3.11.77 \(\int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx\)

Optimal. Leaf size=102 \[ \frac {3 a^4 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{4 b}+\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2} \]

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Rubi [A]  time = 0.04, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {38, 63, 217, 203} \begin {gather*} \frac {3 a^4 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{4 b}+\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2),x]

[Out]

(3*a^2*c*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/8 + (x*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/4 + (3*a^4*c^(3/2)*Arc
Tan[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[c*(a - b*x)]])/(4*b)

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1)
, x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx &=\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{4} \left (3 a^2 c\right ) \int \sqrt {a+b x} \sqrt {a c-b c x} \, dx\\ &=\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{8} \left (3 a^4 c^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx\\ &=\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {\left (3 a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 a c-c x^2}} \, dx,x,\sqrt {a+b x}\right )}{4 b}\\ &=\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {\left (3 a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{4 b}\\ &=\frac {3}{8} a^2 c x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {3 a^4 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{4 b}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 109, normalized size = 1.07 \begin {gather*} \frac {c^2 \left (-6 a^{9/2} \sqrt {a-b x} \sqrt {\frac {b x}{a}+1} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right )+5 a^4 b x-7 a^2 b^3 x^3+2 b^5 x^5\right )}{8 b \sqrt {a+b x} \sqrt {c (a-b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2),x]

[Out]

(c^2*(5*a^4*b*x - 7*a^2*b^3*x^3 + 2*b^5*x^5 - 6*a^(9/2)*Sqrt[a - b*x]*Sqrt[1 + (b*x)/a]*ArcSin[Sqrt[a - b*x]/(
Sqrt[2]*Sqrt[a])]))/(8*b*Sqrt[c*(a - b*x)]*Sqrt[a + b*x])

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IntegrateAlgebraic [A]  time = 0.24, size = 169, normalized size = 1.66 \begin {gather*} \frac {a^4 c^2 \sqrt {a c-b c x} \left (\frac {11 c^2 (a c-b c x)}{a+b x}-\frac {11 c (a c-b c x)^2}{(a+b x)^2}-\frac {3 (a c-b c x)^3}{(a+b x)^3}+3 c^3\right )}{4 b \sqrt {a+b x} \left (\frac {a c-b c x}{a+b x}+c\right )^4}-\frac {3 a^4 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {a c-b c x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2),x]

[Out]

(a^4*c^2*Sqrt[a*c - b*c*x]*(3*c^3 + (11*c^2*(a*c - b*c*x))/(a + b*x) - (11*c*(a*c - b*c*x)^2)/(a + b*x)^2 - (3
*(a*c - b*c*x)^3)/(a + b*x)^3))/(4*b*Sqrt[a + b*x]*(c + (a*c - b*c*x)/(a + b*x))^4) - (3*a^4*c^(3/2)*ArcTan[Sq
rt[a*c - b*c*x]/(Sqrt[c]*Sqrt[a + b*x])])/(4*b)

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fricas [A]  time = 1.43, size = 193, normalized size = 1.89 \begin {gather*} \left [\frac {3 \, a^{4} \sqrt {-c} c \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) - 2 \, {\left (2 \, b^{3} c x^{3} - 5 \, a^{2} b c x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{16 \, b}, -\frac {3 \, a^{4} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (2 \, b^{3} c x^{3} - 5 \, a^{2} b c x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{8 \, b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x, algorithm="fricas")

[Out]

[1/16*(3*a^4*sqrt(-c)*c*log(2*b^2*c*x^2 + 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(-c)*x - a^2*c) - 2*(2*b^3*
c*x^3 - 5*a^2*b*c*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b, -1/8*(3*a^4*c^(3/2)*arctan(sqrt(-b*c*x + a*c)*sqrt(b
*x + a)*b*sqrt(c)*x/(b^2*c*x^2 - a^2*c)) + (2*b^3*c*x^3 - 5*a^2*b*c*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.01, size = 185, normalized size = 1.81 \begin {gather*} \frac {3 \sqrt {\left (b x +a \right ) \left (-b c x +a c \right )}\, a^{4} c^{2} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right )}{8 \sqrt {-b c x +a c}\, \sqrt {b x +a}\, \sqrt {b^{2} c}}+\frac {3 \sqrt {-b c x +a c}\, \sqrt {b x +a}\, a^{3} c}{8 b}+\frac {\left (-b c x +a c \right )^{\frac {3}{2}} \sqrt {b x +a}\, a^{2}}{8 b}-\frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{\frac {5}{2}} a}{4 b c}-\frac {\left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {5}{2}}}{4 b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x)

[Out]

-1/4/b/c*(b*x+a)^(3/2)*(-b*c*x+a*c)^(5/2)-1/4*a/b/c*(b*x+a)^(1/2)*(-b*c*x+a*c)^(5/2)+1/8*a^2/b*(-b*c*x+a*c)^(3
/2)*(b*x+a)^(1/2)+3/8*a^3*c/b*(-b*c*x+a*c)^(1/2)*(b*x+a)^(1/2)+3/8*a^4*c^2*((b*x+a)*(-b*c*x+a*c))^(1/2)/(-b*c*
x+a*c)^(1/2)/(b*x+a)^(1/2)/(b^2*c)^(1/2)*arctan((b^2*c)^(1/2)/(-b^2*c*x^2+a^2*c)^(1/2)*x)

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maxima [A]  time = 3.02, size = 63, normalized size = 0.62 \begin {gather*} \frac {3 \, a^{4} c^{\frac {3}{2}} \arcsin \left (\frac {b x}{a}\right )}{8 \, b} + \frac {3}{8} \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{2} c x + \frac {1}{4} \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x, algorithm="maxima")

[Out]

3/8*a^4*c^(3/2)*arcsin(b*x/a)/b + 3/8*sqrt(-b^2*c*x^2 + a^2*c)*a^2*c*x + 1/4*(-b^2*c*x^2 + a^2*c)^(3/2)*x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a\,c-b\,c\,x\right )}^{3/2}\,{\left (a+b\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2),x)

[Out]

int((a*c - b*c*x)^(3/2)*(a + b*x)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- c \left (- a + b x\right )\right )^{\frac {3}{2}} \left (a + b x\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(-b*c*x+a*c)**(3/2),x)

[Out]

Integral((-c*(-a + b*x))**(3/2)*(a + b*x)**(3/2), x)

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