Optimal. Leaf size=259 \[ \frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B) (b d-a e)^3}{4 b^5}+\frac{B e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.322227, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 77} \[ \frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B) (b d-a e)^3}{4 b^5}+\frac{B e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 77
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^3 \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{(A b-a B) (b d-a e)^3 \left (a b+b^2 x\right )^3}{b^4}+\frac{(b d-a e)^2 (b B d+3 A b e-4 a B e) \left (a b+b^2 x\right )^4}{b^5}+\frac{3 e (b d-a e) (b B d+A b e-2 a B e) \left (a b+b^2 x\right )^5}{b^6}+\frac{e^2 (3 b B d+A b e-4 a B e) \left (a b+b^2 x\right )^6}{b^7}+\frac{B e^3 \left (a b+b^2 x\right )^7}{b^8}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{(A b-a B) (b d-a e)^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b^5}+\frac{(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{5 b^5}+\frac{e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{2 b^5}+\frac{e^2 (3 b B d+A b e-4 a B e) (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac{B e^3 (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{8 b^5}\\ \end{align*}
Mathematica [A] time = 0.164407, size = 320, normalized size = 1.24 \[ \frac{x \sqrt{(a+b x)^2} \left (14 a^2 b x \left (3 A \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )\right )+14 a^3 \left (5 A \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+B x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )\right )+2 a b^2 x^2 \left (7 A \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+3 B x \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )\right )+b^3 x^3 \left (2 A \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )+B x \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )\right )\right )}{280 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.007, size = 428, normalized size = 1.7 \begin{align*}{\frac{x \left ( 35\,B{e}^{3}{b}^{3}{x}^{7}+40\,{x}^{6}A{b}^{3}{e}^{3}+120\,{x}^{6}B{e}^{3}{b}^{2}a+120\,{x}^{6}B{b}^{3}d{e}^{2}+140\,{x}^{5}Aa{b}^{2}{e}^{3}+140\,{x}^{5}A{b}^{3}d{e}^{2}+140\,{x}^{5}B{e}^{3}{a}^{2}b+420\,{x}^{5}Ba{b}^{2}d{e}^{2}+140\,{x}^{5}B{b}^{3}{d}^{2}e+168\,{x}^{4}A{a}^{2}b{e}^{3}+504\,{x}^{4}Aa{b}^{2}d{e}^{2}+168\,{x}^{4}A{b}^{3}{d}^{2}e+56\,{x}^{4}B{e}^{3}{a}^{3}+504\,{x}^{4}B{a}^{2}bd{e}^{2}+504\,{x}^{4}Ba{b}^{2}{d}^{2}e+56\,{x}^{4}B{b}^{3}{d}^{3}+70\,{x}^{3}A{a}^{3}{e}^{3}+630\,{x}^{3}A{a}^{2}bd{e}^{2}+630\,{x}^{3}Aa{b}^{2}{d}^{2}e+70\,{x}^{3}A{d}^{3}{b}^{3}+210\,{x}^{3}B{a}^{3}d{e}^{2}+630\,{x}^{3}B{a}^{2}b{d}^{2}e+210\,{x}^{3}Ba{b}^{2}{d}^{3}+280\,A{a}^{3}d{e}^{2}{x}^{2}+840\,A{a}^{2}b{d}^{2}e{x}^{2}+280\,Aa{b}^{2}{d}^{3}{x}^{2}+280\,B{a}^{3}{d}^{2}e{x}^{2}+280\,B{a}^{2}b{d}^{3}{x}^{2}+420\,xA{a}^{3}{d}^{2}e+420\,xA{d}^{3}{a}^{2}b+140\,xB{a}^{3}{d}^{3}+280\,A{d}^{3}{a}^{3} \right ) }{280\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \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 = 1.53471, size = 682, normalized size = 2.63 \begin{align*} \frac{1}{8} \, B b^{3} e^{3} x^{8} + A a^{3} d^{3} x + \frac{1}{7} \,{\left (3 \, B b^{3} d e^{2} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (B b^{3} d^{2} e +{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} +{\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} d^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 9 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (A a^{3} e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} + 9 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{2}\right )} x^{4} +{\left (A a^{3} d e^{2} +{\left (B a^{2} b + A a b^{2}\right )} d^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, A a^{3} d^{2} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B x\right ) \left (d + e x\right )^{3} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.20061, size = 802, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]