Optimal. Leaf size=482 \[ \frac{4 a b d x \sqrt{c^2 d x^2+d}}{35 c^3 \sqrt{c^2 x^2+1}}-\frac{2 b c^3 d x^7 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt{c^2 x^2+1}}-\frac{16 b c d x^5 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt{c^2 x^2+1}}+\frac{1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{35} d x^4 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 b d x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{105 c \sqrt{c^2 x^2+1}}+\frac{d x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}-\frac{2 d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^4}+\frac{2 b^2 d \left (c^2 x^2+1\right )^3 \sqrt{c^2 d x^2+d}}{343 c^4}-\frac{38 b^2 d \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d}}{6125 c^4}-\frac{304 b^2 d \sqrt{c^2 d x^2+d}}{3675 c^4}-\frac{152 b^2 d \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}}{11025 c^4}+\frac{4 b^2 d x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{35 c^3 \sqrt{c^2 x^2+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.81108, antiderivative size = 482, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 14, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5744, 5742, 5758, 5717, 5653, 261, 5661, 266, 43, 14, 5730, 12, 446, 77} \[ \frac{4 a b d x \sqrt{c^2 d x^2+d}}{35 c^3 \sqrt{c^2 x^2+1}}-\frac{2 b c^3 d x^7 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt{c^2 x^2+1}}-\frac{16 b c d x^5 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt{c^2 x^2+1}}+\frac{1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{35} d x^4 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 b d x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{105 c \sqrt{c^2 x^2+1}}+\frac{d x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}-\frac{2 d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^4}+\frac{2 b^2 d \left (c^2 x^2+1\right )^3 \sqrt{c^2 d x^2+d}}{343 c^4}-\frac{38 b^2 d \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d}}{6125 c^4}-\frac{304 b^2 d \sqrt{c^2 d x^2+d}}{3675 c^4}-\frac{152 b^2 d \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}}{11025 c^4}+\frac{4 b^2 d x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{35 c^3 \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5744
Rule 5742
Rule 5758
Rule 5717
Rule 5653
Rule 261
Rule 5661
Rule 266
Rule 43
Rule 14
Rule 5730
Rule 12
Rule 446
Rule 77
Rubi steps
\begin{align*} \int x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} (3 d) \int x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{\left (2 b c d \sqrt{d+c^2 d x^2}\right ) \int x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{7 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 b c d x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{35 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^7 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt{1+c^2 x^2}}+\frac{3}{35} d x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (3 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{35 \sqrt{1+c^2 x^2}}-\frac{\left (6 b c d \sqrt{d+c^2 d x^2}\right ) \int x^4 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{35 \sqrt{1+c^2 x^2}}+\frac{\left (2 b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^5 \left (7+5 c^2 x^2\right )}{35 \sqrt{1+c^2 x^2}} \, dx}{7 \sqrt{1+c^2 x^2}}\\ &=-\frac{16 b c d x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^7 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt{1+c^2 x^2}}+\frac{d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{35 c^2 \sqrt{1+c^2 x^2}}-\frac{\left (2 b d \sqrt{d+c^2 d x^2}\right ) \int x^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{35 c \sqrt{1+c^2 x^2}}+\frac{\left (2 b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^5 \left (7+5 c^2 x^2\right )}{\sqrt{1+c^2 x^2}} \, dx}{245 \sqrt{1+c^2 x^2}}+\frac{\left (6 b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^5}{\sqrt{1+c^2 x^2}} \, dx}{175 \sqrt{1+c^2 x^2}}\\ &=-\frac{2 b d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c \sqrt{1+c^2 x^2}}-\frac{16 b c d x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^7 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt{1+c^2 x^2}}-\frac{2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^4}+\frac{d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (2 b^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^3}{\sqrt{1+c^2 x^2}} \, dx}{105 \sqrt{1+c^2 x^2}}+\frac{\left (4 b d \sqrt{d+c^2 d x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{35 c^3 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (7+5 c^2 x\right )}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{245 \sqrt{1+c^2 x^2}}+\frac{\left (3 b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{175 \sqrt{1+c^2 x^2}}\\ &=\frac{4 a b d x \sqrt{d+c^2 d x^2}}{35 c^3 \sqrt{1+c^2 x^2}}-\frac{2 b d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c \sqrt{1+c^2 x^2}}-\frac{16 b c d x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^7 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt{1+c^2 x^2}}-\frac{2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^4}+\frac{d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (b^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{105 \sqrt{1+c^2 x^2}}+\frac{\left (4 b^2 d \sqrt{d+c^2 d x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{35 c^3 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{2}{c^4 \sqrt{1+c^2 x}}+\frac{\sqrt{1+c^2 x}}{c^4}-\frac{8 \left (1+c^2 x\right )^{3/2}}{c^4}+\frac{5 \left (1+c^2 x\right )^{5/2}}{c^4}\right ) \, dx,x,x^2\right )}{245 \sqrt{1+c^2 x^2}}+\frac{\left (3 b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^4 \sqrt{1+c^2 x}}-\frac{2 \sqrt{1+c^2 x}}{c^4}+\frac{\left (1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{175 \sqrt{1+c^2 x^2}}\\ &=\frac{62 b^2 d \sqrt{d+c^2 d x^2}}{1225 c^4}+\frac{4 a b d x \sqrt{d+c^2 d x^2}}{35 c^3 \sqrt{1+c^2 x^2}}-\frac{74 b^2 d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{3675 c^4}-\frac{38 b^2 d \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2}}{6125 c^4}+\frac{2 b^2 d \left (1+c^2 x^2\right )^3 \sqrt{d+c^2 d x^2}}{343 c^4}+\frac{4 b^2 d x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{35 c^3 \sqrt{1+c^2 x^2}}-\frac{2 b d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c \sqrt{1+c^2 x^2}}-\frac{16 b c d x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^7 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt{1+c^2 x^2}}-\frac{2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^4}+\frac{d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (b^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2 \sqrt{1+c^2 x}}+\frac{\sqrt{1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{105 \sqrt{1+c^2 x^2}}-\frac{\left (4 b^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{35 c^2 \sqrt{1+c^2 x^2}}\\ &=-\frac{304 b^2 d \sqrt{d+c^2 d x^2}}{3675 c^4}+\frac{4 a b d x \sqrt{d+c^2 d x^2}}{35 c^3 \sqrt{1+c^2 x^2}}-\frac{152 b^2 d \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}}{11025 c^4}-\frac{38 b^2 d \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2}}{6125 c^4}+\frac{2 b^2 d \left (1+c^2 x^2\right )^3 \sqrt{d+c^2 d x^2}}{343 c^4}+\frac{4 b^2 d x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{35 c^3 \sqrt{1+c^2 x^2}}-\frac{2 b d x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c \sqrt{1+c^2 x^2}}-\frac{16 b c d x^5 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{175 \sqrt{1+c^2 x^2}}-\frac{2 b c^3 d x^7 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 \sqrt{1+c^2 x^2}}-\frac{2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^4}+\frac{d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end{align*}
Mathematica [A] time = 0.383811, size = 251, normalized size = 0.52 \[ \frac{d \sqrt{c^2 d x^2+d} \left (11025 a^2 \left (5 c^2 x^2-2\right ) \left (c^2 x^2+1\right )^3-210 a b c x \left (75 c^6 x^6+168 c^4 x^4+35 c^2 x^2-210\right ) \sqrt{c^2 x^2+1}-210 b \sinh ^{-1}(c x) \left (b c x \sqrt{c^2 x^2+1} \left (75 c^6 x^6+168 c^4 x^4+35 c^2 x^2-210\right )-105 a \left (c^2 x^2+1\right )^3 \left (5 c^2 x^2-2\right )\right )+2 b^2 \left (1125 c^8 x^8+3303 c^6 x^6+499 c^4 x^4-20371 c^2 x^2-18692\right )+11025 b^2 \left (5 c^2 x^2-2\right ) \left (c^2 x^2+1\right )^3 \sinh ^{-1}(c x)^2\right )}{385875 c^4 \left (c^2 x^2+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.38, size = 1766, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.98113, size = 942, normalized size = 1.95 \begin{align*} \frac{11025 \,{\left (5 \, b^{2} c^{8} d x^{8} + 13 \, b^{2} c^{6} d x^{6} + 9 \, b^{2} c^{4} d x^{4} - b^{2} c^{2} d x^{2} - 2 \, b^{2} d\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 210 \,{\left (525 \, a b c^{8} d x^{8} + 1365 \, a b c^{6} d x^{6} + 945 \, a b c^{4} d x^{4} - 105 \, a b c^{2} d x^{2} - 210 \, a b d -{\left (75 \, b^{2} c^{7} d x^{7} + 168 \, b^{2} c^{5} d x^{5} + 35 \, b^{2} c^{3} d x^{3} - 210 \, b^{2} c d x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (1125 \,{\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{8} d x^{8} + 9 \,{\left (15925 \, a^{2} + 734 \, b^{2}\right )} c^{6} d x^{6} +{\left (99225 \, a^{2} + 998 \, b^{2}\right )} c^{4} d x^{4} -{\left (11025 \, a^{2} + 40742 \, b^{2}\right )} c^{2} d x^{2} - 2 \,{\left (11025 \, a^{2} + 18692 \, b^{2}\right )} d - 210 \,{\left (75 \, a b c^{7} d x^{7} + 168 \, a b c^{5} d x^{5} + 35 \, a b c^{3} d x^{3} - 210 \, a b c d x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \sqrt{c^{2} d x^{2} + d}}{385875 \,{\left (c^{6} x^{2} + c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]