3.12.45 \(\int (a+b x)^{5/2} (a c-b c x)^{5/2} \, dx\) [1145]

Optimal. Leaf size=135 \[ \frac {5}{16} a^4 c^2 x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {5 a^6 c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{8 b} \]

[Out]

5/24*a^2*c*x*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)+1/6*x*(b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2)+5/8*a^6*c^(5/2)*arctan(c^
(1/2)*(b*x+a)^(1/2)/(c*(-b*x+a))^(1/2))/b+5/16*a^4*c^2*x*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {38, 65, 223, 209} \begin {gather*} \frac {5 a^6 c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{8 b}+\frac {5}{16} a^4 c^2 x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2),x]

[Out]

(5*a^4*c^2*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/16 + (5*a^2*c*x*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/24 + (x*(a
+ b*x)^(5/2)*(a*c - b*c*x)^(5/2))/6 + (5*a^6*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[c*(a - b*x)]])/(8*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 65

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 209

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

Rule 223

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)^{5/2} (a c-b c x)^{5/2} \, dx &=\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {1}{6} \left (5 a^2 c\right ) \int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx\\ &=\frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {1}{8} \left (5 a^4 c^2\right ) \int \sqrt {a+b x} \sqrt {a c-b c x} \, dx\\ &=\frac {5}{16} a^4 c^2 x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {1}{16} \left (5 a^6 c^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx\\ &=\frac {5}{16} a^4 c^2 x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {\left (5 a^6 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 a c-c x^2}} \, dx,x,\sqrt {a+b x}\right )}{8 b}\\ &=\frac {5}{16} a^4 c^2 x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {\left (5 a^6 c^3\right ) \text {Subst}\left (\int \frac {1}{1+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{8 b}\\ &=\frac {5}{16} a^4 c^2 x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac {1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac {5 a^6 c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{8 b}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 103, normalized size = 0.76 \begin {gather*} \frac {(c (a-b x))^{5/2} \left (b x \sqrt {a-b x} \sqrt {a+b x} \left (33 a^4-26 a^2 b^2 x^2+8 b^4 x^4\right )+30 a^6 \tan ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )\right )}{48 b (a-b x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2),x]

[Out]

((c*(a - b*x))^(5/2)*(b*x*Sqrt[a - b*x]*Sqrt[a + b*x]*(33*a^4 - 26*a^2*b^2*x^2 + 8*b^4*x^4) + 30*a^6*ArcTan[Sq
rt[a + b*x]/Sqrt[a - b*x]]))/(48*b*(a - b*x)^(5/2))

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2),x]')

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4846 deep

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(241\) vs. \(2(107)=214\).
time = 0.16, size = 242, normalized size = 1.79

method	result	size
risch	\(\frac {x \left (8 b^{4} x^{4}-26 a^{2} b^{2} x^{2}+33 a^{4}\right ) \sqrt {b x +a}\, \left (-b x +a \right ) c^{3}}{48 \sqrt {-c \left (b x -a \right )}}+\frac {5 a^{6} \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}\, c^{3}}{16 \sqrt {b^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(136\)
default	\(-\frac {\left (b x +a \right )^{\frac {5}{2}} \left (-b c x +a c \right )^{\frac {7}{2}}}{6 b c}+\frac {5 a \left (-\frac {\left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {7}{2}}}{5 b c}+\frac {3 a \left (-\frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{\frac {7}{2}}}{4 b c}+\frac {a \left (\frac {\left (-b c x +a c \right )^{\frac {5}{2}} \sqrt {b x +a}}{3 b}+\frac {5 a c \left (\frac {\left (-b c x +a c \right )^{\frac {3}{2}} \sqrt {b x +a}}{2 b}+\frac {3 a c \left (\frac {\sqrt {-b c x +a c}\, \sqrt {b x +a}}{b}+\frac {a c \sqrt {\left (b x +a \right ) \left (-b c x +a c \right )}\, \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right )}{\sqrt {-b c x +a c}\, \sqrt {b x +a}\, \sqrt {b^{2} c}}\right )}{2}\right )}{3}\right )}{4}\right )}{5}\right )}{6}\)	\(242\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/6/b/c*(b*x+a)^(5/2)*(-b*c*x+a*c)^(7/2)+5/6*a*(-1/5/b/c*(b*x+a)^(3/2)*(-b*c*x+a*c)^(7/2)+3/5*a*(-1/4/b/c*(b*
x+a)^(1/2)*(-b*c*x+a*c)^(7/2)+1/4*a*(1/3/b*(-b*c*x+a*c)^(5/2)*(b*x+a)^(1/2)+5/3*a*c*(1/2/b*(-b*c*x+a*c)^(3/2)*
(b*x+a)^(1/2)+3/2*a*c*(1/b*(-b*c*x+a*c)^(1/2)*(b*x+a)^(1/2)+a*c*((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)*x/(-b^2*c*x^2+a^2*c)^(1/2)))))))

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Maxima [A]
time = 0.36, size = 89, normalized size = 0.66 \begin {gather*} \frac {5 \, a^{6} c^{\frac {5}{2}} \arcsin \left (\frac {b x}{a}\right )}{16 \, b} + \frac {5}{16} \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{4} c^{2} x + \frac {5}{24} \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{2} c x + \frac {1}{6} \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2),x, algorithm="maxima")

[Out]

5/16*a^6*c^(5/2)*arcsin(b*x/a)/b + 5/16*sqrt(-b^2*c*x^2 + a^2*c)*a^4*c^2*x + 5/24*(-b^2*c*x^2 + a^2*c)^(3/2)*a
^2*c*x + 1/6*(-b^2*c*x^2 + a^2*c)^(5/2)*x

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Fricas [A]
time = 0.31, size = 232, normalized size = 1.72 \begin {gather*} \left [\frac {15 \, a^{6} \sqrt {-c} c^{2} \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 (8 \, b^{5} c^{2} x^{5} - 26 \, a^{2} b^{3} c^{2} x^{3} + 33 \, a^{4} b c^{2} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{96 \, b}, -\frac {15 \, a^{6} c^{\frac {5}{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 (8 \, b^{5} c^{2} x^{5} - 26 \, a^{2} b^{3} c^{2} x^{3} + 33 \, a^{4} b c^{2} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{48 \, b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2),x, algorithm="fricas")

[Out]

[1/96*(15*a^6*sqrt(-c)*c^2*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*(8*b
^5*c^2*x^5 - 26*a^2*b^3*c^2*x^3 + 33*a^4*b*c^2*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b, -1/48*(15*a^6*c^(5/2)*a
rctan(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(c)*x/(b^2*c*x^2 - a^2*c)) - (8*b^5*c^2*x^5 - 26*a^2*b^3*c^2*x^3
+ 33*a^4*b*c^2*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b]

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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 {5}{2}} \left (a + b x\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(-b*c*x+a*c)**(5/2),x)

[Out]

Integral((-c*(-a + b*x))**(5/2)*(a + b*x)**(5/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (109) = 218\).
time = 0.11, size = 951, normalized size = 7.04 \begin {gather*} \frac {2 a^{4} c^{2} \left (2 \left (\frac {1}{8} \sqrt {a+b x} \sqrt {a+b x}-\frac {12}{32} a\right ) \sqrt {a+b x} \sqrt {2 a c-c \left (a+b x\right )}+\frac {2 a^{2} c \ln \left |\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right |}{4 \sqrt {-c}}\right )+2 a^{5} c^{2} \left (\frac {1}{2} \sqrt {a+b x} \sqrt {2 a c-c \left (a+b x\right )}-\frac {2 a c \ln \left |\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right |}{2 \sqrt {-c}}\right )+2 c^{2} \left (2 \left (\left (\left (\left (\left (\frac {1}{24} \sqrt {a+b x} \sqrt {a+b x}-\frac {89994240}{348364800} a\right ) \sqrt {a+b x} \sqrt {a+b x}+\frac {232968960}{348364800} a^{2}\right ) \sqrt {a+b x} \sqrt {a+b x}-\frac {327317760}{348364800} a^{3}\right ) \sqrt {a+b x} \sqrt {a+b x}+\frac {270345600}{348364800} a^{4}\right ) \sqrt {a+b x} \sqrt {a+b x}-\frac {146966400}{348364800} a^{5}\right ) \sqrt {a+b x} \sqrt {2 a c-c \left (a+b x\right )}+\frac {10 a^{6} c \ln \left |\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right |}{32 \sqrt {-c}}\right )+2 a c^{2} \left (2 \left (\left (\left (\left (\frac {1}{20} \sqrt {a+b x} \sqrt {a+b x}-\frac {211680}{806400} a\right ) \sqrt {a+b x} \sqrt {a+b x}+\frac {446880}{806400} a^{2}\right ) \sqrt {a+b x} \sqrt {a+b x}-\frac {495600}{806400} a^{3}\right ) \sqrt {a+b x} \sqrt {a+b x}+\frac {327600}{806400} a^{4}\right ) \sqrt {a+b x} \sqrt {2 a c-c \left (a+b x\right )}-\frac {6 a^{5} c \ln \left |\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right |}{16 \sqrt {-c}}\right )-4 a^{2} c^{2} \left (2 \left (\left (\left (\frac {1}{16} \sqrt {a+b x} \sqrt {a+b x}-\frac {24960}{92160} a\right ) \sqrt {a+b x} \sqrt {a+b x}+\frac {41280}{92160} a^{2}\right ) \sqrt {a+b x} \sqrt {a+b x}-\frac {37440}{92160} a^{3}\right ) \sqrt {a+b x} \sqrt {2 a c-c \left (a+b x\right )}+\frac {6 a^{4} c \ln \left |\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right |}{16 \sqrt {-c}}\right )-4 a^{3} c^{2} \left (2 \left (\left (\frac {1}{12} \sqrt {a+b x} \sqrt {a+b x}-\frac {84}{288} a\right ) \sqrt {a+b x} \sqrt {a+b x}+\frac {108}{288} a^{2}\right ) \sqrt {a+b x} \sqrt {2 a c-c \left (a+b x\right )}-\frac {2 a^{3} c \ln \left |\sqrt {2 a c-c \left (a+b x\right )}-\sqrt {-c} \sqrt {a+b x}\right |}{4 \sqrt {-c}}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2),x)

[Out]

-1/240*(240*(2*a*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - sqrt(-(b*x + a)*c
+ 2*a*c)*sqrt(b*x + a))*a^5*c^2 - 120*(2*a^2*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))
/sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)*(b*x - 2*a))*a^4*c^2 - 80*(6*a^3*c*log(abs(-sqrt(b*x + a)
*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - ((2*b*x - 5*a)*(b*x + a) + 9*a^2)*sqrt(-(b*x + a)*c + 2*a*
c)*sqrt(b*x + a))*a^3*c^2 + 20*(18*a^4*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-
c) - (39*a^3 - (2*(3*b*x - 10*a)*(b*x + a) + 43*a^2)*(b*x + a))*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a))*a^2*
c^2 + 2*(90*a^5*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - (195*a^4 - (295*a^
3 - 2*(3*(4*b*x - 17*a)*(b*x + a) + 133*a^2)*(b*x + a))*(b*x + a))*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a))*a
*c^2 - (150*a^6*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - (405*a^5 - (745*a^
4 - 2*(451*a^3 - (4*(5*b*x - 26*a)*(b*x + a) + 321*a^2)*(b*x + a))*(b*x + a))*(b*x + a))*sqrt(-(b*x + a)*c + 2
*a*c)*sqrt(b*x + a))*c^2)/b

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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 )}^{5/2}\,{\left (a+b\,x\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c - b*c*x)^(5/2)*(a + b*x)^(5/2),x)

[Out]

int((a*c - b*c*x)^(5/2)*(a + b*x)^(5/2), x)

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