3.259 \(\int \frac {(a+b \sinh ^{-1}(c+d x))^4}{(c e+d e x)^{7/2}} \, dx\)

Optimal. Leaf size=82 \[ \frac {8 b \text {Int}\left (\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {(c+d x)^2+1} (e (c+d x))^{5/2}},x\right )}{5 e}-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{5 d e (e (c+d x))^{5/2}} \]

[Out]

-2/5*(a+b*arcsinh(d*x+c))^4/d/e/(e*(d*x+c))^(5/2)+8/5*b*Unintegrable((a+b*arcsinh(d*x+c))^3/(e*(d*x+c))^(5/2)/
(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 = {} \[ \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{(c e+d e x)^{7/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(7/2),x]

[Out]

(-2*(a + b*ArcSinh[c + d*x])^4)/(5*d*e*(e*(c + d*x))^(5/2)) + (8*b*Defer[Subst][Defer[Int][(a + b*ArcSinh[x])^
3/((e*x)^(5/2)*Sqrt[1 + x^2]), x], x, c + d*x])/(5*d*e)

Rubi steps

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

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Mathematica [A]  time = 112.48, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{(c e+d e x)^{7/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(7/2),x]

[Out]

Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(7/2), x]

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fricas [A]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{4} \operatorname {arsinh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arsinh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arsinh}\left (d x + c\right ) + a^{4}\right )} \sqrt {d e x + c e}}{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^(7/2),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^4*e^4*x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4*x + c^4*e^
4), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^(7/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^4/(d*e*x + c*e)^(7/2), x)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsinh \left (d x +c \right )\right )^{4}}{\left (d e x +c e \right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^(7/2),x)

[Out]

int((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^(7/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^(7/2),x, algorithm="maxima")

[Out]

-2/5*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^4*e^4*x^3 + 3*c*d^3*e^4*x
^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) - 2/5*a^4/((d*e*x + c*e)^(5/2)*d*e) + integrate(2/5*(2*((2*b^4*c^2*sqrt(e) +
5*(c^2*sqrt(e) + sqrt(e))*a*b^3 + (5*a*b^3*d^2*sqrt(e) + 2*b^4*d^2*sqrt(e))*x^2 + 2*(5*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) + (5*(c^3*sqrt(e) + c*sqrt(e))*a*b^3 + 2*(
c^3*sqrt(e) + c*sqrt(e))*b^4 + (5*a*b^3*d^3*sqrt(e) + 2*b^4*d^3*sqrt(e))*x^3 + 3*(5*a*b^3*c*d^2*sqrt(e) + 2*b^
4*c*d^2*sqrt(e))*x^2 + (5*(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 + 15*((a^2*b^2*d^2*sqrt(e)*x^2 + 2*a^2*b^2*c*d*sqr
t(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*sqrt(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 + 10*((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*s
qrt(e)*x^3 + 3*a^3*b*c*d^2*sqrt(e)*x^2 + (3*c^2*d*sqrt(e) + d*sqrt(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^7*e^4*x^7 + 7*c*d^6*e^4*x^6 + c^7*e^4
+ c^5*e^4 + (21*c^2*d^5*e^4 + d^5*e^4)*x^5 + 5*(7*c^3*d^4*e^4 + c*d^4*e^4)*x^4 + 5*(7*c^4*d^3*e^4 + 2*c^2*d^3*
e^4)*x^3 + (21*c^5*d^2*e^4 + 10*c^3*d^2*e^4)*x^2 + (7*c^6*d*e^4 + 5*c^4*d*e^4)*x + (d^6*e^4*x^6 + 6*c*d^5*e^4*
x^5 + c^6*e^4 + c^4*e^4 + (15*c^2*d^4*e^4 + d^4*e^4)*x^4 + 4*(5*c^3*d^3*e^4 + c*d^3*e^4)*x^3 + 3*(5*c^4*d^2*e^
4 + 2*c^2*d^2*e^4)*x^2 + 2*(3*c^5*d*e^4 + 2*c^3*d*e^4)*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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c + d*x))^4/(c*e + d*e*x)^(7/2),x)

[Out]

int((a + b*asinh(c + d*x))^4/(c*e + d*e*x)^(7/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(d*x+c))**4/(d*e*x+c*e)**(7/2),x)

[Out]

Timed out

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