### 3.818 $$\int (a+b x)^3 \sqrt{-\frac{a^2 c}{b^2}+c x^2} \, dx$$

Optimal. Leaf size=167 $\frac{7}{8} a^3 x \sqrt{c x^2-\frac{a^2 c}{b^2}}+\frac{7 a^2 b \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{12 c}+\frac{7 a b (a+b x) \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{20 c}+\frac{b (a+b x)^2 \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{5 c}-\frac{7 a^5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2-\frac{a^2 c}{b^2}}}\right )}{8 b^2}$

[Out]

(7*a^3*x*Sqrt[-((a^2*c)/b^2) + c*x^2])/8 + (7*a^2*b*(-((a^2*c)/b^2) + c*x^2)^(3/2))/(12*c) + (7*a*b*(a + b*x)*
(-((a^2*c)/b^2) + c*x^2)^(3/2))/(20*c) + (b*(a + b*x)^2*(-((a^2*c)/b^2) + c*x^2)^(3/2))/(5*c) - (7*a^5*Sqrt[c]
*ArcTanh[(Sqrt[c]*x)/Sqrt[-((a^2*c)/b^2) + c*x^2]])/(8*b^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0783444, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.185, Rules used = {671, 641, 195, 217, 206} $\frac{7}{8} a^3 x \sqrt{c x^2-\frac{a^2 c}{b^2}}+\frac{7 a^2 b \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{12 c}+\frac{7 a b (a+b x) \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{20 c}+\frac{b (a+b x)^2 \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{5 c}-\frac{7 a^5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2-\frac{a^2 c}{b^2}}}\right )}{8 b^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(7*a^3*x*Sqrt[-((a^2*c)/b^2) + c*x^2])/8 + (7*a^2*b*(-((a^2*c)/b^2) + c*x^2)^(3/2))/(12*c) + (7*a*b*(a + b*x)*
(-((a^2*c)/b^2) + c*x^2)^(3/2))/(20*c) + (b*(a + b*x)^2*(-((a^2*c)/b^2) + c*x^2)^(3/2))/(5*c) - (7*a^5*Sqrt[c]
*ArcTanh[(Sqrt[c]*x)/Sqrt[-((a^2*c)/b^2) + c*x^2]])/(8*b^2)

Rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*(m + p))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rule 641

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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]

Rule 206

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

Rubi steps

\begin{align*} \int (a+b x)^3 \sqrt{-\frac{a^2 c}{b^2}+c x^2} \, dx &=\frac{b (a+b x)^2 \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}+\frac{1}{5} (7 a) \int (a+b x)^2 \sqrt{-\frac{a^2 c}{b^2}+c x^2} \, dx\\ &=\frac{7 a b (a+b x) \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{20 c}+\frac{b (a+b x)^2 \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}+\frac{1}{4} \left (7 a^2\right ) \int (a+b x) \sqrt{-\frac{a^2 c}{b^2}+c x^2} \, dx\\ &=\frac{7 a^2 b \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{12 c}+\frac{7 a b (a+b x) \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{20 c}+\frac{b (a+b x)^2 \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}+\frac{1}{4} \left (7 a^3\right ) \int \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{7 a^2 b \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{12 c}+\frac{7 a b (a+b x) \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{20 c}+\frac{b (a+b x)^2 \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}-\frac{\left (7 a^5 c\right ) \int \frac{1}{\sqrt{-\frac{a^2 c}{b^2}+c x^2}} \, dx}{8 b^2}\\ &=\frac{7}{8} a^3 x \sqrt{-\frac{a^2 c}{b^2}+c x^2}+\frac{7 a^2 b \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{12 c}+\frac{7 a b (a+b x) \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{20 c}+\frac{b (a+b x)^2 \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}-\frac{\left (7 a^5 c\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{-\frac{a^2 c}{b^2}+c x^2}}\right )}{8 b^2}\\ &=\frac{7}{8} a^3 x \sqrt{-\frac{a^2 c}{b^2}+c x^2}+\frac{7 a^2 b \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{12 c}+\frac{7 a b (a+b x) \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{20 c}+\frac{b (a+b x)^2 \left (-\frac{a^2 c}{b^2}+c x^2\right )^{3/2}}{5 c}-\frac{7 a^5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{-\frac{a^2 c}{b^2}+c x^2}}\right )}{8 b^2}\\ \end{align*}

Mathematica [A]  time = 0.177469, size = 114, normalized size = 0.68 $\frac{\sqrt{c \left (x^2-\frac{a^2}{b^2}\right )} \left (\sqrt{1-\frac{b^2 x^2}{a^2}} \left (112 a^2 b^2 x^2+15 a^3 b x-136 a^4+90 a b^3 x^3+24 b^4 x^4\right )+105 a^4 \sin ^{-1}\left (\frac{b x}{a}\right )\right )}{120 b \sqrt{1-\frac{b^2 x^2}{a^2}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[c*(-(a^2/b^2) + x^2)]*(Sqrt[1 - (b^2*x^2)/a^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^4*ArcSin[(b*x)/a]))/(120*b*Sqrt[1 - (b^2*x^2)/a^2])

________________________________________________________________________________________

Maple [A]  time = 0.154, size = 169, normalized size = 1. \begin{align*}{\frac{{b}^{3}{x}^{2}}{5\,c} \left ( -{\frac{{a}^{2}c}{{b}^{2}}}+c{x}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{2\,b{a}^{2}}{15\,c} \left ( -{\frac{{a}^{2}c}{{b}^{2}}}+c{x}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{b}^{2}ax}{4\,c} \left ( -{\frac{{a}^{2}c}{{b}^{2}}}+c{x}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{7\,x{a}^{3}}{8}\sqrt{-{\frac{{a}^{2}c}{{b}^{2}}}+c{x}^{2}}}-{\frac{7\,{a}^{5}}{8\,{b}^{2}}\sqrt{c}\ln \left ( x\sqrt{c}+\sqrt{-{\frac{{a}^{2}c}{{b}^{2}}}+c{x}^{2}} \right ) }+{\frac{b{a}^{2}}{c} \left ({\frac{c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }{{b}^{2}}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.31945, size = 568, normalized size = 3.4 \begin{align*} \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 ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[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 [C]  time = 6.71576, size = 495, normalized size = 2.96 \begin{align*} - \frac{2 a^{4} \sqrt{- \frac{a^{2} c}{b^{2}} + c x^{2}}}{15 b} + a^{3} \left (\begin{cases} - \frac{a^{2} \sqrt{c} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{2 b^{2}} - \frac{a \sqrt{c} x}{2 b \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{b \sqrt{c} x^{3}}{2 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \frac{\left |{b^{2} x^{2}}\right |}{\left |{a^{2}}\right |} > 1 \\\frac{i a^{2} \sqrt{c} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{2 b^{2}} + \frac{i a \sqrt{c} x \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}}{2 b} & \text{otherwise} \end{cases}\right ) - \frac{a^{2} b x^{2} \sqrt{- \frac{a^{2} c}{b^{2}} + c x^{2}}}{15} + 3 a^{2} b \left (\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\left (- \frac{a^{2} c}{b^{2}} + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + 3 a b^{2} \left (\begin{cases} - \frac{a^{4} \sqrt{c} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{8 b^{4}} + \frac{a^{3} \sqrt{c} x}{8 b^{3} \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} - \frac{3 a \sqrt{c} x^{3}}{8 b \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{b \sqrt{c} x^{5}}{4 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \frac{\left |{b^{2} x^{2}}\right |}{\left |{a^{2}}\right |} > 1 \\\frac{i a^{4} \sqrt{c} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{8 b^{4}} - \frac{i a^{3} \sqrt{c} x}{8 b^{3} \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} + \frac{3 i a \sqrt{c} x^{3}}{8 b \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} - \frac{i b \sqrt{c} x^{5}}{4 a \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\right ) + \frac{b^{3} x^{4} \sqrt{- \frac{a^{2} c}{b^{2}} + c x^{2}}}{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**4*sqrt(-a**2*c/b**2 + c*x**2)/(15*b) + a**3*Piecewise((-a**2*sqrt(c)*acosh(b*x/a)/(2*b**2) - a*sqrt(c)*x
/(2*b*sqrt(-1 + b**2*x**2/a**2)) + b*sqrt(c)*x**3/(2*a*sqrt(-1 + b**2*x**2/a**2)), Abs(b**2*x**2)/Abs(a**2) >
1), (I*a**2*sqrt(c)*asin(b*x/a)/(2*b**2) + I*a*sqrt(c)*x*sqrt(1 - b**2*x**2/a**2)/(2*b), True)) - a**2*b*x**2*
sqrt(-a**2*c/b**2 + c*x**2)/15 + 3*a**2*b*Piecewise((0, Eq(c, 0)), ((-a**2*c/b**2 + c*x**2)**(3/2)/(3*c), True
)) + 3*a*b**2*Piecewise((-a**4*sqrt(c)*acosh(b*x/a)/(8*b**4) + a**3*sqrt(c)*x/(8*b**3*sqrt(-1 + b**2*x**2/a**2
)) - 3*a*sqrt(c)*x**3/(8*b*sqrt(-1 + b**2*x**2/a**2)) + b*sqrt(c)*x**5/(4*a*sqrt(-1 + b**2*x**2/a**2)), Abs(b*
*2*x**2)/Abs(a**2) > 1), (I*a**4*sqrt(c)*asin(b*x/a)/(8*b**4) - I*a**3*sqrt(c)*x/(8*b**3*sqrt(1 - b**2*x**2/a*
*2)) + 3*I*a*sqrt(c)*x**3/(8*b*sqrt(1 - b**2*x**2/a**2)) - I*b*sqrt(c)*x**5/(4*a*sqrt(1 - b**2*x**2/a**2)), Tr
ue)) + b**3*x**4*sqrt(-a**2*c/b**2 + c*x**2)/5

________________________________________________________________________________________

Giac [A]  time = 1.31103, size = 153, normalized size = 0.92 \begin{align*} \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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*(13
6*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