### 3.537 $$\int (d+e x) (a+c x^2)^{3/2} \, dx$$

Optimal. Leaf size=87 $\frac{3 a^2 d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{4} d x \left (a+c x^2\right )^{3/2}+\frac{3}{8} a d x \sqrt{a+c x^2}+\frac{e \left (a+c x^2\right )^{5/2}}{5 c}$

[Out]

(3*a*d*x*Sqrt[a + c*x^2])/8 + (d*x*(a + c*x^2)^(3/2))/4 + (e*(a + c*x^2)^(5/2))/(5*c) + (3*a^2*d*ArcTanh[(Sqrt
[c]*x)/Sqrt[a + c*x^2]])/(8*Sqrt[c])

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Rubi [A]  time = 0.0253526, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.235, Rules used = {641, 195, 217, 206} $\frac{3 a^2 d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c}}+\frac{1}{4} d x \left (a+c x^2\right )^{3/2}+\frac{3}{8} a d x \sqrt{a+c x^2}+\frac{e \left (a+c x^2\right )^{5/2}}{5 c}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*(a + c*x^2)^(3/2),x]

[Out]

(3*a*d*x*Sqrt[a + c*x^2])/8 + (d*x*(a + c*x^2)^(3/2))/4 + (e*(a + c*x^2)^(5/2))/(5*c) + (3*a^2*d*ArcTanh[(Sqrt
[c]*x)/Sqrt[a + c*x^2]])/(8*Sqrt[c])

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

Mathematica [A]  time = 0.0515484, size = 88, normalized size = 1.01 $\frac{\sqrt{a+c x^2} \left (8 a^2 e+a c x (25 d+16 e x)+2 c^2 x^3 (5 d+4 e x)\right )+15 a^2 \sqrt{c} d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{40 c}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)*(a + c*x^2)^(3/2),x]

[Out]

(Sqrt[a + c*x^2]*(8*a^2*e + 2*c^2*x^3*(5*d + 4*e*x) + a*c*x*(25*d + 16*e*x)) + 15*a^2*Sqrt[c]*d*Log[c*x + Sqrt
[c]*Sqrt[a + c*x^2]])/(40*c)

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Maple [A]  time = 0.046, size = 69, normalized size = 0.8 \begin{align*}{\frac{e}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{dx}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,adx}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,{a}^{2}d}{8}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(c*x^2+a)^(3/2),x)

[Out]

1/5*e*(c*x^2+a)^(5/2)/c+1/4*d*x*(c*x^2+a)^(3/2)+3/8*a*d*x*(c*x^2+a)^(1/2)+3/8*d*a^2/c^(1/2)*ln(x*c^(1/2)+(c*x^
2+a)^(1/2))

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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((e*x+d)*(c*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.15479, size = 425, normalized size = 4.89 \begin{align*} \left [\frac{15 \, a^{2} \sqrt{c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (8 \, c^{2} e x^{4} + 10 \, c^{2} d x^{3} + 16 \, a c e x^{2} + 25 \, a c d x + 8 \, a^{2} e\right )} \sqrt{c x^{2} + a}}{80 \, c}, -\frac{15 \, a^{2} \sqrt{-c} d \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (8 \, c^{2} e x^{4} + 10 \, c^{2} d x^{3} + 16 \, a c e x^{2} + 25 \, a c d x + 8 \, a^{2} e\right )} \sqrt{c x^{2} + a}}{40 \, c}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/80*(15*a^2*sqrt(c)*d*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(8*c^2*e*x^4 + 10*c^2*d*x^3 + 16*a
*c*e*x^2 + 25*a*c*d*x + 8*a^2*e)*sqrt(c*x^2 + a))/c, -1/40*(15*a^2*sqrt(-c)*d*arctan(sqrt(-c)*x/sqrt(c*x^2 + a
)) - (8*c^2*e*x^4 + 10*c^2*d*x^3 + 16*a*c*e*x^2 + 25*a*c*d*x + 8*a^2*e)*sqrt(c*x^2 + a))/c]

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Sympy [A]  time = 6.85815, size = 219, normalized size = 2.52 \begin{align*} \frac{a^{\frac{3}{2}} d x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{a^{\frac{3}{2}} d x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 \sqrt{a} c d x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 \sqrt{c}} + a e \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + c e \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + \frac{c^{2} d x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x**2+a)**(3/2),x)

[Out]

a**(3/2)*d*x*sqrt(1 + c*x**2/a)/2 + a**(3/2)*d*x/(8*sqrt(1 + c*x**2/a)) + 3*sqrt(a)*c*d*x**3/(8*sqrt(1 + c*x**
2/a)) + 3*a**2*d*asinh(sqrt(c)*x/sqrt(a))/(8*sqrt(c)) + a*e*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2
)**(3/2)/(3*c), True)) + c*e*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) +
x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + c**2*d*x**5/(4*sqrt(a)*sqrt(1 + c*x**2/a))

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Giac [A]  time = 1.24093, size = 107, normalized size = 1.23 \begin{align*} -\frac{3 \, a^{2} d \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, \sqrt{c}} + \frac{1}{40} \, \sqrt{c x^{2} + a}{\left ({\left (25 \, a d + 2 \,{\left ({\left (4 \, c x e + 5 \, c d\right )} x + 8 \, a e\right )} x\right )} x + \frac{8 \, a^{2} e}{c}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*a^2*d*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/sqrt(c) + 1/40*sqrt(c*x^2 + a)*((25*a*d + 2*((4*c*x*e + 5*c*
d)*x + 8*a*e)*x)*x + 8*a^2*e/c)