Optimal. Leaf size=370 \[ \frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}{e^7 (a+b x)}+\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^{3/2}}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^7 (a+b x)}+\frac{6 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^7 (a+b x)} \]
[Out]
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Rubi [A] time = 0.406624, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}{e^7 (a+b x)}+\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^{3/2}}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^7 (a+b x)}+\frac{6 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^7 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 58.6624, size = 320, normalized size = 0.86 \[ \frac{80 b^{2} \left (a + b x\right ) \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{9 e^{3}} + \frac{640 b^{2} \sqrt{d + e x} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{63 e^{4}} + \frac{256 b^{2} \left (3 a + 3 b x\right ) \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{5}} + \frac{1024 b^{2} \sqrt{d + e x} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{6}} + \frac{2048 b^{2} \sqrt{d + e x} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{7} \left (a + b x\right )} - \frac{8 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{e^{2} \sqrt{d + e x}} - \frac{2 \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{3 e \left (d + e x\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(5/2),x)
[Out]
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Mathematica [A] time = 0.519242, size = 238, normalized size = 0.64 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (3 b^4 e^2 x^2 \left (63 a^2 e^2-72 a b d e+23 b^2 d^2\right )+2 b^3 e x \left (210 a^3 e^3-441 a^2 b d e^2+333 a b^2 d^2 e-88 b^3 d^3\right )+b^2 \left (945 a^4 e^4-3360 a^3 b d e^3+4599 a^2 b^2 d^2 e^2-2844 a b^3 d^3 e+667 b^4 d^4\right )-2 b^5 e^3 x^3 (13 b d-27 a e)+\frac{378 b (b d-a e)^5}{d+e x}-\frac{21 (b d-a e)^6}{(d+e x)^2}+7 b^6 e^4 x^4\right )}{63 e^7 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(5/2),x]
[Out]
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Maple [A] time = 0.013, size = 393, normalized size = 1.1 \[ -{\frac{-14\,{x}^{6}{b}^{6}{e}^{6}-108\,{x}^{5}a{b}^{5}{e}^{6}+24\,{x}^{5}{b}^{6}d{e}^{5}-378\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+216\,{x}^{4}a{b}^{5}d{e}^{5}-48\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-840\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+1008\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-576\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+128\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-1890\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+5040\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-6048\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+3456\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-768\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+756\,x{a}^{5}b{e}^{6}-7560\,x{a}^{4}{b}^{2}d{e}^{5}+20160\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-24192\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+13824\,xa{b}^{5}{d}^{4}{e}^{2}-3072\,x{b}^{6}{d}^{5}e+42\,{a}^{6}{e}^{6}+504\,{a}^{5}bd{e}^{5}-5040\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+13440\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-16128\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+9216\,{d}^{5}a{b}^{5}e-2048\,{b}^{6}{d}^{6}}{63\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(5/2),x)
[Out]
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Maxima [A] time = 0.749102, size = 844, normalized size = 2.28 \[ \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \,{\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \,{\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} a}{21 \,{\left (e^{7} x + d e^{6}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (7 \, b^{5} e^{6} x^{6} + 1024 \, b^{5} d^{6} - 3840 \, a b^{4} d^{5} e + 5376 \, a^{2} b^{3} d^{4} e^{2} - 3360 \, a^{3} b^{2} d^{3} e^{3} + 840 \, a^{4} b d^{2} e^{4} - 42 \, a^{5} d e^{5} - 3 \,{\left (4 \, b^{5} d e^{5} - 15 \, a b^{4} e^{6}\right )} x^{5} + 6 \,{\left (4 \, b^{5} d^{2} e^{4} - 15 \, a b^{4} d e^{5} + 21 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \,{\left (32 \, b^{5} d^{3} e^{3} - 120 \, a b^{4} d^{2} e^{4} + 168 \, a^{2} b^{3} d e^{5} - 105 \, a^{3} b^{2} e^{6}\right )} x^{3} + 3 \,{\left (128 \, b^{5} d^{4} e^{2} - 480 \, a b^{4} d^{3} e^{3} + 672 \, a^{2} b^{3} d^{2} e^{4} - 420 \, a^{3} b^{2} d e^{5} + 105 \, a^{4} b e^{6}\right )} x^{2} + 3 \,{\left (512 \, b^{5} d^{5} e - 1920 \, a b^{4} d^{4} e^{2} + 2688 \, a^{2} b^{3} d^{3} e^{3} - 1680 \, a^{3} b^{2} d^{2} e^{4} + 420 \, a^{4} b d e^{5} - 21 \, a^{5} e^{6}\right )} x\right )} b}{63 \,{\left (e^{8} x + d e^{7}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.291351, size = 494, normalized size = 1.34 \[ \frac{2 \,{\left (7 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 4608 \, a b^{5} d^{5} e + 8064 \, a^{2} b^{4} d^{4} e^{2} - 6720 \, a^{3} b^{3} d^{3} e^{3} + 2520 \, a^{4} b^{2} d^{2} e^{4} - 252 \, a^{5} b d e^{5} - 21 \, a^{6} e^{6} - 6 \,{\left (2 \, b^{6} d e^{5} - 9 \, a b^{5} e^{6}\right )} x^{5} + 3 \,{\left (8 \, b^{6} d^{2} e^{4} - 36 \, a b^{5} d e^{5} + 63 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \,{\left (16 \, b^{6} d^{3} e^{3} - 72 \, a b^{5} d^{2} e^{4} + 126 \, a^{2} b^{4} d e^{5} - 105 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{4} e^{2} - 576 \, a b^{5} d^{3} e^{3} + 1008 \, a^{2} b^{4} d^{2} e^{4} - 840 \, a^{3} b^{3} d e^{5} + 315 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \,{\left (256 \, b^{6} d^{5} e - 1152 \, a b^{5} d^{4} e^{2} + 2016 \, a^{2} b^{4} d^{3} e^{3} - 1680 \, a^{3} b^{3} d^{2} e^{4} + 630 \, a^{4} b^{2} d e^{5} - 63 \, a^{5} b e^{6}\right )} x\right )}}{63 \,{\left (e^{8} x + d e^{7}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.310197, size = 851, normalized size = 2.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^(5/2),x, algorithm="giac")
[Out]