3.301 \(\int \frac{(a+b x)^2 (A+B x)}{\sqrt{x}} \, dx\)

Optimal. Leaf size=61 \[ 2 a^2 A \sqrt{x}+\frac{2}{5} b x^{5/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{7} b^2 B x^{7/2} \]

[Out]

2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(3/2))/3 + (2*b*(A*b + 2*a*B)*x^(5/2))/5
+ (2*b^2*B*x^(7/2))/7

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Rubi [A]  time = 0.0769108, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ 2 a^2 A \sqrt{x}+\frac{2}{5} b x^{5/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{7} b^2 B x^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^2*(A + B*x))/Sqrt[x],x]

[Out]

2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(3/2))/3 + (2*b*(A*b + 2*a*B)*x^(5/2))/5
+ (2*b^2*B*x^(7/2))/7

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Rubi in Sympy [A]  time = 8.98865, size = 61, normalized size = 1. \[ 2 A a^{2} \sqrt{x} + \frac{2 B b^{2} x^{\frac{7}{2}}}{7} + \frac{2 a x^{\frac{3}{2}} \left (2 A b + B a\right )}{3} + \frac{2 b x^{\frac{5}{2}} \left (A b + 2 B a\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(B*x+A)/x**(1/2),x)

[Out]

2*A*a**2*sqrt(x) + 2*B*b**2*x**(7/2)/7 + 2*a*x**(3/2)*(2*A*b + B*a)/3 + 2*b*x**(
5/2)*(A*b + 2*B*a)/5

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Mathematica [A]  time = 0.0272869, size = 51, normalized size = 0.84 \[ \frac{2}{105} \sqrt{x} \left (35 a^2 (3 A+B x)+14 a b x (5 A+3 B x)+3 b^2 x^2 (7 A+5 B x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^2*(A + B*x))/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(35*a^2*(3*A + B*x) + 14*a*b*x*(5*A + 3*B*x) + 3*b^2*x^2*(7*A + 5*B*x
)))/105

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Maple [A]  time = 0.009, size = 52, normalized size = 0.9 \[{\frac{30\,B{b}^{2}{x}^{3}+42\,A{b}^{2}{x}^{2}+84\,B{x}^{2}ab+140\,aAbx+70\,{a}^{2}Bx+210\,{a}^{2}A}{105}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(B*x+A)/x^(1/2),x)

[Out]

2/105*x^(1/2)*(15*B*b^2*x^3+21*A*b^2*x^2+42*B*a*b*x^2+70*A*a*b*x+35*B*a^2*x+105*
A*a^2)

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Maxima [A]  time = 1.36247, size = 69, normalized size = 1.13 \[ \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + 2 \, A a^{2} \sqrt{x} + \frac{2}{5} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2/sqrt(x),x, algorithm="maxima")

[Out]

2/7*B*b^2*x^(7/2) + 2*A*a^2*sqrt(x) + 2/5*(2*B*a*b + A*b^2)*x^(5/2) + 2/3*(B*a^2
+ 2*A*a*b)*x^(3/2)

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Fricas [A]  time = 0.205841, size = 69, normalized size = 1.13 \[ \frac{2}{105} \,{\left (15 \, B b^{2} x^{3} + 105 \, A a^{2} + 21 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 35 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2/sqrt(x),x, algorithm="fricas")

[Out]

2/105*(15*B*b^2*x^3 + 105*A*a^2 + 21*(2*B*a*b + A*b^2)*x^2 + 35*(B*a^2 + 2*A*a*b
)*x)*sqrt(x)

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Sympy [A]  time = 7.0341, size = 78, normalized size = 1.28 \[ 2 A a^{2} \sqrt{x} + \frac{4 A a b x^{\frac{3}{2}}}{3} + \frac{2 A b^{2} x^{\frac{5}{2}}}{5} + \frac{2 B a^{2} x^{\frac{3}{2}}}{3} + \frac{4 B a b x^{\frac{5}{2}}}{5} + \frac{2 B b^{2} x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2*(B*x+A)/x**(1/2),x)

[Out]

2*A*a**2*sqrt(x) + 4*A*a*b*x**(3/2)/3 + 2*A*b**2*x**(5/2)/5 + 2*B*a**2*x**(3/2)/
3 + 4*B*a*b*x**(5/2)/5 + 2*B*b**2*x**(7/2)/7

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GIAC/XCAS [A]  time = 0.259753, size = 72, normalized size = 1.18 \[ \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{4}{5} \, B a b x^{\frac{5}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B a^{2} x^{\frac{3}{2}} + \frac{4}{3} \, A a b x^{\frac{3}{2}} + 2 \, A a^{2} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2/sqrt(x),x, algorithm="giac")

[Out]

2/7*B*b^2*x^(7/2) + 4/5*B*a*b*x^(5/2) + 2/5*A*b^2*x^(5/2) + 2/3*B*a^2*x^(3/2) +
4/3*A*a*b*x^(3/2) + 2*A*a^2*sqrt(x)