\(\int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx\) [110]

Optimal result

Integrand size = 27, antiderivative size = 140 \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {7}{8} a^3 x \sqrt {-\frac {a^2 c}{b^2}+c x^2}+\frac {b (a+b x)^2 \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}+\frac {7 a b (8 a+3 b x) \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{60 c}-\frac {7 a^5 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {-\frac {a^2 c}{b^2}+c x^2}}\right )}{8 b^2} \] Output:

7/8*a^3*x*(-a^2*c/b^2+c*x^2)^(1/2)+1/5*b*(b*x+a)^2*(-a^2*c/b^2+c*x^2)^(3/2 
)/c+7/60*a*b*(3*b*x+8*a)*(-a^2*c/b^2+c*x^2)^(3/2)/c-7/8*a^5*c^(1/2)*arctan 
h(c^(1/2)*x/(-a^2*c/b^2+c*x^2)^(1/2))/b^2
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {\sqrt {c \left (-\frac {a^2}{b^2}+x^2\right )} \left (b \sqrt {-\frac {a^2}{b^2}+x^2} \left (-136 a^4+15 a^3 b x+112 a^2 b^2 x^2+90 a b^3 x^3+24 b^4 x^4\right )+105 a^5 \log \left (-x+\sqrt {-\frac {a^2}{b^2}+x^2}\right )\right )}{120 b^2 \sqrt {-\frac {a^2}{b^2}+x^2}} \] Input:

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

Output:

(Sqrt[c*(-(a^2/b^2) + x^2)]*(b*Sqrt[-(a^2/b^2) + x^2]*(-136*a^4 + 15*a^3*b 
*x + 112*a^2*b^2*x^2 + 90*a*b^3*x^3 + 24*b^4*x^4) + 105*a^5*Log[-x + Sqrt[ 
-(a^2/b^2) + x^2]]))/(120*b^2*Sqrt[-(a^2/b^2) + x^2])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {469, 469, 455, 211, 224, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (a+b x)^3 \sqrt {c x^2-\frac {a^2 c}{b^2}} \, dx\)

\(\Big \downarrow \) 469

\(\displaystyle \frac {7}{5} a \int (a+b x)^2 \sqrt {c x^2-\frac {a^2 c}{b^2}}dx+\frac {b (a+b x)^2 \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{5 c}\)

\(\Big \downarrow \) 469

\(\displaystyle \frac {7}{5} a \left (\frac {5}{4} a \int (a+b x) \sqrt {c x^2-\frac {a^2 c}{b^2}}dx+\frac {b (a+b x) \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{4 c}\right )+\frac {b (a+b x)^2 \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{5 c}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {7}{5} a \left (\frac {5}{4} a \left (a \int \sqrt {c x^2-\frac {a^2 c}{b^2}}dx+\frac {b \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{3 c}\right )+\frac {b (a+b x) \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{4 c}\right )+\frac {b (a+b x)^2 \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{5 c}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {7}{5} a \left (\frac {5}{4} a \left (a \left (\frac {1}{2} x \sqrt {c x^2-\frac {a^2 c}{b^2}}-\frac {a^2 c \int \frac {1}{\sqrt {c x^2-\frac {a^2 c}{b^2}}}dx}{2 b^2}\right )+\frac {b \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{3 c}\right )+\frac {b (a+b x) \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{4 c}\right )+\frac {b (a+b x)^2 \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{5 c}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {7}{5} a \left (\frac {5}{4} a \left (a \left (\frac {1}{2} x \sqrt {c x^2-\frac {a^2 c}{b^2}}-\frac {a^2 c \int \frac {1}{1-\frac {c x^2}{c x^2-\frac {a^2 c}{b^2}}}d\frac {x}{\sqrt {c x^2-\frac {a^2 c}{b^2}}}}{2 b^2}\right )+\frac {b \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{3 c}\right )+\frac {b (a+b x) \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{4 c}\right )+\frac {b (a+b x)^2 \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{5 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {7}{5} a \left (\frac {5}{4} a \left (a \left (\frac {1}{2} x \sqrt {c x^2-\frac {a^2 c}{b^2}}-\frac {a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {c x^2-\frac {a^2 c}{b^2}}}\right )}{2 b^2}\right )+\frac {b \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{3 c}\right )+\frac {b (a+b x) \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{4 c}\right )+\frac {b (a+b x)^2 \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{5 c}\)

Input:

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

Output:

(b*(a + b*x)^2*(-((a^2*c)/b^2) + c*x^2)^(3/2))/(5*c) + (7*a*((b*(a + b*x)* 
(-((a^2*c)/b^2) + c*x^2)^(3/2))/(4*c) + (5*a*((b*(-((a^2*c)/b^2) + c*x^2)^ 
(3/2))/(3*c) + a*((x*Sqrt[-((a^2*c)/b^2) + c*x^2])/2 - (a^2*Sqrt[c]*ArcTan 
h[(Sqrt[c]*x)/Sqrt[-((a^2*c)/b^2) + c*x^2]])/(2*b^2))))/4))/5
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 219

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

rule 224

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]
 

rule 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 469

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* 
((n + p)/(n + 2*p + 1))   Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; Fr 
eeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[n, 0] && NeQ[n + 2* 
p + 1, 0] && IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.41

method	result	size
risch	\(-\frac {\left (-24 b^{4} x^{4}-90 a \,b^{3} x^{3}-112 a^{2} b^{2} x^{2}-15 a^{3} b x +136 a^{4}\right ) \sqrt {-\frac {c \left (-b^{2} x^{2}+a^{2}\right )}{b^{2}}}\, \sqrt {-c \left (-b^{2} x^{2}+a^{2}\right )}}{120 b \sqrt {c \left (b^{2} x^{2}-a^{2}\right )}}+\frac {7 a^{5} \ln \left (\frac {b^{2} c x}{\sqrt {b^{2} c}}+\sqrt {b^{2} c \,x^{2}-a^{2} c}\right ) \sqrt {-\frac {c \left (-b^{2} x^{2}+a^{2}\right )}{b^{2}}}\, \sqrt {-c \left (-b^{2} x^{2}+a^{2}\right )}}{8 \sqrt {b^{2} c}\, \left (-b^{2} x^{2}+a^{2}\right )}\)	\(197\)
default	\(a^{3} \left (\frac {x \sqrt {-\frac {a^{2} c}{b^{2}}+c \,x^{2}}}{2}-\frac {\sqrt {c}\, a^{2} \ln \left (\sqrt {c}\, x +\sqrt {-\frac {a^{2} c}{b^{2}}+c \,x^{2}}\right )}{2 b^{2}}\right )+b^{3} \left (\frac {x^{2} \left (-\frac {a^{2} c}{b^{2}}+c \,x^{2}\right )^{\frac {3}{2}}}{5 c}+\frac {2 a^{2} \left (-\frac {a^{2} c}{b^{2}}+c \,x^{2}\right )^{\frac {3}{2}}}{15 c \,b^{2}}\right )+3 a \,b^{2} \left (\frac {x \left (-\frac {a^{2} c}{b^{2}}+c \,x^{2}\right )^{\frac {3}{2}}}{4 c}+\frac {a^{2} \left (\frac {x \sqrt {-\frac {a^{2} c}{b^{2}}+c \,x^{2}}}{2}-\frac {\sqrt {c}\, a^{2} \ln \left (\sqrt {c}\, x +\sqrt {-\frac {a^{2} c}{b^{2}}+c \,x^{2}}\right )}{2 b^{2}}\right )}{4 b^{2}}\right )+\frac {a^{2} b {\left (-\frac {c \left (-b^{2} x^{2}+a^{2}\right )}{b^{2}}\right )}^{\frac {3}{2}}}{c}\)	\(242\)

Input:

int((b*x+a)^3*(-a^2*c/b^2+c*x^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/120*(-24*b^4*x^4-90*a*b^3*x^3-112*a^2*b^2*x^2-15*a^3*b*x+136*a^4)/b/(c* 
(b^2*x^2-a^2))^(1/2)*(-c*(-b^2*x^2+a^2)/b^2)^(1/2)*(-c*(-b^2*x^2+a^2))^(1/ 
2)+7/8*a^5*ln(b^2*c*x/(b^2*c)^(1/2)+(b^2*c*x^2-a^2*c)^(1/2))/(b^2*c)^(1/2) 
*(-c*(-b^2*x^2+a^2)/b^2)^(1/2)*(-c*(-b^2*x^2+a^2))^(1/2)/(-b^2*x^2+a^2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.86 \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\left [\frac {105 \, a^{5} \sqrt {c} \log \left (2 \, b^{2} c x^{2} - 2 \, b^{2} \sqrt {c} x \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}} - a^{2} c\right ) + 2 \, {\left (24 \, b^{5} x^{4} + 90 \, a b^{4} x^{3} + 112 \, a^{2} b^{3} x^{2} + 15 \, a^{3} b^{2} x - 136 \, a^{4} b\right )} \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}}}{240 \, b^{2}}, \frac {105 \, a^{5} \sqrt {-c} \arctan \left (\frac {b^{2} \sqrt {-c} x \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}}}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (24 \, b^{5} x^{4} + 90 \, a b^{4} x^{3} + 112 \, a^{2} b^{3} x^{2} + 15 \, a^{3} b^{2} x - 136 \, a^{4} b\right )} \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}}}{120 \, b^{2}}\right ] \] Input:

integrate((b*x+a)^3*(-a^2*c/b^2+c*x^2)^(1/2),x, algorithm="fricas")
 

Output:

[1/240*(105*a^5*sqrt(c)*log(2*b^2*c*x^2 - 2*b^2*sqrt(c)*x*sqrt((b^2*c*x^2 
- a^2*c)/b^2) - a^2*c) + 2*(24*b^5*x^4 + 90*a*b^4*x^3 + 112*a^2*b^3*x^2 + 
15*a^3*b^2*x - 136*a^4*b)*sqrt((b^2*c*x^2 - a^2*c)/b^2))/b^2, 1/120*(105*a 
^5*sqrt(-c)*arctan(b^2*sqrt(-c)*x*sqrt((b^2*c*x^2 - a^2*c)/b^2)/(b^2*c*x^2 
 - a^2*c)) + (24*b^5*x^4 + 90*a*b^4*x^3 + 112*a^2*b^3*x^2 + 15*a^3*b^2*x - 
 136*a^4*b)*sqrt((b^2*c*x^2 - a^2*c)/b^2))/b^2]
 

Sympy [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.14 \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\begin {cases} - \frac {7 a^{5} c \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {- \frac {a^{2} c}{b^{2}} + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {a^{2} c}{b^{2}} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right )}{8 b^{2}} + \sqrt {- \frac {a^{2} c}{b^{2}} + c x^{2}} \left (- \frac {17 a^{4}}{15 b} + \frac {a^{3} x}{8} + \frac {14 a^{2} b x^{2}}{15} + \frac {3 a b^{2} x^{3}}{4} + \frac {b^{3} x^{4}}{5}\right ) & \text {for}\: c \neq 0 \\\sqrt {- \frac {a^{2} c}{b^{2}}} \left (\begin {cases} a^{3} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{4}}{4 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((b*x+a)**3*(-a**2*c/b**2+c*x**2)**(1/2),x)
 

Output:

Piecewise((-7*a**5*c*Piecewise((log(2*sqrt(c)*sqrt(-a**2*c/b**2 + c*x**2) 
+ 2*c*x)/sqrt(c), Ne(a**2*c/b**2, 0)), (x*log(x)/sqrt(c*x**2), True))/(8*b 
**2) + sqrt(-a**2*c/b**2 + c*x**2)*(-17*a**4/(15*b) + a**3*x/8 + 14*a**2*b 
*x**2/15 + 3*a*b**2*x**3/4 + b**3*x**4/5), Ne(c, 0)), (sqrt(-a**2*c/b**2)* 
Piecewise((a**3*x, Eq(b, 0)), ((a + b*x)**4/(4*b), True)), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.03 \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {{\left (c x^{2} - \frac {a^{2} c}{b^{2}}\right )}^{\frac {3}{2}} b^{3} x^{2}}{5 \, c} + \frac {7}{8} \, \sqrt {c x^{2} - \frac {a^{2} c}{b^{2}}} a^{3} x + \frac {3 \, {\left (c x^{2} - \frac {a^{2} c}{b^{2}}\right )}^{\frac {3}{2}} a b^{2} x}{4 \, c} - \frac {7 \, a^{5} \sqrt {c} \log \left (2 \, c x + 2 \, \sqrt {c x^{2} - \frac {a^{2} c}{b^{2}}} \sqrt {c}\right )}{8 \, b^{2}} + \frac {17 \, {\left (c x^{2} - \frac {a^{2} c}{b^{2}}\right )}^{\frac {3}{2}} a^{2} b}{15 \, c} \] Input:

integrate((b*x+a)^3*(-a^2*c/b^2+c*x^2)^(1/2),x, algorithm="maxima")
 

Output:

1/5*(c*x^2 - a^2*c/b^2)^(3/2)*b^3*x^2/c + 7/8*sqrt(c*x^2 - a^2*c/b^2)*a^3* 
x + 3/4*(c*x^2 - a^2*c/b^2)^(3/2)*a*b^2*x/c - 7/8*a^5*sqrt(c)*log(2*c*x + 
2*sqrt(c*x^2 - a^2*c/b^2)*sqrt(c))/b^2 + 17/15*(c*x^2 - a^2*c/b^2)^(3/2)*a 
^2*b/c
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.81 \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {{\left (\frac {105 \, a^{5} \sqrt {c} \log \left ({\left | -\sqrt {b^{2} c} x + \sqrt {b^{2} c x^{2} - a^{2} c} \right |}\right )}{{\left | b \right |}} - \sqrt {b^{2} c x^{2} - a^{2} c} {\left (\frac {136 \, a^{4}}{b} - {\left (15 \, a^{3} + 2 \, {\left (56 \, a^{2} b + 3 \, {\left (4 \, b^{3} x + 15 \, a b^{2}\right )} x\right )} x\right )} x\right )}\right )} {\left | b \right |}}{120 \, b^{2}} \] Input:

integrate((b*x+a)^3*(-a^2*c/b^2+c*x^2)^(1/2),x, algorithm="giac")
 

Output:

1/120*(105*a^5*sqrt(c)*log(abs(-sqrt(b^2*c)*x + sqrt(b^2*c*x^2 - a^2*c)))/ 
abs(b) - sqrt(b^2*c*x^2 - a^2*c)*(136*a^4/b - (15*a^3 + 2*(56*a^2*b + 3*(4 
*b^3*x + 15*a*b^2)*x)*x)*x))*abs(b)/b^2
 

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\int \sqrt {c\,x^2-\frac {a^2\,c}{b^2}}\,{\left (a+b\,x\right )}^3 \,d x \] Input:

int((c*x^2 - (a^2*c)/b^2)^(1/2)*(a + b*x)^3,x)
 

Output:

int((c*x^2 - (a^2*c)/b^2)^(1/2)*(a + b*x)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04 \[ \int (a+b x)^3 \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {\sqrt {c}\, \left (-136 \sqrt {b^{2} x^{2}-a^{2}}\, a^{4}+15 \sqrt {b^{2} x^{2}-a^{2}}\, a^{3} b x +112 \sqrt {b^{2} x^{2}-a^{2}}\, a^{2} b^{2} x^{2}+90 \sqrt {b^{2} x^{2}-a^{2}}\, a \,b^{3} x^{3}+24 \sqrt {b^{2} x^{2}-a^{2}}\, b^{4} x^{4}-105 \,\mathrm {log}\left (\frac {\sqrt {b^{2} x^{2}-a^{2}}+b x}{a}\right ) a^{5}\right )}{120 b^{2}} \] Input:

int((b*x+a)^3*(-a^2*c/b^2+c*x^2)^(1/2),x)
 

Output:

(sqrt(c)*( - 136*sqrt( - a**2 + b**2*x**2)*a**4 + 15*sqrt( - a**2 + b**2*x 
**2)*a**3*b*x + 112*sqrt( - a**2 + b**2*x**2)*a**2*b**2*x**2 + 90*sqrt( - 
a**2 + b**2*x**2)*a*b**3*x**3 + 24*sqrt( - a**2 + b**2*x**2)*b**4*x**4 - 1 
05*log((sqrt( - a**2 + b**2*x**2) + b*x)/a)*a**5))/(120*b**2)