3.1968 \(\int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=254 \[ \frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^2}{5 e^5 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^3}{9 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^4}{8 e^5 (a+b x)}+\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{12}}{12 e^5 (a+b x)}-\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)}{11 e^5 (a+b x)} \]

[Out]

((b*d - a*e)^4*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^5*(a + b*x)) - (4
*b*(b*d - a*e)^3*(d + e*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)) +
(3*b^2*(b*d - a*e)^2*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x
)) - (4*b^3*(b*d - a*e)*(d + e*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a +
b*x)) + (b^4*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*e^5*(a + b*x))

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Rubi [A]  time = 0.766789, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^2}{5 e^5 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^3}{9 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^4}{8 e^5 (a+b x)}+\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{12}}{12 e^5 (a+b x)}-\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)}{11 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((b*d - a*e)^4*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^5*(a + b*x)) - (4
*b*(b*d - a*e)^3*(d + e*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)) +
(3*b^2*(b*d - a*e)^2*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x
)) - (4*b^3*(b*d - a*e)*(d + e*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a +
b*x)) + (b^4*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*e^5*(a + b*x))

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Rubi in Sympy [A]  time = 46.2132, size = 209, normalized size = 0.82 \[ \frac{\left (a + b x\right ) \left (d + e x\right )^{8} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{12 e} + \frac{\left (d + e x\right )^{8} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{33 e^{2}} + \frac{\left (3 a + 3 b x\right ) \left (d + e x\right )^{8} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{330 e^{3}} + \frac{\left (d + e x\right )^{8} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{495 e^{4}} + \frac{\left (d + e x\right )^{8} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3960 e^{5} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**7*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

(a + b*x)*(d + e*x)**8*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(12*e) + (d + e*x)**8
*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(33*e**2) + (3*a + 3*b*x)*(d +
e*x)**8*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(330*e**3) + (d + e*x)**
8*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(495*e**4) + (d + e*x)**8*(a*e
- b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3960*e**5*(a + b*x))

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Mathematica [A]  time = 0.380932, size = 432, normalized size = 1.7 \[ \frac{x \sqrt{(a+b x)^2} \left (495 a^4 \left (8 d^7+28 d^6 e x+56 d^5 e^2 x^2+70 d^4 e^3 x^3+56 d^3 e^4 x^4+28 d^2 e^5 x^5+8 d e^6 x^6+e^7 x^7\right )+220 a^3 b x \left (36 d^7+168 d^6 e x+378 d^5 e^2 x^2+504 d^4 e^3 x^3+420 d^3 e^4 x^4+216 d^2 e^5 x^5+63 d e^6 x^6+8 e^7 x^7\right )+66 a^2 b^2 x^2 \left (120 d^7+630 d^6 e x+1512 d^5 e^2 x^2+2100 d^4 e^3 x^3+1800 d^3 e^4 x^4+945 d^2 e^5 x^5+280 d e^6 x^6+36 e^7 x^7\right )+12 a b^3 x^3 \left (330 d^7+1848 d^6 e x+4620 d^5 e^2 x^2+6600 d^4 e^3 x^3+5775 d^3 e^4 x^4+3080 d^2 e^5 x^5+924 d e^6 x^6+120 e^7 x^7\right )+b^4 x^4 \left (792 d^7+4620 d^6 e x+11880 d^5 e^2 x^2+17325 d^4 e^3 x^3+15400 d^3 e^4 x^4+8316 d^2 e^5 x^5+2520 d e^6 x^6+330 e^7 x^7\right )\right )}{3960 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(495*a^4*(8*d^7 + 28*d^6*e*x + 56*d^5*e^2*x^2 + 70*d^4*e^3*
x^3 + 56*d^3*e^4*x^4 + 28*d^2*e^5*x^5 + 8*d*e^6*x^6 + e^7*x^7) + 220*a^3*b*x*(36
*d^7 + 168*d^6*e*x + 378*d^5*e^2*x^2 + 504*d^4*e^3*x^3 + 420*d^3*e^4*x^4 + 216*d
^2*e^5*x^5 + 63*d*e^6*x^6 + 8*e^7*x^7) + 66*a^2*b^2*x^2*(120*d^7 + 630*d^6*e*x +
1512*d^5*e^2*x^2 + 2100*d^4*e^3*x^3 + 1800*d^3*e^4*x^4 + 945*d^2*e^5*x^5 + 280*
d*e^6*x^6 + 36*e^7*x^7) + 12*a*b^3*x^3*(330*d^7 + 1848*d^6*e*x + 4620*d^5*e^2*x^
2 + 6600*d^4*e^3*x^3 + 5775*d^3*e^4*x^4 + 3080*d^2*e^5*x^5 + 924*d*e^6*x^6 + 120
*e^7*x^7) + b^4*x^4*(792*d^7 + 4620*d^6*e*x + 11880*d^5*e^2*x^2 + 17325*d^4*e^3*
x^3 + 15400*d^3*e^4*x^4 + 8316*d^2*e^5*x^5 + 2520*d*e^6*x^6 + 330*e^7*x^7)))/(39
60*(a + b*x))

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Maple [B]  time = 0.013, size = 564, normalized size = 2.2 \[{\frac{x \left ( 330\,{b}^{4}{e}^{7}{x}^{11}+1440\,{x}^{10}a{b}^{3}{e}^{7}+2520\,{x}^{10}{b}^{4}d{e}^{6}+2376\,{x}^{9}{a}^{2}{b}^{2}{e}^{7}+11088\,{x}^{9}a{b}^{3}d{e}^{6}+8316\,{x}^{9}{b}^{4}{d}^{2}{e}^{5}+1760\,{x}^{8}{a}^{3}b{e}^{7}+18480\,{x}^{8}{a}^{2}{b}^{2}d{e}^{6}+36960\,{x}^{8}a{b}^{3}{d}^{2}{e}^{5}+15400\,{x}^{8}{b}^{4}{d}^{3}{e}^{4}+495\,{x}^{7}{a}^{4}{e}^{7}+13860\,{x}^{7}{a}^{3}bd{e}^{6}+62370\,{x}^{7}{a}^{2}{b}^{2}{d}^{2}{e}^{5}+69300\,{x}^{7}a{b}^{3}{d}^{3}{e}^{4}+17325\,{x}^{7}{b}^{4}{d}^{4}{e}^{3}+3960\,{a}^{4}d{e}^{6}{x}^{6}+47520\,{a}^{3}b{d}^{2}{e}^{5}{x}^{6}+118800\,{a}^{2}{b}^{2}{d}^{3}{e}^{4}{x}^{6}+79200\,a{b}^{3}{d}^{4}{e}^{3}{x}^{6}+11880\,{b}^{4}{d}^{5}{e}^{2}{x}^{6}+13860\,{x}^{5}{a}^{4}{d}^{2}{e}^{5}+92400\,{x}^{5}{a}^{3}b{d}^{3}{e}^{4}+138600\,{x}^{5}{a}^{2}{b}^{2}{d}^{4}{e}^{3}+55440\,{x}^{5}a{b}^{3}{d}^{5}{e}^{2}+4620\,{x}^{5}{b}^{4}{d}^{6}e+27720\,{x}^{4}{a}^{4}{d}^{3}{e}^{4}+110880\,{x}^{4}{a}^{3}b{d}^{4}{e}^{3}+99792\,{x}^{4}{a}^{2}{b}^{2}{d}^{5}{e}^{2}+22176\,{x}^{4}a{b}^{3}{d}^{6}e+792\,{x}^{4}{b}^{4}{d}^{7}+34650\,{x}^{3}{a}^{4}{d}^{4}{e}^{3}+83160\,{x}^{3}{a}^{3}b{d}^{5}{e}^{2}+41580\,{x}^{3}{a}^{2}{b}^{2}{d}^{6}e+3960\,{x}^{3}a{b}^{3}{d}^{7}+27720\,{x}^{2}{a}^{4}{d}^{5}{e}^{2}+36960\,{x}^{2}{a}^{3}b{d}^{6}e+7920\,{x}^{2}{a}^{2}{b}^{2}{d}^{7}+13860\,x{a}^{4}{d}^{6}e+7920\,x{a}^{3}b{d}^{7}+3960\,{a}^{4}{d}^{7} \right ) }{3960\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/3960*x*(330*b^4*e^7*x^11+1440*a*b^3*e^7*x^10+2520*b^4*d*e^6*x^10+2376*a^2*b^2*
e^7*x^9+11088*a*b^3*d*e^6*x^9+8316*b^4*d^2*e^5*x^9+1760*a^3*b*e^7*x^8+18480*a^2*
b^2*d*e^6*x^8+36960*a*b^3*d^2*e^5*x^8+15400*b^4*d^3*e^4*x^8+495*a^4*e^7*x^7+1386
0*a^3*b*d*e^6*x^7+62370*a^2*b^2*d^2*e^5*x^7+69300*a*b^3*d^3*e^4*x^7+17325*b^4*d^
4*e^3*x^7+3960*a^4*d*e^6*x^6+47520*a^3*b*d^2*e^5*x^6+118800*a^2*b^2*d^3*e^4*x^6+
79200*a*b^3*d^4*e^3*x^6+11880*b^4*d^5*e^2*x^6+13860*a^4*d^2*e^5*x^5+92400*a^3*b*
d^3*e^4*x^5+138600*a^2*b^2*d^4*e^3*x^5+55440*a*b^3*d^5*e^2*x^5+4620*b^4*d^6*e*x^
5+27720*a^4*d^3*e^4*x^4+110880*a^3*b*d^4*e^3*x^4+99792*a^2*b^2*d^5*e^2*x^4+22176
*a*b^3*d^6*e*x^4+792*b^4*d^7*x^4+34650*a^4*d^4*e^3*x^3+83160*a^3*b*d^5*e^2*x^3+4
1580*a^2*b^2*d^6*e*x^3+3960*a*b^3*d^7*x^3+27720*a^4*d^5*e^2*x^2+36960*a^3*b*d^6*
e*x^2+7920*a^2*b^2*d^7*x^2+13860*a^4*d^6*e*x+7920*a^3*b*d^7*x+3960*a^4*d^7)*((b*
x+a)^2)^(3/2)/(b*x+a)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)*(e*x + d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28002, size = 660, normalized size = 2.6 \[ \frac{1}{12} \, b^{4} e^{7} x^{12} + a^{4} d^{7} x + \frac{1}{11} \,{\left (7 \, b^{4} d e^{6} + 4 \, a b^{3} e^{7}\right )} x^{11} + \frac{1}{10} \,{\left (21 \, b^{4} d^{2} e^{5} + 28 \, a b^{3} d e^{6} + 6 \, a^{2} b^{2} e^{7}\right )} x^{10} + \frac{1}{9} \,{\left (35 \, b^{4} d^{3} e^{4} + 84 \, a b^{3} d^{2} e^{5} + 42 \, a^{2} b^{2} d e^{6} + 4 \, a^{3} b e^{7}\right )} x^{9} + \frac{1}{8} \,{\left (35 \, b^{4} d^{4} e^{3} + 140 \, a b^{3} d^{3} e^{4} + 126 \, a^{2} b^{2} d^{2} e^{5} + 28 \, a^{3} b d e^{6} + a^{4} e^{7}\right )} x^{8} +{\left (3 \, b^{4} d^{5} e^{2} + 20 \, a b^{3} d^{4} e^{3} + 30 \, a^{2} b^{2} d^{3} e^{4} + 12 \, a^{3} b d^{2} e^{5} + a^{4} d e^{6}\right )} x^{7} + \frac{7}{6} \,{\left (b^{4} d^{6} e + 12 \, a b^{3} d^{5} e^{2} + 30 \, a^{2} b^{2} d^{4} e^{3} + 20 \, a^{3} b d^{3} e^{4} + 3 \, a^{4} d^{2} e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{7} + 28 \, a b^{3} d^{6} e + 126 \, a^{2} b^{2} d^{5} e^{2} + 140 \, a^{3} b d^{4} e^{3} + 35 \, a^{4} d^{3} e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, a b^{3} d^{7} + 42 \, a^{2} b^{2} d^{6} e + 84 \, a^{3} b d^{5} e^{2} + 35 \, a^{4} d^{4} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, a^{2} b^{2} d^{7} + 28 \, a^{3} b d^{6} e + 21 \, a^{4} d^{5} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d^{7} + 7 \, a^{4} d^{6} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)*(e*x + d)^7,x, algorithm="fricas")

[Out]

1/12*b^4*e^7*x^12 + a^4*d^7*x + 1/11*(7*b^4*d*e^6 + 4*a*b^3*e^7)*x^11 + 1/10*(21
*b^4*d^2*e^5 + 28*a*b^3*d*e^6 + 6*a^2*b^2*e^7)*x^10 + 1/9*(35*b^4*d^3*e^4 + 84*a
*b^3*d^2*e^5 + 42*a^2*b^2*d*e^6 + 4*a^3*b*e^7)*x^9 + 1/8*(35*b^4*d^4*e^3 + 140*a
*b^3*d^3*e^4 + 126*a^2*b^2*d^2*e^5 + 28*a^3*b*d*e^6 + a^4*e^7)*x^8 + (3*b^4*d^5*
e^2 + 20*a*b^3*d^4*e^3 + 30*a^2*b^2*d^3*e^4 + 12*a^3*b*d^2*e^5 + a^4*d*e^6)*x^7
+ 7/6*(b^4*d^6*e + 12*a*b^3*d^5*e^2 + 30*a^2*b^2*d^4*e^3 + 20*a^3*b*d^3*e^4 + 3*
a^4*d^2*e^5)*x^6 + 1/5*(b^4*d^7 + 28*a*b^3*d^6*e + 126*a^2*b^2*d^5*e^2 + 140*a^3
*b*d^4*e^3 + 35*a^4*d^3*e^4)*x^5 + 1/4*(4*a*b^3*d^7 + 42*a^2*b^2*d^6*e + 84*a^3*
b*d^5*e^2 + 35*a^4*d^4*e^3)*x^4 + 1/3*(6*a^2*b^2*d^7 + 28*a^3*b*d^6*e + 21*a^4*d
^5*e^2)*x^3 + 1/2*(4*a^3*b*d^7 + 7*a^4*d^6*e)*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right ) \left (d + e x\right )^{7} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**7*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**7*((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.301087, size = 1027, normalized size = 4.04 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(b*x + a)*(e*x + d)^7,x, algorithm="giac")

[Out]

1/12*b^4*x^12*e^7*sign(b*x + a) + 7/11*b^4*d*x^11*e^6*sign(b*x + a) + 21/10*b^4*
d^2*x^10*e^5*sign(b*x + a) + 35/9*b^4*d^3*x^9*e^4*sign(b*x + a) + 35/8*b^4*d^4*x
^8*e^3*sign(b*x + a) + 3*b^4*d^5*x^7*e^2*sign(b*x + a) + 7/6*b^4*d^6*x^6*e*sign(
b*x + a) + 1/5*b^4*d^7*x^5*sign(b*x + a) + 4/11*a*b^3*x^11*e^7*sign(b*x + a) + 1
4/5*a*b^3*d*x^10*e^6*sign(b*x + a) + 28/3*a*b^3*d^2*x^9*e^5*sign(b*x + a) + 35/2
*a*b^3*d^3*x^8*e^4*sign(b*x + a) + 20*a*b^3*d^4*x^7*e^3*sign(b*x + a) + 14*a*b^3
*d^5*x^6*e^2*sign(b*x + a) + 28/5*a*b^3*d^6*x^5*e*sign(b*x + a) + a*b^3*d^7*x^4*
sign(b*x + a) + 3/5*a^2*b^2*x^10*e^7*sign(b*x + a) + 14/3*a^2*b^2*d*x^9*e^6*sign
(b*x + a) + 63/4*a^2*b^2*d^2*x^8*e^5*sign(b*x + a) + 30*a^2*b^2*d^3*x^7*e^4*sign
(b*x + a) + 35*a^2*b^2*d^4*x^6*e^3*sign(b*x + a) + 126/5*a^2*b^2*d^5*x^5*e^2*sig
n(b*x + a) + 21/2*a^2*b^2*d^6*x^4*e*sign(b*x + a) + 2*a^2*b^2*d^7*x^3*sign(b*x +
a) + 4/9*a^3*b*x^9*e^7*sign(b*x + a) + 7/2*a^3*b*d*x^8*e^6*sign(b*x + a) + 12*a
^3*b*d^2*x^7*e^5*sign(b*x + a) + 70/3*a^3*b*d^3*x^6*e^4*sign(b*x + a) + 28*a^3*b
*d^4*x^5*e^3*sign(b*x + a) + 21*a^3*b*d^5*x^4*e^2*sign(b*x + a) + 28/3*a^3*b*d^6
*x^3*e*sign(b*x + a) + 2*a^3*b*d^7*x^2*sign(b*x + a) + 1/8*a^4*x^8*e^7*sign(b*x
+ a) + a^4*d*x^7*e^6*sign(b*x + a) + 7/2*a^4*d^2*x^6*e^5*sign(b*x + a) + 7*a^4*d
^3*x^5*e^4*sign(b*x + a) + 35/4*a^4*d^4*x^4*e^3*sign(b*x + a) + 7*a^4*d^5*x^3*e^
2*sign(b*x + a) + 7/2*a^4*d^6*x^2*e*sign(b*x + a) + a^4*d^7*x*sign(b*x + a)