x}}{\sqrt{d+e x}} \, dx}{512 b^3 e^3}\\ &=-\frac{(b d-a e)^4 (7 b B d-12 A b e+5 a B e) \sqrt{a+b x} \sqrt{d+e x}}{512 b^3 e^4}+\frac{(b d-a e)^3 (7 b B d-12 A b e+5 a B e) (a+b x)^{3/2} \sqrt{d+e x}}{768 b^3 e^3}-\frac{(b d-a e)^2 (7 b B d-12 A b e+5 a B e) (a+b x)^{5/2} \sqrt{d+e x}}{960 b^3 e^2}-\frac{(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{160 b^3 e}-\frac{(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}+\frac{\left ((b d-a e)^5 (7 b B d-12 A b e+5 a B e)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{1024 b^3 e^4}\\ &=-\frac{(b d-a e)^4 (7 b B d-12 A b e+5 a B e) \sqrt{a+b x} \sqrt{d+e x}}{512 b^3 e^4}+\frac{(b d-a e)^3 (7 b B d-12 A b e+5 a B e) (a+b x)^{3/2} \sqrt{d+e x}}{768 b^3 e^3}-\frac{(b d-a e)^2 (7 b B d-12 A b e+5 a B e) (a+b x)^{5/2} \sqrt{d+e x}}{960 b^3 e^2}-\frac{(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{160 b^3 e}-\frac{(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}+\frac{\left ((b d-a e)^5 (7 b B d-12 A b e+5 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^4 e^4}\\ &=-\frac{(b d-a e)^4 (7 b B d-12 A b e+5 a B e) \sqrt{a+b x} \sqrt{d+e x}}{512 b^3 e^4}+\frac{(b d-a e)^3 (7 b B d-12 A b e+5 a B e) (a+b x)^{3/2} \sqrt{d+e x}}{768 b^3 e^3}-\frac{(b d-a e)^2 (7 b B d-12 A b e+5 a B e) (a+b x)^{5/2} \sqrt{d+e x}}{960 b^3 e^2}-\frac{(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{160 b^3 e}-\frac{(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}+\frac{\left ((b d-a e)^5 (7 b B d-12 A b e+5 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^4 e^4}\\ &=-\frac{(b d-a e)^4 (7 b B d-12 A b e+5 a B e) \sqrt{a+b x} \sqrt{d+e x}}{512 b^3 e^4}+\frac{(b d-a e)^3 (7 b B d-12 A b e+5 a B e) (a+b x)^{3/2} \sqrt{d+e x}}{768 b^3 e^3}-\frac{(b d-a e)^2 (7 b B d-12 A b e+5 a B e) (a+b x)^{5/2} \sqrt{d+e x}}{960 b^3 e^2}-\frac{(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{160 b^3 e}-\frac{(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}+\frac{(b d-a e)^5 (7 b B d-12 A b e+5 a B e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{512 b^{7/2} e^{9/2}}\\ \end{align*}

Mathematica [A]  time = 4.00545, size = 360, normalized size = 1.01 \[ \frac{\sqrt{b d-a e} \left (\frac{b (d+e x)}{b d-a e}\right )^{3/2} \left (-\frac{5 a B e}{2}+6 A b e-\frac{7}{2} b B d\right ) \left (-10 e^{3/2} (a+b x)^2 (b d-a e)^{9/2} \sqrt{\frac{b (d+e x)}{b d-a e}}+8 e^{5/2} (a+b x)^3 (b d-a e)^{7/2} \sqrt{\frac{b (d+e x)}{b d-a e}}+16 e^{7/2} (a+b x)^4 (b d-a e)^{3/2} \sqrt{\frac{b (d+e x)}{b d-a e}} (-3 a e+11 b d+8 b e x)+15 \sqrt{e} (a+b x) (b d-a e)^{11/2} \sqrt{\frac{b (d+e x)}{b d-a e}}-15 \sqrt{a+b x} (b d-a e)^6 \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )\right )+640 b^4 B e^{7/2} (a+b x)^4 (d+e x)^4}{3840 b^5 e^{9/2} \sqrt{a+b x} (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(640*b^4*B*e^(7/2)*(a + b*x)^4*(d + e*x)^4 + Sqrt[b*d - a*e]*((-7*b*B*d)/2 + 6*A*b*e - (5*a*B*e)/2)*((b*(d + e
*x))/(b*d - a*e))^(3/2)*(15*Sqrt[e]*(b*d - a*e)^(11/2)*(a + b*x)*Sqrt[(b*(d + e*x))/(b*d - a*e)] - 10*e^(3/2)*
(b*d - a*e)^(9/2)*(a + b*x)^2*Sqrt[(b*(d + e*x))/(b*d - a*e)] + 8*e^(5/2)*(b*d - a*e)^(7/2)*(a + b*x)^3*Sqrt[(
b*(d + e*x))/(b*d - a*e)] + 16*e^(7/2)*(b*d - a*e)^(3/2)*(a + b*x)^4*Sqrt[(b*(d + e*x))/(b*d - a*e)]*(11*b*d -
3*a*e + 8*b*e*x) - 15*(b*d - a*e)^6*Sqrt[a + b*x]*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]]))/(3840*b^
5*e^(9/2)*Sqrt[a + b*x]*(d + e*x)^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.019, size = 2198, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15360*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(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)-490*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+6400*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)+96*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)+5952*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)+192*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)+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
^2-100*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+140*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^2-210*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*b^5*d^5-75*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^6*e^6+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))*b^6*d^6+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^3-544*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+320*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+11184*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+1104*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+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)+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)+6768*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)+432*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)+80*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)-112*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)+300*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^3-1092*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+830*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+3328*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)+8064*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)+4224*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)+4320*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)+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)+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+1800*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^3*d^2*e^4-1800*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^4*d^3*e^3+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
*b^5*d^4*e^2+270*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*b*d
*e^5-225*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^2*d^2*e^4
-300*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^3*d^3*e^3+675
*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^4*d^4*e^2-450*B*l
n(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^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^5-900*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^4*b^2*d*e^5-180*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^6*d^5*e+150*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*a^5*e^5+180*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^5*b*e^6)/b^3/e^4/(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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.68871, size = 3062, normalized size = 8.55 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/30720*(15*(7*B*b^6*d^6 - 6*(5*B*a*b^5 + 2*A*b^6)*d^5*e + 15*(3*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*(B*a^4*b^2 - 8*A*a^3*b^3)*d^2*e^4 + 6*(3*B*a^5*b - 10*A*a^4*b^2)*d*e^5 - (5*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)*sq
rt(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 - 105*B*b^6*d^5*e + 5*(
83*B*a*b^5 + 36*A*b^6)*d^4*e^2 - 42*(13*B*a^2*b^4 + 20*A*a*b^5)*d^3*e^3 + 6*(25*B*a^3*b^3 + 256*A*a^2*b^4)*d^2
*e^4 - 35*(7*B*a^4*b^2 - 24*A*a^3*b^3)*d*e^5 + 15*(5*B*a^5*b - 12*A*a^4*b^2)*e^6 + 128*(13*B*b^6*d*e^5 + (25*B
*a*b^5 + 12*A*b^6)*e^6)*x^4 + 16*(3*B*b^6*d^2*e^4 + 2*(139*B*a*b^5 + 66*A*b^6)*d*e^5 + 9*(15*B*a^2*b^4 + 28*A*
a*b^5)*e^6)*x^3 - 8*(7*B*b^6*d^3*e^3 - 3*(9*B*a*b^5 + 4*A*b^6)*d^2*e^4 - 3*(141*B*a^2*b^4 + 256*A*a*b^5)*d*e^5
- (5*B*a^3*b^3 + 372*A*a^2*b^4)*e^6)*x^2 + 2*(35*B*b^6*d^4*e^2 - 4*(34*B*a*b^5 + 15*A*b^6)*d^3*e^3 + 6*(29*B*
a^2*b^4 + 46*A*a*b^5)*d^2*e^4 + 4*(20*B*a^3*b^3 + 699*A*a^2*b^4)*d*e^5 - 5*(5*B*a^4*b^2 - 12*A*a^3*b^3)*e^6)*x
)*sqrt(b*x + a)*sqrt(e*x + d))/(b^4*e^5), -1/15360*(15*(7*B*b^6*d^6 - 6*(5*B*a*b^5 + 2*A*b^6)*d^5*e + 15*(3*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*(B*a^4*b^2 - 8*A*a^3*b^3)*d^2*e^4 + 6
*(3*B*a^5*b - 10*A*a^4*b^2)*d*e^5 - (5*B*a^6 - 12*A*a^5*b)*e^6)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sq
rt(-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
- 105*B*b^6*d^5*e + 5*(83*B*a*b^5 + 36*A*b^6)*d^4*e^2 - 42*(13*B*a^2*b^4 + 20*A*a*b^5)*d^3*e^3 + 6*(25*B*a^3*b
^3 + 256*A*a^2*b^4)*d^2*e^4 - 35*(7*B*a^4*b^2 - 24*A*a^3*b^3)*d*e^5 + 15*(5*B*a^5*b - 12*A*a^4*b^2)*e^6 + 128*
(13*B*b^6*d*e^5 + (25*B*a*b^5 + 12*A*b^6)*e^6)*x^4 + 16*(3*B*b^6*d^2*e^4 + 2*(139*B*a*b^5 + 66*A*b^6)*d*e^5 +
9*(15*B*a^2*b^4 + 28*A*a*b^5)*e^6)*x^3 - 8*(7*B*b^6*d^3*e^3 - 3*(9*B*a*b^5 + 4*A*b^6)*d^2*e^4 - 3*(141*B*a^2*b
^4 + 256*A*a*b^5)*d*e^5 - (5*B*a^3*b^3 + 372*A*a^2*b^4)*e^6)*x^2 + 2*(35*B*b^6*d^4*e^2 - 4*(34*B*a*b^5 + 15*A*
b^6)*d^3*e^3 + 6*(29*B*a^2*b^4 + 46*A*a*b^5)*d^2*e^4 + 4*(20*B*a^3*b^3 + 699*A*a^2*b^4)*d*e^5 - 5*(5*B*a^4*b^2
- 12*A*a^3*b^3)*e^6)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^4*e^5)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 3.53764, size = 4425, normalized size = 12.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7680*(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^2 + (b^7*d*e^5 - 17*a
*b^6*e^6)*e^(-6)/b^8) - (5*b^8*d^2*e^4 + 6*a*b^7*d*e^5 - 59*a^2*b^6*e^6)*e^(-6)/b^8) + 3*(5*b^9*d^3*e^3 + a*b^
8*d^2*e^4 - a^2*b^7*d*e^5 - 5*a^3*b^6*e^6)*e^(-6)/b^8)*sqrt(b*x + a) + 3*(5*b^4*d^4 - 4*a*b^3*d^3*e - 2*a^2*b^
2*d^2*e^2 - 4*a^3*b*d*e^3 + 5*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^(3/2))*A*d*abs(b) + 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^3 + (b^13*d*e^7 - 31*a*b^12*e^8)*e^(-8)/b^15) - (7*b^14*d^2*e^6 + 16*a*b^13*d*e^7 - 263*a^2*b^12*e
^8)*e^(-8)/b^15) + 5*(7*b^15*d^3*e^5 + 9*a*b^14*d^2*e^6 + 9*a^2*b^13*d*e^7 - 121*a^3*b^12*e^8)*e^(-8)/b^15)*(b
*x + a) - 15*(7*b^16*d^4*e^4 + 2*a*b^15*d^3*e^5 - 2*a^3*b^13*d*e^7 - 7*a^4*b^12*e^8)*e^(-8)/b^15)*sqrt(b*x + a
) - 15*(7*b^5*d^5 - 5*a*b^4*d^4*e - 2*a^2*b^3*d^3*e^2 - 2*a^3*b^2*d^2*e^3 - 5*a^4*b*d*e^4 + 7*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^(5/2))*B*d*abs(b) + 80*(sqr
t(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*e^(-2)/b^4 + (b*d*e - a*e^2)*e^(-4)/b^4) + (b^2*d^
2 - 2*a*b*d*e + a^2*e^2)*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^(7/2))*A*a^2*d*abs(b)/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^2 + (b^7*d*e^5 - 17*a*b^6*e^6)*e^(-6)/b^8) - (5*b^8*d^2*e^4 + 6*a*b^7*d*e^5 - 59*a^2*b^6*e^6)*e^(-6)/b^8)
+ 3*(5*b^9*d^3*e^3 + a*b^8*d^2*e^4 - a^2*b^7*d*e^5 - 5*a^3*b^6*e^6)*e^(-6)/b^8)*sqrt(b*x + a) + 3*(5*b^4*d^4 -
4*a*b^3*d^3*e - 2*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + 5*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^(3/2))*B*a*d*abs(b)/b + 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^3 + (b^13*d*e^7 - 31*a*b^12*e^8)*e^(-8)/b^15) - (7*b^14*d^2*e^6 + 16*
a*b^13*d*e^7 - 263*a^2*b^12*e^8)*e^(-8)/b^15) + 5*(7*b^15*d^3*e^5 + 9*a*b^14*d^2*e^6 + 9*a^2*b^13*d*e^7 - 121*
a^3*b^12*e^8)*e^(-8)/b^15)*(b*x + a) - 15*(7*b^16*d^4*e^4 + 2*a*b^15*d^3*e^5 - 2*a^3*b^13*d*e^7 - 7*a^4*b^12*e
^8)*e^(-8)/b^15)*sqrt(b*x + a) - 15*(7*b^5*d^5 - 5*a*b^4*d^4*e - 2*a^2*b^3*d^3*e^2 - 2*a^3*b^2*d^2*e^3 - 5*a^4
*b*d*e^4 + 7*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^(5/2))*A*abs(b)*e + (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^4 +
(b^21*d*e^9 - 49*a*b^20*e^10)*e^(-10)/b^24) - 3*(3*b^22*d^2*e^8 + 10*a*b^21*d*e^9 - 253*a^2*b^20*e^10)*e^(-10
)/b^24) + (21*b^23*d^3*e^7 + 49*a*b^22*d^2*e^8 + 79*a^2*b^21*d*e^9 - 1429*a^3*b^20*e^10)*e^(-10)/b^24)*(b*x +
a) - 5*(21*b^24*d^4*e^6 + 28*a*b^23*d^3*e^7 + 30*a^2*b^22*d^2*e^8 + 28*a^3*b^21*d*e^9 - 491*a^4*b^20*e^10)*e^(
-10)/b^24)*(b*x + a) + 15*(21*b^25*d^5*e^5 + 7*a*b^24*d^4*e^6 + 2*a^2*b^23*d^3*e^7 - 2*a^3*b^22*d^2*e^8 - 7*a^
4*b^21*d*e^9 - 21*a^5*b^20*e^10)*e^(-10)/b^24)*sqrt(b*x + a) + 15*(21*b^6*d^6 - 14*a*b^5*d^5*e - 5*a^2*b^4*d^4
*e^2 - 4*a^3*b^3*d^3*e^3 - 5*a^4*b^2*d^2*e^4 - 14*a^5*b*d*e^5 + 21*a^6*e^6)*e^(-11/2)*log(abs(-sqrt(b*x + a)*s
qrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(7/2))*B*abs(b)*e + 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^2 + (b^7*d*e^5 - 17*a*b^6*e^6)*e^(-6)/b^8) - (5*b^8*d^2*e^4 +
6*a*b^7*d*e^5 - 59*a^2*b^6*e^6)*e^(-6)/b^8) + 3*(5*b^9*d^3*e^3 + a*b^8*d^2*e^4 - a^2*b^7*d*e^5 - 5*a^3*b^6*e^6
)*e^(-6)/b^8)*sqrt(b*x + a) + 3*(5*b^4*d^4 - 4*a*b^3*d^3*e - 2*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + 5*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^(3/2))*B*a^2*abs(b)*e/
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^2 + (b^7*d*e^5 - 17*a*b
^6*e^6)*e^(-6)/b^8) - (5*b^8*d^2*e^4 + 6*a*b^7*d*e^5 - 59*a^2*b^6*e^6)*e^(-6)/b^8) + 3*(5*b^9*d^3*e^3 + a*b^8*
d^2*e^4 - a^2*b^7*d*e^5 - 5*a^3*b^6*e^6)*e^(-6)/b^8)*sqrt(b*x + a) + 3*(5*b^4*d^4 - 4*a*b^3*d^3*e - 2*a^2*b^2*
d^2*e^2 - 4*a^3*b*d*e^3 + 5*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^(3/2))*A*a*abs(b)*e/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^3 + (b^13*d*e^7 - 31*a*b^12*e^8)*e^(-8)/b^15) - (7*b^14*d^2*e^6 + 16*a*b^13*d*e^7 - 263*a^2*b^12
*e^8)*e^(-8)/b^15) + 5*(7*b^15*d^3*e^5 + 9*a*b^14*d^2*e^6 + 9*a^2*b^13*d*e^7 - 121*a^3*b^12*e^8)*e^(-8)/b^15)*
(b*x + a) - 15*(7*b^16*d^4*e^4 + 2*a*b^15*d^3*e^5 - 2*a^3*b^13*d*e^7 - 7*a^4*b^12*e^8)*e^(-8)/b^15)*sqrt(b*x +
a) - 15*(7*b^5*d^5 - 5*a*b^4*d^4*e - 2*a^2*b^3*d^3*e^2 - 2*a^3*b^2*d^2*e^3 - 5*a^4*b*d*e^4 + 7*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^(5/2))*B*a*abs(b)*e/b + 4
*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*e^(-2)/b^6 + (b*d*e^3 - 7*a*e^4)
*e^(-6)/b^6) - 3*(b^2*d^2*e^2 - a^2*e^4)*e^(-6)/b^6) - 3*(b^3*d^3 - a*b^2*d^2*e - a^2*b*d*e^2 + a^3*e^3)*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^(11/2))*B*a^2*d*abs(b)/b^
3 + 8*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*e^(-2)/b^6 + (b*d*e^3 - 7*a
*e^4)*e^(-6)/b^6) - 3*(b^2*d^2*e^2 - a^2*e^4)*e^(-6)/b^6) - 3*(b^3*d^3 - a*b^2*d^2*e - a^2*b*d*e^2 + a^3*e^3)*
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^(11/2))*A*a*d*abs(b)
/b^2 + 4*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*e^(-2)/b^6 + (b*d*e^3 -
7*a*e^4)*e^(-6)/b^6) - 3*(b^2*d^2*e^2 - a^2*e^4)*e^(-6)/b^6) - 3*(b^3*d^3 - a*b^2*d^2*e - a^2*b*d*e^2 + a^3*e^
3)*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^(11/2))*A*a^2*abs
(b)*e/b^3)/b