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

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.319616, 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} \[ \frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^4 (7 a B e-12 A b e+5 b B d)}{512 b^4 e^3}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^3 (7 a B e-12 A b e+5 b B d)}{768 b^4 e^2}-\frac{(b d-a e)^5 (7 a B e-12 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{512 b^{9/2} e^{7/2}}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e)^2 (7 a B e-12 A b e+5 b B d)}{192 b^4 e}-\frac{(a+b x)^{5/2} (d+e x)^{3/2} (b d-a e) (7 a B e-12 A b e+5 b B d)}{96 b^3 e}-\frac{(a+b x)^{5/2} (d+e x)^{5/2} (7 a B e-12 A b e+5 b B d)}{60 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e} \]

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 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)^{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*}

Mathematica [A]  time = 3.91124, size = 357, normalized size = 1. \[ \frac{(a+b x)^{5/2} \sqrt{d+e x} \left (\frac{\left (-\frac{7 a B e}{2}+6 A b e-\frac{5}{2} b B d\right ) \left (8 b^6 e^3 (a+b x)^3 (b d-a e)^{3/2} \sqrt{\frac{b (d+e x)}{b d-a e}} \left (5 a^2 e^2-10 a b e (2 d+e x)+b^2 \left (31 d^2+42 d e x+16 e^2 x^2\right )\right )+5 b^6 \sqrt{e} \left (2 e^{3/2} (a+b x)^2 (b d-a e)^{9/2} \sqrt{\frac{b (d+e x)}{b d-a e}}-3 \sqrt{e} (a+b x) (b d-a e)^{11/2} \sqrt{\frac{b (d+e x)}{b d-a e}}+3 \sqrt{a+b x} (b d-a e)^6 \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )\right )\right )}{128 b^6 e^3 (a+b x)^3 (b d-a e)^{3/2} \sqrt{\frac{b (d+e x)}{b d-a e}}}+5 b^3 B (d+e x)^3\right )}{30 b^4 e} \]

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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Maple [B]  time = 0.022, 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)^(3/2)*(B*x+A)*(e*x+d)^(5/2),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)-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^4-3328*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)-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)+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^2-140*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+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^2-150*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)*b^5*d^5-105*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^6*e^6+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))*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^3-320*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+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^4-11184*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^3-12288*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)-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)+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)-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)+1
092*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-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)-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)-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+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^5*b
*d*e^5-675*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^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+2
25*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^2-270*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*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+210*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^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., 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)^(3/2)*(B*x+A)*(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.06088, size = 3052, normalized size = 8.53 \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)^(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)]

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

[Out]

Timed out

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Giac [B]  time = 4.39897, size = 4447, normalized size = 12.42 \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)^(3/2)*(B*x+A)*(e*x+d)^(5/2),x, algorithm="giac")

[Out]

1/7680*(80*(sqrt(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*d^2*abs(b)/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^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*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^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)*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*d*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*d*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*d^2*abs
(b)/b^3 + 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*d^2*
abs(b)/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^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^2/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^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 - 2
63*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^2/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^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^2/b + (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 + 7
9*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)*sqr
t(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)*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 + 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)*e/b^3)/b