3.1858 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=448 \[ -\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (-5 a B e-A b e+6 b B d)}{5 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x) (d+e x)^{5/2}}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^7 (a+b x)} \]

[Out]

(-2*(b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^(5/2)) + (2*(b*d - a*e
)^4*(6*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(3/2)) - (10*b*(b*d
- a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) - (20*b^2*(b
*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (10*b^3*(
b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (2*b
^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (2*b^5*B*(d
+ e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x))

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Rubi [A]  time = 0.204824, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {770, 77} \[ -\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (-5 a B e-A b e+6 b B d)}{5 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x) \sqrt{d+e x}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x) (d+e x)^{5/2}}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(7/2),x]

[Out]

(-2*(b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^(5/2)) + (2*(b*d - a*e
)^4*(6*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(3/2)) - (10*b*(b*d
- a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) - (20*b^2*(b
*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (10*b^3*(
b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (2*b
^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (2*b^5*B*(d
+ e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^{7/2}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^{7/2}}+\frac{b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{5/2}}-\frac{5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 (d+e x)^{3/2}}+\frac{10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 \sqrt{d+e x}}-\frac{5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) \sqrt{d+e x}}{e^6}+\frac{b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{3/2}}{e^6}+\frac{b^{10} B (d+e x)^{5/2}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{2 (b d-a e)^5 (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{5/2}}+\frac{2 (b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac{10 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt{d+e x}}-\frac{20 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac{10 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac{2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac{2 b^5 B (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.348601, size = 239, normalized size = 0.53 \[ \frac{2 \sqrt{(a+b x)^2} \left (-21 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+175 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-1050 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)-525 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)+35 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)-21 (b d-a e)^5 (B d-A e)+15 b^5 B (d+e x)^6\right )}{105 e^7 (a+b x) (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(7/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-21*(b*d - a*e)^5*(B*d - A*e) + 35*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x) -
525*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 1050*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e
)*(d + e*x)^3 + 175*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 - 21*b^4*(6*b*B*d - A*b*e - 5*a*B*
e)*(d + e*x)^5 + 15*b^5*B*(d + e*x)^6))/(105*e^7*(a + b*x)*(d + e*x)^(5/2))

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Maple [A]  time = 0.008, size = 689, normalized size = 1.5 \begin{align*} -{\frac{-30\,B{x}^{6}{b}^{5}{e}^{6}-42\,A{x}^{5}{b}^{5}{e}^{6}-210\,B{x}^{5}a{b}^{4}{e}^{6}+72\,B{x}^{5}{b}^{5}d{e}^{5}-350\,A{x}^{4}a{b}^{4}{e}^{6}+140\,A{x}^{4}{b}^{5}d{e}^{5}-700\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+700\,B{x}^{4}a{b}^{4}d{e}^{5}-240\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}-2100\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+2800\,A{x}^{3}a{b}^{4}d{e}^{5}-1120\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}-2100\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+5600\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}-5600\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+1920\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+2100\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}-12600\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+16800\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}-6720\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+1050\,B{x}^{2}{a}^{4}b{e}^{6}-12600\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+33600\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}-33600\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+11520\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+350\,Ax{a}^{4}b{e}^{6}+2800\,Ax{a}^{3}{b}^{2}d{e}^{5}-16800\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+22400\,Axa{b}^{4}{d}^{3}{e}^{3}-8960\,Ax{b}^{5}{d}^{4}{e}^{2}+70\,Bx{a}^{5}{e}^{6}+1400\,Bx{a}^{4}bd{e}^{5}-16800\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+44800\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}-44800\,Bxa{b}^{4}{d}^{4}{e}^{2}+15360\,Bx{b}^{5}{d}^{5}e+42\,A{a}^{5}{e}^{6}+140\,Ad{e}^{5}{a}^{4}b+1120\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}-6720\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+8960\,Aa{b}^{4}{d}^{4}{e}^{2}-3584\,A{b}^{5}{d}^{5}e+28\,Bd{e}^{5}{a}^{5}+560\,B{a}^{4}b{d}^{2}{e}^{4}-6720\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+17920\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}-17920\,Ba{b}^{4}{d}^{5}e+6144\,B{b}^{5}{d}^{6}}{105\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(7/2),x)

[Out]

-2/105/(e*x+d)^(5/2)*(-15*B*b^5*e^6*x^6-21*A*b^5*e^6*x^5-105*B*a*b^4*e^6*x^5+36*B*b^5*d*e^5*x^5-175*A*a*b^4*e^
6*x^4+70*A*b^5*d*e^5*x^4-350*B*a^2*b^3*e^6*x^4+350*B*a*b^4*d*e^5*x^4-120*B*b^5*d^2*e^4*x^4-1050*A*a^2*b^3*e^6*
x^3+1400*A*a*b^4*d*e^5*x^3-560*A*b^5*d^2*e^4*x^3-1050*B*a^3*b^2*e^6*x^3+2800*B*a^2*b^3*d*e^5*x^3-2800*B*a*b^4*
d^2*e^4*x^3+960*B*b^5*d^3*e^3*x^3+1050*A*a^3*b^2*e^6*x^2-6300*A*a^2*b^3*d*e^5*x^2+8400*A*a*b^4*d^2*e^4*x^2-336
0*A*b^5*d^3*e^3*x^2+525*B*a^4*b*e^6*x^2-6300*B*a^3*b^2*d*e^5*x^2+16800*B*a^2*b^3*d^2*e^4*x^2-16800*B*a*b^4*d^3
*e^3*x^2+5760*B*b^5*d^4*e^2*x^2+175*A*a^4*b*e^6*x+1400*A*a^3*b^2*d*e^5*x-8400*A*a^2*b^3*d^2*e^4*x+11200*A*a*b^
4*d^3*e^3*x-4480*A*b^5*d^4*e^2*x+35*B*a^5*e^6*x+700*B*a^4*b*d*e^5*x-8400*B*a^3*b^2*d^2*e^4*x+22400*B*a^2*b^3*d
^3*e^3*x-22400*B*a*b^4*d^4*e^2*x+7680*B*b^5*d^5*e*x+21*A*a^5*e^6+70*A*a^4*b*d*e^5+560*A*a^3*b^2*d^2*e^4-3360*A
*a^2*b^3*d^3*e^3+4480*A*a*b^4*d^4*e^2-1792*A*b^5*d^5*e+14*B*a^5*d*e^5+280*B*a^4*b*d^2*e^4-3360*B*a^3*b^2*d^3*e
^3+8960*B*a^2*b^3*d^4*e^2-8960*B*a*b^4*d^5*e+3072*B*b^5*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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Maxima [A]  time = 1.24421, size = 873, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 640 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} - 80 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{2} e^{3} - 20 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 30 \,{\left (16 \, b^{5} d^{3} e^{2} - 40 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} - 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 5 \,{\left (128 \, b^{5} d^{4} e - 320 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} - 40 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} A}{15 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (15 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 8960 \, a b^{4} d^{5} e - 8960 \, a^{2} b^{3} d^{4} e^{2} + 3360 \, a^{3} b^{2} d^{3} e^{3} - 280 \, a^{4} b d^{2} e^{4} - 14 \, a^{5} d e^{5} - 3 \,{\left (12 \, b^{5} d e^{5} - 35 \, a b^{4} e^{6}\right )} x^{5} + 10 \,{\left (12 \, b^{5} d^{2} e^{4} - 35 \, a b^{4} d e^{5} + 35 \, a^{2} b^{3} e^{6}\right )} x^{4} - 10 \,{\left (96 \, b^{5} d^{3} e^{3} - 280 \, a b^{4} d^{2} e^{4} + 280 \, a^{2} b^{3} d e^{5} - 105 \, a^{3} b^{2} e^{6}\right )} x^{3} - 15 \,{\left (384 \, b^{5} d^{4} e^{2} - 1120 \, a b^{4} d^{3} e^{3} + 1120 \, a^{2} b^{3} d^{2} e^{4} - 420 \, a^{3} b^{2} d e^{5} + 35 \, a^{4} b e^{6}\right )} x^{2} - 5 \,{\left (1536 \, b^{5} d^{5} e - 4480 \, a b^{4} d^{4} e^{2} + 4480 \, a^{2} b^{3} d^{3} e^{3} - 1680 \, a^{3} b^{2} d^{2} e^{4} + 140 \, a^{4} b d e^{5} + 7 \, a^{5} e^{6}\right )} x\right )} B}{105 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )} \sqrt{e x + d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/15*(3*b^5*e^5*x^5 + 256*b^5*d^5 - 640*a*b^4*d^4*e + 480*a^2*b^3*d^3*e^2 - 80*a^3*b^2*d^2*e^3 - 10*a^4*b*d*e^
4 - 3*a^5*e^5 - 5*(2*b^5*d*e^4 - 5*a*b^4*e^5)*x^4 + 10*(8*b^5*d^2*e^3 - 20*a*b^4*d*e^4 + 15*a^2*b^3*e^5)*x^3 +
30*(16*b^5*d^3*e^2 - 40*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 - 5*a^3*b^2*e^5)*x^2 + 5*(128*b^5*d^4*e - 320*a*b^4*
d^3*e^2 + 240*a^2*b^3*d^2*e^3 - 40*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)*A/((e^8*x^2 + 2*d*e^7*x + d^2*e^6)*sqrt(e*x
+ d)) + 2/105*(15*b^5*e^6*x^6 - 3072*b^5*d^6 + 8960*a*b^4*d^5*e - 8960*a^2*b^3*d^4*e^2 + 3360*a^3*b^2*d^3*e^3
- 280*a^4*b*d^2*e^4 - 14*a^5*d*e^5 - 3*(12*b^5*d*e^5 - 35*a*b^4*e^6)*x^5 + 10*(12*b^5*d^2*e^4 - 35*a*b^4*d*e^
5 + 35*a^2*b^3*e^6)*x^4 - 10*(96*b^5*d^3*e^3 - 280*a*b^4*d^2*e^4 + 280*a^2*b^3*d*e^5 - 105*a^3*b^2*e^6)*x^3 -
15*(384*b^5*d^4*e^2 - 1120*a*b^4*d^3*e^3 + 1120*a^2*b^3*d^2*e^4 - 420*a^3*b^2*d*e^5 + 35*a^4*b*e^6)*x^2 - 5*(1
536*b^5*d^5*e - 4480*a*b^4*d^4*e^2 + 4480*a^2*b^3*d^3*e^3 - 1680*a^3*b^2*d^2*e^4 + 140*a^4*b*d*e^5 + 7*a^5*e^6
)*x)*B/((e^9*x^2 + 2*d*e^8*x + d^2*e^7)*sqrt(e*x + d))

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Fricas [A]  time = 1.49902, size = 1278, normalized size = 2.85 \begin{align*} \frac{2 \,{\left (15 \, B b^{5} e^{6} x^{6} - 3072 \, B b^{5} d^{6} - 21 \, A a^{5} e^{6} + 1792 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 4480 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 3360 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 280 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - 14 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 3 \,{\left (12 \, B b^{5} d e^{5} - 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 5 \,{\left (24 \, B b^{5} d^{2} e^{4} - 14 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 35 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 10 \,{\left (96 \, B b^{5} d^{3} e^{3} - 56 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 140 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 105 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} - 15 \,{\left (384 \, B b^{5} d^{4} e^{2} - 224 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 560 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 420 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 35 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - 5 \,{\left (1536 \, B b^{5} d^{5} e - 896 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 2240 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 1680 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 140 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )} \sqrt{e x + d}}{105 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/105*(15*B*b^5*e^6*x^6 - 3072*B*b^5*d^6 - 21*A*a^5*e^6 + 1792*(5*B*a*b^4 + A*b^5)*d^5*e - 4480*(2*B*a^2*b^3 +
A*a*b^4)*d^4*e^2 + 3360*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 - 280*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - 14*(B*a^5 + 5
*A*a^4*b)*d*e^5 - 3*(12*B*b^5*d*e^5 - 7*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 5*(24*B*b^5*d^2*e^4 - 14*(5*B*a*b^4 + A
*b^5)*d*e^5 + 35*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 10*(96*B*b^5*d^3*e^3 - 56*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 14
0*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 - 105*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 - 15*(384*B*b^5*d^4*e^2 - 224*(5*B*a*b^
4 + A*b^5)*d^3*e^3 + 560*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 420*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 35*(B*a^4*b + 2
*A*a^3*b^2)*e^6)*x^2 - 5*(1536*B*b^5*d^5*e - 896*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 2240*(2*B*a^2*b^3 + A*a*b^4)*d^
3*e^3 - 1680*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 140*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + 7*(B*a^5 + 5*A*a^4*b)*e^6)*
x)*sqrt(e*x + d)/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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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)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

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Giac [B]  time = 1.27588, size = 1486, normalized size = 3.32 \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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/105*(15*(x*e + d)^(7/2)*B*b^5*e^42*sgn(b*x + a) - 126*(x*e + d)^(5/2)*B*b^5*d*e^42*sgn(b*x + a) + 525*(x*e +
d)^(3/2)*B*b^5*d^2*e^42*sgn(b*x + a) - 2100*sqrt(x*e + d)*B*b^5*d^3*e^42*sgn(b*x + a) + 105*(x*e + d)^(5/2)*B
*a*b^4*e^43*sgn(b*x + a) + 21*(x*e + d)^(5/2)*A*b^5*e^43*sgn(b*x + a) - 875*(x*e + d)^(3/2)*B*a*b^4*d*e^43*sgn
(b*x + a) - 175*(x*e + d)^(3/2)*A*b^5*d*e^43*sgn(b*x + a) + 5250*sqrt(x*e + d)*B*a*b^4*d^2*e^43*sgn(b*x + a) +
1050*sqrt(x*e + d)*A*b^5*d^2*e^43*sgn(b*x + a) + 350*(x*e + d)^(3/2)*B*a^2*b^3*e^44*sgn(b*x + a) + 175*(x*e +
d)^(3/2)*A*a*b^4*e^44*sgn(b*x + a) - 4200*sqrt(x*e + d)*B*a^2*b^3*d*e^44*sgn(b*x + a) - 2100*sqrt(x*e + d)*A*
a*b^4*d*e^44*sgn(b*x + a) + 1050*sqrt(x*e + d)*B*a^3*b^2*e^45*sgn(b*x + a) + 1050*sqrt(x*e + d)*A*a^2*b^3*e^45
*sgn(b*x + a))*e^(-49) - 2/15*(225*(x*e + d)^2*B*b^5*d^4*sgn(b*x + a) - 30*(x*e + d)*B*b^5*d^5*sgn(b*x + a) +
3*B*b^5*d^6*sgn(b*x + a) - 750*(x*e + d)^2*B*a*b^4*d^3*e*sgn(b*x + a) - 150*(x*e + d)^2*A*b^5*d^3*e*sgn(b*x +
a) + 125*(x*e + d)*B*a*b^4*d^4*e*sgn(b*x + a) + 25*(x*e + d)*A*b^5*d^4*e*sgn(b*x + a) - 15*B*a*b^4*d^5*e*sgn(b
*x + a) - 3*A*b^5*d^5*e*sgn(b*x + a) + 900*(x*e + d)^2*B*a^2*b^3*d^2*e^2*sgn(b*x + a) + 450*(x*e + d)^2*A*a*b^
4*d^2*e^2*sgn(b*x + a) - 200*(x*e + d)*B*a^2*b^3*d^3*e^2*sgn(b*x + a) - 100*(x*e + d)*A*a*b^4*d^3*e^2*sgn(b*x
+ a) + 30*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 15*A*a*b^4*d^4*e^2*sgn(b*x + a) - 450*(x*e + d)^2*B*a^3*b^2*d*e^3*s
gn(b*x + a) - 450*(x*e + d)^2*A*a^2*b^3*d*e^3*sgn(b*x + a) + 150*(x*e + d)*B*a^3*b^2*d^2*e^3*sgn(b*x + a) + 15
0*(x*e + d)*A*a^2*b^3*d^2*e^3*sgn(b*x + a) - 30*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 30*A*a^2*b^3*d^3*e^3*sgn(b*x
+ a) + 75*(x*e + d)^2*B*a^4*b*e^4*sgn(b*x + a) + 150*(x*e + d)^2*A*a^3*b^2*e^4*sgn(b*x + a) - 50*(x*e + d)*B*a
^4*b*d*e^4*sgn(b*x + a) - 100*(x*e + d)*A*a^3*b^2*d*e^4*sgn(b*x + a) + 15*B*a^4*b*d^2*e^4*sgn(b*x + a) + 30*A*
a^3*b^2*d^2*e^4*sgn(b*x + a) + 5*(x*e + d)*B*a^5*e^5*sgn(b*x + a) + 25*(x*e + d)*A*a^4*b*e^5*sgn(b*x + a) - 3*
B*a^5*d*e^5*sgn(b*x + a) - 15*A*a^4*b*d*e^5*sgn(b*x + a) + 3*A*a^5*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^(5/2)