3.30 \(\int (d+e x)^m (a+b \sinh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=75 \[ \frac {(d+e x)^{m+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{e (m+1)}-\frac {2 b c \text {Int}\left (\frac {(d+e x)^{m+1} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}},x\right )}{e (m+1)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.26, 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 = {} \[ \int (d+e x)^m \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 4.17, size = 0, normalized size = 0.00 \[ \int (d+e x)^m \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}\right )} {\left (e x + d\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)*(e*x + d)^m, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arcsinh(c*x) + a)^2*(e*x + d)^m, x)

________________________________________________________________________________________

maple [A]  time = 3.60, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{m} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(a+b*arcsinh(c*x))^2,x)

[Out]

int((e*x+d)^m*(a+b*arcsinh(c*x))^2,x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c*x))^2*(d + e*x)^m,x)

[Out]

int((a + b*asinh(c*x))^2*(d + e*x)^m, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(a+b*asinh(c*x))**2,x)

[Out]

Integral((a + b*asinh(c*x))**2*(d + e*x)**m, x)

________________________________________________________________________________________