3.2.57 \(\int \frac {(A+B x) (b x+c x^2)^2}{x^{9/2}} \, dx\) [157]

Optimal. Leaf size=59 \[ -\frac {2 A b^2}{3 x^{3/2}}-\frac {2 b (b B+2 A c)}{\sqrt {x}}+2 c (2 b B+A c) \sqrt {x}+\frac {2}{3} B c^2 x^{3/2} \]

[Out]

-2/3*A*b^2/x^(3/2)+2/3*B*c^2*x^(3/2)-2*b*(2*A*c+B*b)/x^(1/2)+2*c*(A*c+2*B*b)*x^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {779} \begin {gather*} -\frac {2 A b^2}{3 x^{3/2}}-\frac {2 b (2 A c+b B)}{\sqrt {x}}+2 c \sqrt {x} (A c+2 b B)+\frac {2}{3} B c^2 x^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/x^(9/2),x]

[Out]

(-2*A*b^2)/(3*x^(3/2)) - (2*b*(b*B + 2*A*c))/Sqrt[x] + 2*c*(2*b*B + A*c)*Sqrt[x] + (2*B*c^2*x^(3/2))/3

Rule 779

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^2}{x^{9/2}} \, dx &=\int \left (\frac {A b^2}{x^{5/2}}+\frac {b (b B+2 A c)}{x^{3/2}}+\frac {c (2 b B+A c)}{\sqrt {x}}+B c^2 \sqrt {x}\right ) \, dx\\ &=-\frac {2 A b^2}{3 x^{3/2}}-\frac {2 b (b B+2 A c)}{\sqrt {x}}+2 c (2 b B+A c) \sqrt {x}+\frac {2}{3} B c^2 x^{3/2}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 54, normalized size = 0.92 \begin {gather*} -\frac {2 \left (A b^2+3 b^2 B x+6 A b c x-6 b B c x^2-3 A c^2 x^2-B c^2 x^3\right )}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^(9/2),x]

[Out]

(-2*(A*b^2 + 3*b^2*B*x + 6*A*b*c*x - 6*b*B*c*x^2 - 3*A*c^2*x^2 - B*c^2*x^3))/(3*x^(3/2))

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Maple [A]
time = 0.49, size = 51, normalized size = 0.86

method	result	size
gosper	\(-\frac {2 \left (-B \,c^{2} x^{3}-3 A \,c^{2} x^{2}-6 b B \,x^{2} c +6 A b c x +3 b^{2} B x +b^{2} A \right )}{3 x^{\frac {3}{2}}}\)	\(51\)
derivativedivides	\(\frac {2 B \,c^{2} x^{\frac {3}{2}}}{3}+2 A \,c^{2} \sqrt {x}+4 b B c \sqrt {x}-\frac {2 A \,b^{2}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (2 A c +B b \right )}{\sqrt {x}}\)	\(51\)
default	\(\frac {2 B \,c^{2} x^{\frac {3}{2}}}{3}+2 A \,c^{2} \sqrt {x}+4 b B c \sqrt {x}-\frac {2 A \,b^{2}}{3 x^{\frac {3}{2}}}-\frac {2 b \left (2 A c +B b \right )}{\sqrt {x}}\)	\(51\)
trager	\(-\frac {2 \left (-B \,c^{2} x^{3}-3 A \,c^{2} x^{2}-6 b B \,x^{2} c +6 A b c x +3 b^{2} B x +b^{2} A \right )}{3 x^{\frac {3}{2}}}\)	\(51\)
risch	\(-\frac {2 \left (-B \,c^{2} x^{3}-3 A \,c^{2} x^{2}-6 b B \,x^{2} c +6 A b c x +3 b^{2} B x +b^{2} A \right )}{3 x^{\frac {3}{2}}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/x^(9/2),x,method=_RETURNVERBOSE)

[Out]

2/3*B*c^2*x^(3/2)+2*A*c^2*x^(1/2)+4*b*B*c*x^(1/2)-2/3*A*b^2/x^(3/2)-2*b*(2*A*c+B*b)/x^(1/2)

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Maxima [A]
time = 0.28, size = 51, normalized size = 0.86 \begin {gather*} \frac {2}{3} \, B c^{2} x^{\frac {3}{2}} + 2 \, {\left (2 \, B b c + A c^{2}\right )} \sqrt {x} - \frac {2 \, {\left (A b^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^(9/2),x, algorithm="maxima")

[Out]

2/3*B*c^2*x^(3/2) + 2*(2*B*b*c + A*c^2)*sqrt(x) - 2/3*(A*b^2 + 3*(B*b^2 + 2*A*b*c)*x)/x^(3/2)

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Fricas [A]
time = 3.23, size = 50, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (B c^{2} x^{3} - A b^{2} + 3 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} - 3 \, {\left (B b^{2} + 2 \, A b c\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^(9/2),x, algorithm="fricas")

[Out]

2/3*(B*c^2*x^3 - A*b^2 + 3*(2*B*b*c + A*c^2)*x^2 - 3*(B*b^2 + 2*A*b*c)*x)/x^(3/2)

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Sympy [A]
time = 0.57, size = 73, normalized size = 1.24 \begin {gather*} - \frac {2 A b^{2}}{3 x^{\frac {3}{2}}} - \frac {4 A b c}{\sqrt {x}} + 2 A c^{2} \sqrt {x} - \frac {2 B b^{2}}{\sqrt {x}} + 4 B b c \sqrt {x} + \frac {2 B c^{2} x^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**2/x**(9/2),x)

[Out]

-2*A*b**2/(3*x**(3/2)) - 4*A*b*c/sqrt(x) + 2*A*c**2*sqrt(x) - 2*B*b**2/sqrt(x) + 4*B*b*c*sqrt(x) + 2*B*c**2*x*
*(3/2)/3

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Giac [A]
time = 0.64, size = 51, normalized size = 0.86 \begin {gather*} \frac {2}{3} \, B c^{2} x^{\frac {3}{2}} + 4 \, B b c \sqrt {x} + 2 \, A c^{2} \sqrt {x} - \frac {2 \, {\left (3 \, B b^{2} x + 6 \, A b c x + A b^{2}\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^(9/2),x, algorithm="giac")

[Out]

2/3*B*c^2*x^(3/2) + 4*B*b*c*sqrt(x) + 2*A*c^2*sqrt(x) - 2/3*(3*B*b^2*x + 6*A*b*c*x + A*b^2)/x^(3/2)

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Mupad [B]
time = 0.05, size = 51, normalized size = 0.86 \begin {gather*} -\frac {6\,B\,b^2\,x+2\,A\,b^2-12\,B\,b\,c\,x^2+12\,A\,b\,c\,x-2\,B\,c^2\,x^3-6\,A\,c^2\,x^2}{3\,x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^2*(A + B*x))/x^(9/2),x)

[Out]

-(2*A*b^2 - 6*A*c^2*x^2 - 2*B*c^2*x^3 + 6*B*b^2*x - 12*B*b*c*x^2 + 12*A*b*c*x)/(3*x^(3/2))

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