3.1995 \(\int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=125 \[ \frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{4 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^3}+\frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^3} \]

[Out]

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

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Rubi [A]  time = 0.424232, antiderivative size = 125, 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 \[ \frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{4 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^3}+\frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.1698, size = 100, normalized size = 0.8 \[ \frac{\left (d + e x\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{9 b} - \frac{\left (d + e x\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{36 b^{2}} + \frac{\left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{252 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**2*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

(d + e*x)**2*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(9*b) - (d + e*x)*(a*e - b*d)*(
a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(36*b**2) + (a*e - b*d)**2*(a**2 + 2*a*b*x +
b**2*x**2)**(7/2)/(252*b**3)

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Mathematica [A]  time = 0.143236, size = 217, normalized size = 1.74 \[ \frac{x \sqrt{(a+b x)^2} \left (84 a^6 \left (3 d^2+3 d e x+e^2 x^2\right )+126 a^5 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+126 a^4 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+84 a^3 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+36 a^2 b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+9 a b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+b^6 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )}{252 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[(a + b*x)^2]*(84*a^6*(3*d^2 + 3*d*e*x + e^2*x^2) + 126*a^5*b*x*(6*d^2 +
8*d*e*x + 3*e^2*x^2) + 126*a^4*b^2*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + 84*a^3*
b^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 36*a^2*b^4*x^4*(21*d^2 + 35*d*e*x + 1
5*e^2*x^2) + 9*a*b^5*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + b^6*x^6*(36*d^2 + 63
*d*e*x + 28*e^2*x^2)))/(252*(a + b*x))

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Maple [B]  time = 0.012, size = 271, normalized size = 2.2 \[{\frac{x \left ( 28\,{e}^{2}{b}^{6}{x}^{8}+189\,{x}^{7}{e}^{2}a{b}^{5}+63\,{x}^{7}de{b}^{6}+540\,{x}^{6}{a}^{2}{b}^{4}{e}^{2}+432\,{x}^{6}a{b}^{5}de+36\,{x}^{6}{b}^{6}{d}^{2}+840\,{x}^{5}{e}^{2}{a}^{3}{b}^{3}+1260\,{x}^{5}de{a}^{2}{b}^{4}+252\,{x}^{5}{d}^{2}a{b}^{5}+756\,{a}^{4}{b}^{2}{e}^{2}{x}^{4}+2016\,{a}^{3}{b}^{3}de{x}^{4}+756\,{a}^{2}{b}^{4}{d}^{2}{x}^{4}+378\,{x}^{3}{e}^{2}{a}^{5}b+1890\,{x}^{3}de{b}^{2}{a}^{4}+1260\,{x}^{3}{d}^{2}{a}^{3}{b}^{3}+84\,{x}^{2}{e}^{2}{a}^{6}+1008\,{x}^{2}de{a}^{5}b+1260\,{x}^{2}{d}^{2}{b}^{2}{a}^{4}+252\,{a}^{6}dex+756\,{a}^{5}b{d}^{2}x+252\,{d}^{2}{a}^{6} \right ) }{252\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^2*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/252*x*(28*b^6*e^2*x^8+189*a*b^5*e^2*x^7+63*b^6*d*e*x^7+540*a^2*b^4*e^2*x^6+432
*a*b^5*d*e*x^6+36*b^6*d^2*x^6+840*a^3*b^3*e^2*x^5+1260*a^2*b^4*d*e*x^5+252*a*b^5
*d^2*x^5+756*a^4*b^2*e^2*x^4+2016*a^3*b^3*d*e*x^4+756*a^2*b^4*d^2*x^4+378*a^5*b*
e^2*x^3+1890*a^4*b^2*d*e*x^3+1260*a^3*b^3*d^2*x^3+84*a^6*e^2*x^2+1008*a^5*b*d*e*
x^2+1260*a^4*b^2*d^2*x^2+252*a^6*d*e*x+756*a^5*b*d^2*x+252*a^6*d^2)*((b*x+a)^2)^
(5/2)/(b*x+a)^5

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295628, size = 316, normalized size = 2.53 \[ \frac{1}{9} \, b^{6} e^{2} x^{9} + a^{6} d^{2} x + \frac{1}{4} \,{\left (b^{6} d e + 3 \, a b^{5} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{2} + 12 \, a b^{5} d e + 15 \, a^{2} b^{4} e^{2}\right )} x^{7} + \frac{1}{3} \,{\left (3 \, a b^{5} d^{2} + 15 \, a^{2} b^{4} d e + 10 \, a^{3} b^{3} e^{2}\right )} x^{6} +{\left (3 \, a^{2} b^{4} d^{2} + 8 \, a^{3} b^{3} d e + 3 \, a^{4} b^{2} e^{2}\right )} x^{5} + \frac{1}{2} \,{\left (10 \, a^{3} b^{3} d^{2} + 15 \, a^{4} b^{2} d e + 3 \, a^{5} b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (15 \, a^{4} b^{2} d^{2} + 12 \, a^{5} b d e + a^{6} e^{2}\right )} x^{3} +{\left (3 \, a^{5} b d^{2} + a^{6} d e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*(e*x + d)^2,x, algorithm="fricas")

[Out]

1/9*b^6*e^2*x^9 + a^6*d^2*x + 1/4*(b^6*d*e + 3*a*b^5*e^2)*x^8 + 1/7*(b^6*d^2 + 1
2*a*b^5*d*e + 15*a^2*b^4*e^2)*x^7 + 1/3*(3*a*b^5*d^2 + 15*a^2*b^4*d*e + 10*a^3*b
^3*e^2)*x^6 + (3*a^2*b^4*d^2 + 8*a^3*b^3*d*e + 3*a^4*b^2*e^2)*x^5 + 1/2*(10*a^3*
b^3*d^2 + 15*a^4*b^2*d*e + 3*a^5*b*e^2)*x^4 + 1/3*(15*a^4*b^2*d^2 + 12*a^5*b*d*e
+ a^6*e^2)*x^3 + (3*a^5*b*d^2 + a^6*d*e)*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right ) \left (d + e x\right )^{2} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**2*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.288728, size = 512, normalized size = 4.1 \[ \frac{1}{9} \, b^{6} x^{9} e^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, b^{6} d x^{8} e{\rm sign}\left (b x + a\right ) + \frac{1}{7} \, b^{6} d^{2} x^{7}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, a b^{5} x^{8} e^{2}{\rm sign}\left (b x + a\right ) + \frac{12}{7} \, a b^{5} d x^{7} e{\rm sign}\left (b x + a\right ) + a b^{5} d^{2} x^{6}{\rm sign}\left (b x + a\right ) + \frac{15}{7} \, a^{2} b^{4} x^{7} e^{2}{\rm sign}\left (b x + a\right ) + 5 \, a^{2} b^{4} d x^{6} e{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{2} x^{5}{\rm sign}\left (b x + a\right ) + \frac{10}{3} \, a^{3} b^{3} x^{6} e^{2}{\rm sign}\left (b x + a\right ) + 8 \, a^{3} b^{3} d x^{5} e{\rm sign}\left (b x + a\right ) + 5 \, a^{3} b^{3} d^{2} x^{4}{\rm sign}\left (b x + a\right ) + 3 \, a^{4} b^{2} x^{5} e^{2}{\rm sign}\left (b x + a\right ) + \frac{15}{2} \, a^{4} b^{2} d x^{4} e{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{2} x^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a^{5} b x^{4} e^{2}{\rm sign}\left (b x + a\right ) + 4 \, a^{5} b d x^{3} e{\rm sign}\left (b x + a\right ) + 3 \, a^{5} b d^{2} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, a^{6} x^{3} e^{2}{\rm sign}\left (b x + a\right ) + a^{6} d x^{2} e{\rm sign}\left (b x + a\right ) + a^{6} d^{2} x{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*(e*x + d)^2,x, algorithm="giac")

[Out]

1/9*b^6*x^9*e^2*sign(b*x + a) + 1/4*b^6*d*x^8*e*sign(b*x + a) + 1/7*b^6*d^2*x^7*
sign(b*x + a) + 3/4*a*b^5*x^8*e^2*sign(b*x + a) + 12/7*a*b^5*d*x^7*e*sign(b*x +
a) + a*b^5*d^2*x^6*sign(b*x + a) + 15/7*a^2*b^4*x^7*e^2*sign(b*x + a) + 5*a^2*b^
4*d*x^6*e*sign(b*x + a) + 3*a^2*b^4*d^2*x^5*sign(b*x + a) + 10/3*a^3*b^3*x^6*e^2
*sign(b*x + a) + 8*a^3*b^3*d*x^5*e*sign(b*x + a) + 5*a^3*b^3*d^2*x^4*sign(b*x +
a) + 3*a^4*b^2*x^5*e^2*sign(b*x + a) + 15/2*a^4*b^2*d*x^4*e*sign(b*x + a) + 5*a^
4*b^2*d^2*x^3*sign(b*x + a) + 3/2*a^5*b*x^4*e^2*sign(b*x + a) + 4*a^5*b*d*x^3*e*
sign(b*x + a) + 3*a^5*b*d^2*x^2*sign(b*x + a) + 1/3*a^6*x^3*e^2*sign(b*x + a) +
a^6*d*x^2*e*sign(b*x + a) + a^6*d^2*x*sign(b*x + a)