Optimal. Leaf size=212 \[ -\frac{2 (c+d x)^{3/2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )\right )}{3 d^5}-\frac{2 \sqrt{c+d x} (b c-a d) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^5}+\frac{2 (c+d x)^{5/2} \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{5 d^5}+\frac{2 (c+d x)^{7/2} (a d D-4 b c D+b C d)}{7 d^5}+\frac{2 b D (c+d x)^{9/2}}{9 d^5} \]
[Out]
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Rubi [A] time = 0.346458, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ -\frac{2 (c+d x)^{3/2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )\right )}{3 d^5}-\frac{2 \sqrt{c+d x} (b c-a d) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^5}+\frac{2 (c+d x)^{5/2} \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{5 d^5}+\frac{2 (c+d x)^{7/2} (a d D-4 b c D+b C d)}{7 d^5}+\frac{2 b D (c+d x)^{9/2}}{9 d^5} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]
[Out]
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Rubi in Sympy [A] time = 81.0796, size = 226, normalized size = 1.07 \[ \frac{2 D b \left (c + d x\right )^{\frac{9}{2}}}{9 d^{5}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}} \left (C b d + D a d - 4 D b c\right )}{7 d^{5}} + \frac{2 \left (c + d x\right )^{\frac{5}{2}} \left (B b d^{2} + C a d^{2} - 3 C b c d - 3 D a c d + 6 D b c^{2}\right )}{5 d^{5}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (A b d^{3} + B a d^{3} - 2 B b c d^{2} - 2 C a c d^{2} + 3 C b c^{2} d + 3 D a c^{2} d - 4 D b c^{3}\right )}{3 d^{5}} + \frac{2 \sqrt{c + d x} \left (a d - b c\right ) \left (A d^{3} - B c d^{2} + C c^{2} d - D c^{3}\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.331097, size = 184, normalized size = 0.87 \[ \frac{2 \sqrt{c+d x} \left (3 a d \left (d^3 (105 A+x (35 B+3 x (7 C+5 D x)))-2 c d^2 (35 B+x (14 C+9 D x))-48 c^3 D+8 c^2 d (7 C+3 D x)\right )+b \left (-2 c d^3 (105 A+x (42 B+x (27 C+20 D x)))+d^4 x (105 A+x (63 B+5 x (9 C+7 D x)))+24 c^2 d^2 (7 B+x (3 C+2 D x))+128 c^4 D-16 c^3 d (9 C+4 D x)\right )\right )}{315 d^5} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/Sqrt[c + d*x],x]
[Out]
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Maple [A] time = 0.008, size = 241, normalized size = 1.1 \[{\frac{70\,Db{x}^{4}{d}^{4}+90\,Cb{d}^{4}{x}^{3}+90\,Da{d}^{4}{x}^{3}-80\,Dbc{d}^{3}{x}^{3}+126\,Bb{d}^{4}{x}^{2}+126\,Ca{d}^{4}{x}^{2}-108\,Cbc{d}^{3}{x}^{2}-108\,Dac{d}^{3}{x}^{2}+96\,Db{c}^{2}{d}^{2}{x}^{2}+210\,Ab{d}^{4}x+210\,Ba{d}^{4}x-168\,Bbc{d}^{3}x-168\,Cac{d}^{3}x+144\,Cb{c}^{2}{d}^{2}x+144\,Da{c}^{2}{d}^{2}x-128\,Db{c}^{3}dx+630\,Aa{d}^{4}-420\,Abc{d}^{3}-420\,Bac{d}^{3}+336\,Bb{c}^{2}{d}^{2}+336\,Ca{c}^{2}{d}^{2}-288\,Cb{c}^{3}d-288\,Da{c}^{3}d+256\,Db{c}^{4}}{315\,{d}^{5}}\sqrt{dx+c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x)
[Out]
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Maxima [A] time = 1.35823, size = 267, normalized size = 1.26 \[ \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} D b - 45 \,{\left (4 \, D b c -{\left (D a + C b\right )} d\right )}{\left (d x + c\right )}^{\frac{7}{2}} + 63 \,{\left (6 \, D b c^{2} - 3 \,{\left (D a + C b\right )} c d +{\left (C a + B b\right )} d^{2}\right )}{\left (d x + c\right )}^{\frac{5}{2}} - 105 \,{\left (4 \, D b c^{3} - 3 \,{\left (D a + C b\right )} c^{2} d + 2 \,{\left (C a + B b\right )} c d^{2} -{\left (B a + A b\right )} d^{3}\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 315 \,{\left (D b c^{4} + A a d^{4} -{\left (D a + C b\right )} c^{3} d +{\left (C a + B b\right )} c^{2} d^{2} -{\left (B a + A b\right )} c d^{3}\right )} \sqrt{d x + c}\right )}}{315 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x + a)/sqrt(d*x + c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212277, size = 266, normalized size = 1.25 \[ \frac{2 \,{\left (35 \, D b d^{4} x^{4} + 128 \, D b c^{4} + 315 \, A a d^{4} - 144 \,{\left (D a + C b\right )} c^{3} d + 168 \,{\left (C a + B b\right )} c^{2} d^{2} - 210 \,{\left (B a + A b\right )} c d^{3} - 5 \,{\left (8 \, D b c d^{3} - 9 \,{\left (D a + C b\right )} d^{4}\right )} x^{3} + 3 \,{\left (16 \, D b c^{2} d^{2} - 18 \,{\left (D a + C b\right )} c d^{3} + 21 \,{\left (C a + B b\right )} d^{4}\right )} x^{2} -{\left (64 \, D b c^{3} d - 72 \,{\left (D a + C b\right )} c^{2} d^{2} + 84 \,{\left (C a + B b\right )} c d^{3} - 105 \,{\left (B a + A b\right )} d^{4}\right )} x\right )} \sqrt{d x + c}}{315 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x + a)/sqrt(d*x + c),x, algorithm="fricas")
[Out]
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Sympy [A] time = 57.2831, size = 848, normalized size = 4. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.225034, size = 494, normalized size = 2.33 \[ \frac{2 \,{\left (315 \, \sqrt{d x + c} A a + \frac{105 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} B a}{d} + \frac{105 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} A b}{d} + \frac{21 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} d^{8} - 10 \,{\left (d x + c\right )}^{\frac{3}{2}} c d^{8} + 15 \, \sqrt{d x + c} c^{2} d^{8}\right )} C a}{d^{10}} + \frac{21 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} d^{8} - 10 \,{\left (d x + c\right )}^{\frac{3}{2}} c d^{8} + 15 \, \sqrt{d x + c} c^{2} d^{8}\right )} B b}{d^{10}} + \frac{9 \,{\left (5 \,{\left (d x + c\right )}^{\frac{7}{2}} d^{18} - 21 \,{\left (d x + c\right )}^{\frac{5}{2}} c d^{18} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} d^{18} - 35 \, \sqrt{d x + c} c^{3} d^{18}\right )} D a}{d^{21}} + \frac{9 \,{\left (5 \,{\left (d x + c\right )}^{\frac{7}{2}} d^{18} - 21 \,{\left (d x + c\right )}^{\frac{5}{2}} c d^{18} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} d^{18} - 35 \, \sqrt{d x + c} c^{3} d^{18}\right )} C b}{d^{21}} + \frac{{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} d^{32} - 180 \,{\left (d x + c\right )}^{\frac{7}{2}} c d^{32} + 378 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} d^{32} - 420 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3} d^{32} + 315 \, \sqrt{d x + c} c^{4} d^{32}\right )} D b}{d^{36}}\right )}}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x + a)/sqrt(d*x + c),x, algorithm="giac")
[Out]