3.250 \(\int \frac {(a+b \sinh ^{-1}(c+d x))^3}{(c e+d e x)^{5/2}} \, dx\)

Optimal. Leaf size=80 \[ \frac {2 b \text {Int}\left (\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{\sqrt {(c+d x)^2+1} (e (c+d x))^{3/2}},x\right )}{e}-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d e (e (c+d x))^{3/2}} \]

[Out]

-2/3*(a+b*arcsinh(d*x+c))^3/d/e/(e*(d*x+c))^(3/2)+2*b*Unintegrable((a+b*arcsinh(d*x+c))^2/(e*(d*x+c))^(3/2)/(1
+(d*x+c)^2)^(1/2),x)/e

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Rubi [A]  time = 0.21, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(5/2),x]

[Out]

(-2*(a + b*ArcSinh[c + d*x])^3)/(3*d*e*(e*(c + d*x))^(3/2)) + (2*b*Defer[Subst][Defer[Int][(a + b*ArcSinh[x])^
2/((e*x)^(3/2)*Sqrt[1 + x^2]), x], x, c + d*x])/(d*e)

Rubi steps

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

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Mathematica [A]  time = 23.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(5/2),x]

[Out]

Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(5/2), x]

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fricas [A]  time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \operatorname {arsinh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arsinh}\left (d x + c\right ) + a^{3}\right )} \sqrt {d e x + c e}}{d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(5/2),x, algorithm="fricas")

[Out]

integral((b^3*arcsinh(d*x + c)^3 + 3*a*b^2*arcsinh(d*x + c)^2 + 3*a^2*b*arcsinh(d*x + c) + a^3)*sqrt(d*e*x + c
*e)/(d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^3/(d*e*x + c*e)^(5/2), x)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsinh \left (d x +c \right )\right )^{3}}{\left (d e x +c e \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(5/2),x)

[Out]

int((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(5/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, b^{3} \sqrt {e} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3}}{3 \, {\left (d^{2} e^{3} x + c d e^{3}\right )} \sqrt {d x + c}} - \frac {2 \, a^{3}}{3 \, {\left (d e x + c e\right )}^{\frac {3}{2}} d e} + \int \frac {{\left (2 \, {\left (c^{3} \sqrt {e} + c \sqrt {e}\right )} b^{3} + {\left (3 \, a b^{2} d^{3} \sqrt {e} + 2 \, b^{3} d^{3} \sqrt {e}\right )} x^{3} + 3 \, {\left (a c^{3} \sqrt {e} + a c \sqrt {e}\right )} b^{2} + 3 \, {\left (3 \, a b^{2} c d^{2} \sqrt {e} + 2 \, b^{3} c d^{2} \sqrt {e}\right )} x^{2} + {\left (2 \, {\left (3 \, c^{2} d \sqrt {e} + d \sqrt {e}\right )} b^{3} + 3 \, {\left (3 \, a c^{2} d \sqrt {e} + a d \sqrt {e}\right )} b^{2}\right )} x + {\left (2 \, b^{3} c^{2} \sqrt {e} + 3 \, {\left (a c^{2} \sqrt {e} + a \sqrt {e}\right )} b^{2} + {\left (3 \, a b^{2} d^{2} \sqrt {e} + 2 \, b^{3} d^{2} \sqrt {e}\right )} x^{2} + 2 \, {\left (3 \, a b^{2} c d \sqrt {e} + 2 \, b^{3} c d \sqrt {e}\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 3 \, {\left (a^{2} b d^{3} \sqrt {e} x^{3} + 3 \, a^{2} b c d^{2} \sqrt {e} x^{2} + {\left (3 \, a^{2} c^{2} d \sqrt {e} + a^{2} d \sqrt {e}\right )} b x + {\left (a^{2} c^{3} \sqrt {e} + a^{2} c \sqrt {e}\right )} b + {\left (a^{2} b d^{2} \sqrt {e} x^{2} + 2 \, a^{2} b c d \sqrt {e} x + {\left (a^{2} c^{2} \sqrt {e} + a^{2} \sqrt {e}\right )} b\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )}{{\left (d^{4} e^{3} x^{4} + 4 \, c d^{3} e^{3} x^{3} + c^{4} e^{3} + c^{2} e^{3} + {\left (6 \, c^{2} d^{2} e^{3} + d^{2} e^{3}\right )} x^{2} + 2 \, {\left (2 \, c^{3} d e^{3} + c d e^{3}\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d x + c} + {\left (d^{5} e^{3} x^{5} + 5 \, c d^{4} e^{3} x^{4} + c^{5} e^{3} + c^{3} e^{3} + {\left (10 \, c^{2} d^{3} e^{3} + d^{3} e^{3}\right )} x^{3} + {\left (10 \, c^{3} d^{2} e^{3} + 3 \, c d^{2} e^{3}\right )} x^{2} + {\left (5 \, c^{4} d e^{3} + 3 \, c^{2} d e^{3}\right )} x\right )} \sqrt {d x + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(5/2),x, algorithm="maxima")

[Out]

-2/3*b^3*sqrt(e)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3/((d^2*e^3*x + c*d*e^3)*sqrt(d*x + c)) - 2/
3*a^3/((d*e*x + c*e)^(3/2)*d*e) + integrate(((2*(c^3*sqrt(e) + c*sqrt(e))*b^3 + (3*a*b^2*d^3*sqrt(e) + 2*b^3*d
^3*sqrt(e))*x^3 + 3*(a*c^3*sqrt(e) + a*c*sqrt(e))*b^2 + 3*(3*a*b^2*c*d^2*sqrt(e) + 2*b^3*c*d^2*sqrt(e))*x^2 +
(2*(3*c^2*d*sqrt(e) + d*sqrt(e))*b^3 + 3*(3*a*c^2*d*sqrt(e) + a*d*sqrt(e))*b^2)*x + (2*b^3*c^2*sqrt(e) + 3*(a*
c^2*sqrt(e) + a*sqrt(e))*b^2 + (3*a*b^2*d^2*sqrt(e) + 2*b^3*d^2*sqrt(e))*x^2 + 2*(3*a*b^2*c*d*sqrt(e) + 2*b^3*
c*d*sqrt(e))*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + 3*(a^2
*b*d^3*sqrt(e)*x^3 + 3*a^2*b*c*d^2*sqrt(e)*x^2 + (3*a^2*c^2*d*sqrt(e) + a^2*d*sqrt(e))*b*x + (a^2*c^3*sqrt(e)
+ a^2*c*sqrt(e))*b + (a^2*b*d^2*sqrt(e)*x^2 + 2*a^2*b*c*d*sqrt(e)*x + (a^2*c^2*sqrt(e) + a^2*sqrt(e))*b)*sqrt(
d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)))/((d^4*e^3*x^4 + 4*c*d^3*e^3*x^
3 + c^4*e^3 + c^2*e^3 + (6*c^2*d^2*e^3 + d^2*e^3)*x^2 + 2*(2*c^3*d*e^3 + c*d*e^3)*x)*sqrt(d^2*x^2 + 2*c*d*x +
c^2 + 1)*sqrt(d*x + c) + (d^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + c^5*e^3 + c^3*e^3 + (10*c^2*d^3*e^3 + d^3*e^3)*x^3 +
(10*c^3*d^2*e^3 + 3*c*d^2*e^3)*x^2 + (5*c^4*d*e^3 + 3*c^2*d*e^3)*x)*sqrt(d*x + c)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c + d*x))^3/(c*e + d*e*x)^(5/2),x)

[Out]

int((a + b*asinh(c + d*x))^3/(c*e + d*e*x)^(5/2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{3}}{\left (e \left (c + d x\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(d*x+c))**3/(d*e*x+c*e)**(5/2),x)

[Out]

Integral((a + b*asinh(c + d*x))**3/(e*(c + d*x))**(5/2), x)

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