3.1721 \(\int (A+B x) (d+e x)^4 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=298 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (-3 a B e-A b e+4 b B d)}{8 e^5 (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9}{9 e^5 (a+b x)} \]

[Out]

((b*d - a*e)^3*(B*d - A*e)*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)) - ((b*d - a*e)^2*(4*b*
B*d - 3*A*b*e - a*B*e)*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^5*(a + b*x)) + (3*b*(b*d - a*e)*(2*b*B*
d - A*b*e - a*B*e)*(d + e*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) - (b^2*(4*b*B*d - A*b*e - 3*a*
B*e)*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^5*(a + b*x)) + (b^3*B*(d + e*x)^9*Sqrt[a^2 + 2*a*b*x + b^
2*x^2])/(9*e^5*(a + b*x))

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Rubi [A]  time = 0.398901, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 77} \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (-3 a B e-A b e+4 b B d)}{8 e^5 (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9}{9 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.173777, size = 410, normalized size = 1.38 \[ \frac{x \sqrt{(a+b x)^2} \left (36 a^2 b x \left (7 A \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+2 B x \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )\right )+84 a^3 \left (6 A \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+B x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )\right )+9 a b^2 x^2 \left (8 A \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )+3 B x \left (280 d^2 e^2 x^2+224 d^3 e x+70 d^4+160 d e^3 x^3+35 e^4 x^4\right )\right )+b^3 x^3 \left (9 A \left (280 d^2 e^2 x^2+224 d^3 e x+70 d^4+160 d e^3 x^3+35 e^4 x^4\right )+4 B x \left (540 d^2 e^2 x^2+420 d^3 e x+126 d^4+315 d e^3 x^3+70 e^4 x^4\right )\right )\right )}{2520 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(84*a^3*(6*A*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + B*x*(15*d^4
+ 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4)) + 36*a^2*b*x*(7*A*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2
*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 2*B*x*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4))
+ 9*a*b^2*x^2*(8*A*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4) + 3*B*x*(70*d^4 + 224
*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4)) + b^3*x^3*(9*A*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x
^2 + 160*d*e^3*x^3 + 35*e^4*x^4) + 4*B*x*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4
))))/(2520*(a + b*x))

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Maple [B]  time = 0.007, size = 552, normalized size = 1.9 \begin{align*}{\frac{x \left ( 280\,{b}^{3}B{e}^{4}{x}^{8}+315\,{x}^{7}A{b}^{3}{e}^{4}+945\,{x}^{7}Ba{b}^{2}{e}^{4}+1260\,{x}^{7}{b}^{3}Bd{e}^{3}+1080\,{x}^{6}Aa{b}^{2}{e}^{4}+1440\,{x}^{6}A{b}^{3}d{e}^{3}+1080\,{x}^{6}B{a}^{2}b{e}^{4}+4320\,{x}^{6}Ba{b}^{2}d{e}^{3}+2160\,{x}^{6}{b}^{3}B{d}^{2}{e}^{2}+1260\,{x}^{5}A{a}^{2}b{e}^{4}+5040\,{x}^{5}Aa{b}^{2}d{e}^{3}+2520\,{x}^{5}A{b}^{3}{d}^{2}{e}^{2}+420\,{x}^{5}B{a}^{3}{e}^{4}+5040\,{x}^{5}B{a}^{2}bd{e}^{3}+7560\,{x}^{5}Ba{b}^{2}{d}^{2}{e}^{2}+1680\,{x}^{5}{b}^{3}B{d}^{3}e+504\,{x}^{4}A{a}^{3}{e}^{4}+6048\,{x}^{4}A{a}^{2}bd{e}^{3}+9072\,{x}^{4}Aa{b}^{2}{d}^{2}{e}^{2}+2016\,{x}^{4}A{b}^{3}{d}^{3}e+2016\,{x}^{4}B{a}^{3}d{e}^{3}+9072\,{x}^{4}B{a}^{2}b{d}^{2}{e}^{2}+6048\,{x}^{4}Ba{b}^{2}{d}^{3}e+504\,{x}^{4}{b}^{3}B{d}^{4}+2520\,{x}^{3}A{a}^{3}d{e}^{3}+11340\,{x}^{3}A{a}^{2}b{d}^{2}{e}^{2}+7560\,{x}^{3}Aa{b}^{2}{d}^{3}e+630\,{x}^{3}A{b}^{3}{d}^{4}+3780\,{x}^{3}B{a}^{3}{d}^{2}{e}^{2}+7560\,{x}^{3}B{a}^{2}b{d}^{3}e+1890\,{x}^{3}Ba{b}^{2}{d}^{4}+5040\,{x}^{2}A{a}^{3}{d}^{2}{e}^{2}+10080\,{x}^{2}A{a}^{2}b{d}^{3}e+2520\,{x}^{2}Aa{b}^{2}{d}^{4}+3360\,{x}^{2}B{a}^{3}{d}^{3}e+2520\,{x}^{2}B{a}^{2}b{d}^{4}+5040\,xA{a}^{3}{d}^{3}e+3780\,xA{a}^{2}b{d}^{4}+1260\,xB{a}^{3}{d}^{4}+2520\,A{a}^{3}{d}^{4} \right ) }{2520\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/2520*x*(280*B*b^3*e^4*x^8+315*A*b^3*e^4*x^7+945*B*a*b^2*e^4*x^7+1260*B*b^3*d*e^3*x^7+1080*A*a*b^2*e^4*x^6+14
40*A*b^3*d*e^3*x^6+1080*B*a^2*b*e^4*x^6+4320*B*a*b^2*d*e^3*x^6+2160*B*b^3*d^2*e^2*x^6+1260*A*a^2*b*e^4*x^5+504
0*A*a*b^2*d*e^3*x^5+2520*A*b^3*d^2*e^2*x^5+420*B*a^3*e^4*x^5+5040*B*a^2*b*d*e^3*x^5+7560*B*a*b^2*d^2*e^2*x^5+1
680*B*b^3*d^3*e*x^5+504*A*a^3*e^4*x^4+6048*A*a^2*b*d*e^3*x^4+9072*A*a*b^2*d^2*e^2*x^4+2016*A*b^3*d^3*e*x^4+201
6*B*a^3*d*e^3*x^4+9072*B*a^2*b*d^2*e^2*x^4+6048*B*a*b^2*d^3*e*x^4+504*B*b^3*d^4*x^4+2520*A*a^3*d*e^3*x^3+11340
*A*a^2*b*d^2*e^2*x^3+7560*A*a*b^2*d^3*e*x^3+630*A*b^3*d^4*x^3+3780*B*a^3*d^2*e^2*x^3+7560*B*a^2*b*d^3*e*x^3+18
90*B*a*b^2*d^4*x^3+5040*A*a^3*d^2*e^2*x^2+10080*A*a^2*b*d^3*e*x^2+2520*A*a*b^2*d^4*x^2+3360*B*a^3*d^3*e*x^2+25
20*B*a^2*b*d^4*x^2+5040*A*a^3*d^3*e*x+3780*A*a^2*b*d^4*x+1260*B*a^3*d^4*x+2520*A*a^3*d^4)*((b*x+a)^2)^(3/2)/(b
*x+a)^3

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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)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67275, size = 899, normalized size = 3.02 \begin{align*} \frac{1}{9} \, B b^{3} e^{4} x^{9} + A a^{3} d^{4} x + \frac{1}{8} \,{\left (4 \, B b^{3} d e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{8} + \frac{1}{7} \,{\left (6 \, B b^{3} d^{2} e^{2} + 4 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (4 \, B b^{3} d^{3} e + 6 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} d^{4} + A a^{3} e^{4} + 4 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 18 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 4 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, A a^{3} d e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e + 6 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, A a^{3} d^{2} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} + 4 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, A a^{3} d^{3} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/9*B*b^3*e^4*x^9 + A*a^3*d^4*x + 1/8*(4*B*b^3*d*e^3 + (3*B*a*b^2 + A*b^3)*e^4)*x^8 + 1/7*(6*B*b^3*d^2*e^2 + 4
*(3*B*a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^7 + 1/6*(4*B*b^3*d^3*e + 6*(3*B*a*b^2 + A*b^3)*d^2*e
^2 + 12*(B*a^2*b + A*a*b^2)*d*e^3 + (B*a^3 + 3*A*a^2*b)*e^4)*x^6 + 1/5*(B*b^3*d^4 + A*a^3*e^4 + 4*(3*B*a*b^2 +
A*b^3)*d^3*e + 18*(B*a^2*b + A*a*b^2)*d^2*e^2 + 4*(B*a^3 + 3*A*a^2*b)*d*e^3)*x^5 + 1/4*(4*A*a^3*d*e^3 + (3*B*
a*b^2 + A*b^3)*d^4 + 12*(B*a^2*b + A*a*b^2)*d^3*e + 6*(B*a^3 + 3*A*a^2*b)*d^2*e^2)*x^4 + 1/3*(6*A*a^3*d^2*e^2
+ 3*(B*a^2*b + A*a*b^2)*d^4 + 4*(B*a^3 + 3*A*a^2*b)*d^3*e)*x^3 + 1/2*(4*A*a^3*d^3*e + (B*a^3 + 3*A*a^2*b)*d^4)
*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B x\right ) \left (d + e x\right )^{4} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**4*((a + b*x)**2)**(3/2), x)

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Giac [B]  time = 1.1653, size = 1023, normalized size = 3.43 \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)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

1/9*B*b^3*x^9*e^4*sgn(b*x + a) + 1/2*B*b^3*d*x^8*e^3*sgn(b*x + a) + 6/7*B*b^3*d^2*x^7*e^2*sgn(b*x + a) + 2/3*B
*b^3*d^3*x^6*e*sgn(b*x + a) + 1/5*B*b^3*d^4*x^5*sgn(b*x + a) + 3/8*B*a*b^2*x^8*e^4*sgn(b*x + a) + 1/8*A*b^3*x^
8*e^4*sgn(b*x + a) + 12/7*B*a*b^2*d*x^7*e^3*sgn(b*x + a) + 4/7*A*b^3*d*x^7*e^3*sgn(b*x + a) + 3*B*a*b^2*d^2*x^
6*e^2*sgn(b*x + a) + A*b^3*d^2*x^6*e^2*sgn(b*x + a) + 12/5*B*a*b^2*d^3*x^5*e*sgn(b*x + a) + 4/5*A*b^3*d^3*x^5*
e*sgn(b*x + a) + 3/4*B*a*b^2*d^4*x^4*sgn(b*x + a) + 1/4*A*b^3*d^4*x^4*sgn(b*x + a) + 3/7*B*a^2*b*x^7*e^4*sgn(b
*x + a) + 3/7*A*a*b^2*x^7*e^4*sgn(b*x + a) + 2*B*a^2*b*d*x^6*e^3*sgn(b*x + a) + 2*A*a*b^2*d*x^6*e^3*sgn(b*x +
a) + 18/5*B*a^2*b*d^2*x^5*e^2*sgn(b*x + a) + 18/5*A*a*b^2*d^2*x^5*e^2*sgn(b*x + a) + 3*B*a^2*b*d^3*x^4*e*sgn(b
*x + a) + 3*A*a*b^2*d^3*x^4*e*sgn(b*x + a) + B*a^2*b*d^4*x^3*sgn(b*x + a) + A*a*b^2*d^4*x^3*sgn(b*x + a) + 1/6
*B*a^3*x^6*e^4*sgn(b*x + a) + 1/2*A*a^2*b*x^6*e^4*sgn(b*x + a) + 4/5*B*a^3*d*x^5*e^3*sgn(b*x + a) + 12/5*A*a^2
*b*d*x^5*e^3*sgn(b*x + a) + 3/2*B*a^3*d^2*x^4*e^2*sgn(b*x + a) + 9/2*A*a^2*b*d^2*x^4*e^2*sgn(b*x + a) + 4/3*B*
a^3*d^3*x^3*e*sgn(b*x + a) + 4*A*a^2*b*d^3*x^3*e*sgn(b*x + a) + 1/2*B*a^3*d^4*x^2*sgn(b*x + a) + 3/2*A*a^2*b*d
^4*x^2*sgn(b*x + a) + 1/5*A*a^3*x^5*e^4*sgn(b*x + a) + A*a^3*d*x^4*e^3*sgn(b*x + a) + 2*A*a^3*d^2*x^3*e^2*sgn(
b*x + a) + 2*A*a^3*d^3*x^2*e*sgn(b*x + a) + A*a^3*d^4*x*sgn(b*x + a)