3.257 \(\int \frac {(a+b \sinh ^{-1}(c+d x))^4}{(c e+d e x)^{3/2}} \, dx\)

Optimal. Leaf size=78 \[ \frac {8 b \text {Int}\left (\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {(c+d x)^2+1} \sqrt {e (c+d x)}},x\right )}{e}-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d e \sqrt {e (c+d x)}} \]

[Out]

-2*(a+b*arcsinh(d*x+c))^4/d/e/(e*(d*x+c))^(1/2)+8*b*Unintegrable((a+b*arcsinh(d*x+c))^3/(e*(d*x+c))^(1/2)/(1+(
d*x+c)^2)^(1/2),x)/e

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Rubi [A]  time = 0.19, 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 = {} \[ \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{(c e+d e x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(3/2),x]

[Out]

(-2*(a + b*ArcSinh[c + d*x])^4)/(d*e*Sqrt[e*(c + d*x)]) + (8*b*Defer[Subst][Defer[Int][(a + b*ArcSinh[x])^3/(S
qrt[e*x]*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 )^4}{(c e+d e x)^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^4}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d e \sqrt {e (c+d x)}}+\frac {(8 b) \operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {e x} \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e}\\ \end {align*}

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Mathematica [A]  time = 37.76, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{(c e+d e x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(3/2),x]

[Out]

Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(3/2), x]

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fricas [A]  time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{4} \operatorname {arsinh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arsinh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arsinh}\left (d x + c\right ) + a^{4}\right )} \sqrt {d e x + c e}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^(3/2),x, algorithm="fricas")

[Out]

integral((b^4*arcsinh(d*x + c)^4 + 4*a*b^3*arcsinh(d*x + c)^3 + 6*a^2*b^2*arcsinh(d*x + c)^2 + 4*a^3*b*arcsinh
(d*x + c) + a^4)*sqrt(d*e*x + c*e)/(d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^4/(d*e*x + c*e)^(3/2), x)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsinh \left (d x +c \right )\right )^{4}}{\left (d e x +c e \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^(3/2),x)

[Out]

int((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c + d*x))^4/(c*e + d*e*x)^(3/2),x)

[Out]

int((a + b*asinh(c + d*x))^4/(c*e + d*e*x)^(3/2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{4}}{\left (e \left (c + d x\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(d*x+c))**4/(d*e*x+c*e)**(3/2),x)

[Out]

Integral((a + b*asinh(c + d*x))**4/(e*(c + d*x))**(3/2), x)

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