Optimal. Leaf size=129 \[ -\frac{8 b^3 (d+e x)^{11/2} (b d-a e)}{11 e^5}+\frac{4 b^2 (d+e x)^{9/2} (b d-a e)^2}{3 e^5}-\frac{8 b (d+e x)^{7/2} (b d-a e)^3}{7 e^5}+\frac{2 (d+e x)^{5/2} (b d-a e)^4}{5 e^5}+\frac{2 b^4 (d+e x)^{13/2}}{13 e^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0428612, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {27, 43} \[ -\frac{8 b^3 (d+e x)^{11/2} (b d-a e)}{11 e^5}+\frac{4 b^2 (d+e x)^{9/2} (b d-a e)^2}{3 e^5}-\frac{8 b (d+e x)^{7/2} (b d-a e)^3}{7 e^5}+\frac{2 (d+e x)^{5/2} (b d-a e)^4}{5 e^5}+\frac{2 b^4 (d+e x)^{13/2}}{13 e^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^{3/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^4 (d+e x)^{3/2}}{e^4}-\frac{4 b (b d-a e)^3 (d+e x)^{5/2}}{e^4}+\frac{6 b^2 (b d-a e)^2 (d+e x)^{7/2}}{e^4}-\frac{4 b^3 (b d-a e) (d+e x)^{9/2}}{e^4}+\frac{b^4 (d+e x)^{11/2}}{e^4}\right ) \, dx\\ &=\frac{2 (b d-a e)^4 (d+e x)^{5/2}}{5 e^5}-\frac{8 b (b d-a e)^3 (d+e x)^{7/2}}{7 e^5}+\frac{4 b^2 (b d-a e)^2 (d+e x)^{9/2}}{3 e^5}-\frac{8 b^3 (b d-a e) (d+e x)^{11/2}}{11 e^5}+\frac{2 b^4 (d+e x)^{13/2}}{13 e^5}\\ \end{align*}
Mathematica [A] time = 0.0845709, size = 101, normalized size = 0.78 \[ \frac{2 (d+e x)^{5/2} \left (10010 b^2 (d+e x)^2 (b d-a e)^2-5460 b^3 (d+e x)^3 (b d-a e)-8580 b (d+e x) (b d-a e)^3+3003 (b d-a e)^4+1155 b^4 (d+e x)^4\right )}{15015 e^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 186, normalized size = 1.4 \begin{align*}{\frac{2310\,{x}^{4}{b}^{4}{e}^{4}+10920\,{x}^{3}a{b}^{3}{e}^{4}-1680\,{x}^{3}{b}^{4}d{e}^{3}+20020\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-7280\,{x}^{2}a{b}^{3}d{e}^{3}+1120\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+17160\,x{a}^{3}b{e}^{4}-11440\,x{a}^{2}{b}^{2}d{e}^{3}+4160\,xa{b}^{3}{d}^{2}{e}^{2}-640\,x{b}^{4}{d}^{3}e+6006\,{a}^{4}{e}^{4}-6864\,{a}^{3}bd{e}^{3}+4576\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-1664\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{15015\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.10382, size = 244, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (1155 \,{\left (e x + d\right )}^{\frac{13}{2}} b^{4} - 5460 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 10010 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 8580 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 3003 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{15015 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.49044, size = 702, normalized size = 5.44 \begin{align*} \frac{2 \,{\left (1155 \, b^{4} e^{6} x^{6} + 128 \, b^{4} d^{6} - 832 \, a b^{3} d^{5} e + 2288 \, a^{2} b^{2} d^{4} e^{2} - 3432 \, a^{3} b d^{3} e^{3} + 3003 \, a^{4} d^{2} e^{4} + 210 \,{\left (7 \, b^{4} d e^{5} + 26 \, a b^{3} e^{6}\right )} x^{5} + 35 \,{\left (b^{4} d^{2} e^{4} + 208 \, a b^{3} d e^{5} + 286 \, a^{2} b^{2} e^{6}\right )} x^{4} - 20 \,{\left (2 \, b^{4} d^{3} e^{3} - 13 \, a b^{3} d^{2} e^{4} - 715 \, a^{2} b^{2} d e^{5} - 429 \, a^{3} b e^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{4} d^{4} e^{2} - 104 \, a b^{3} d^{3} e^{3} + 286 \, a^{2} b^{2} d^{2} e^{4} + 4576 \, a^{3} b d e^{5} + 1001 \, a^{4} e^{6}\right )} x^{2} - 2 \,{\left (32 \, b^{4} d^{5} e - 208 \, a b^{3} d^{4} e^{2} + 572 \, a^{2} b^{2} d^{3} e^{3} - 858 \, a^{3} b d^{2} e^{4} - 3003 \, a^{4} d e^{5}\right )} x\right )} \sqrt{e x + d}}{15015 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 18.9131, size = 559, normalized size = 4.33 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.25953, size = 675, normalized size = 5.23 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]