Optimal. Leaf size=227 \[ \frac{8 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}+\frac{4 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}-\frac{2 b c \sqrt{c^2 x^2+1} \left (3 c^2 d-2 e\right )}{15 d^2 \left (c^2 d-e\right )^2 \sqrt{d+e x^2}}-\frac{8 b \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2+1}}{c \sqrt{d+e x^2}}\right )}{15 d^3 \sqrt{e}}-\frac{b c \sqrt{c^2 x^2+1}}{15 d \left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.82047, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {192, 191, 5704, 12, 6715, 949, 78, 63, 217, 206} \[ \frac{8 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}+\frac{4 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}-\frac{2 b c \sqrt{c^2 x^2+1} \left (3 c^2 d-2 e\right )}{15 d^2 \left (c^2 d-e\right )^2 \sqrt{d+e x^2}}-\frac{8 b \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2+1}}{c \sqrt{d+e x^2}}\right )}{15 d^3 \sqrt{e}}-\frac{b c \sqrt{c^2 x^2+1}}{15 d \left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 192
Rule 191
Rule 5704
Rule 12
Rule 6715
Rule 949
Rule 78
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-(b c) \int \frac{x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )}{15 d^3 \sqrt{1+c^2 x^2} \left (d+e x^2\right )^{5/2}} \, dx\\ &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{(b c) \int \frac{x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )}{\sqrt{1+c^2 x^2} \left (d+e x^2\right )^{5/2}} \, dx}{15 d^3}\\ &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{1+c^2 x} (d+e x)^{5/2}} \, dx,x,x^2\right )}{30 d^3}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{15 d \left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{3 d \left (7 c^2 d-6 e\right )+12 \left (c^2 d-e\right ) e x}{\sqrt{1+c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{45 d^3 \left (c^2 d-e\right )}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{15 d \left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}-\frac{2 b c \left (3 c^2 d-2 e\right ) \sqrt{1+c^2 x^2}}{15 d^2 \left (c^2 d-e\right )^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{(4 b c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{15 d^3}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{15 d \left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}-\frac{2 b c \left (3 c^2 d-2 e\right ) \sqrt{1+c^2 x^2}}{15 d^2 \left (c^2 d-e\right )^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{e}{c^2}+\frac{e x^2}{c^2}}} \, dx,x,\sqrt{1+c^2 x^2}\right )}{15 c d^3}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{15 d \left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}-\frac{2 b c \left (3 c^2 d-2 e\right ) \sqrt{1+c^2 x^2}}{15 d^2 \left (c^2 d-e\right )^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{c^2}} \, dx,x,\frac{\sqrt{1+c^2 x^2}}{\sqrt{d+e x^2}}\right )}{15 c d^3}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{15 d \left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}-\frac{2 b c \left (3 c^2 d-2 e\right ) \sqrt{1+c^2 x^2}}{15 d^2 \left (c^2 d-e\right )^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \sinh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{8 b \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{1+c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{15 d^3 \sqrt{e}}\\ \end{align*}
Mathematica [C] time = 0.426648, size = 191, normalized size = 0.84 \[ \frac{a x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )-4 b c x^2 \sqrt{\frac{e x^2}{d}+1} \left (d+e x^2\right )^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-c^2 x^2,-\frac{e x^2}{d}\right )-\frac{b c d \sqrt{c^2 x^2+1} \left (d+e x^2\right ) \left (c^2 d \left (7 d+6 e x^2\right )-e \left (5 d+4 e x^2\right )\right )}{\left (e-c^2 d\right )^2}+b x \sinh ^{-1}(c x) \left (15 d^2+20 d e x^2+8 e^2 x^4\right )}{15 d^3 \left (d+e x^2\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.412, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\it Arcsinh} \left ( cx \right ) ) \left ( e{x}^{2}+d \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{15} \, a{\left (\frac{8 \, x}{\sqrt{e x^{2} + d} d^{3}} + \frac{4 \, x}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{2}} + \frac{3 \, x}{{\left (e x^{2} + d\right )}^{\frac{5}{2}} d}\right )} + b \int \frac{\log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{{\left (e x^{2} + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.29059, size = 2782, normalized size = 12.26 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsinh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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