3.1403 \(\int (a+b x) (c+d x)^{5/2} \, dx\)

Optimal. Leaf size=42 \[ \frac{2 b (c+d x)^{9/2}}{9 d^2}-\frac{2 (c+d x)^{7/2} (b c-a d)}{7 d^2} \]

[Out]

(-2*(b*c - a*d)*(c + d*x)^(7/2))/(7*d^2) + (2*b*(c + d*x)^(9/2))/(9*d^2)

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Rubi [A]  time = 0.0445957, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 b (c+d x)^{9/2}}{9 d^2}-\frac{2 (c+d x)^{7/2} (b c-a d)}{7 d^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(c + d*x)^(5/2),x]

[Out]

(-2*(b*c - a*d)*(c + d*x)^(7/2))/(7*d^2) + (2*b*(c + d*x)^(9/2))/(9*d^2)

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Rubi in Sympy [A]  time = 8.20782, size = 37, normalized size = 0.88 \[ \frac{2 b \left (c + d x\right )^{\frac{9}{2}}}{9 d^{2}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}} \left (a d - b c\right )}{7 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(d*x+c)**(5/2),x)

[Out]

2*b*(c + d*x)**(9/2)/(9*d**2) + 2*(c + d*x)**(7/2)*(a*d - b*c)/(7*d**2)

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Mathematica [A]  time = 0.0465786, size = 30, normalized size = 0.71 \[ \frac{2 (c+d x)^{7/2} (9 a d-2 b c+7 b d x)}{63 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(c + d*x)^(5/2),x]

[Out]

(2*(c + d*x)^(7/2)*(-2*b*c + 9*a*d + 7*b*d*x))/(63*d^2)

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Maple [A]  time = 0.005, size = 27, normalized size = 0.6 \[{\frac{14\,bdx+18\,ad-4\,bc}{63\,{d}^{2}} \left ( dx+c \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(d*x+c)^(5/2),x)

[Out]

2/63*(d*x+c)^(7/2)*(7*b*d*x+9*a*d-2*b*c)/d^2

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Maxima [A]  time = 1.3308, size = 45, normalized size = 1.07 \[ \frac{2 \,{\left (7 \,{\left (d x + c\right )}^{\frac{9}{2}} b - 9 \,{\left (b c - a d\right )}{\left (d x + c\right )}^{\frac{7}{2}}\right )}}{63 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

2/63*(7*(d*x + c)^(9/2)*b - 9*(b*c - a*d)*(d*x + c)^(7/2))/d^2

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Fricas [A]  time = 0.226792, size = 126, normalized size = 3. \[ \frac{2 \,{\left (7 \, b d^{4} x^{4} - 2 \, b c^{4} + 9 \, a c^{3} d +{\left (19 \, b c d^{3} + 9 \, a d^{4}\right )} x^{3} + 3 \,{\left (5 \, b c^{2} d^{2} + 9 \, a c d^{3}\right )} x^{2} +{\left (b c^{3} d + 27 \, a c^{2} d^{2}\right )} x\right )} \sqrt{d x + c}}{63 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

2/63*(7*b*d^4*x^4 - 2*b*c^4 + 9*a*c^3*d + (19*b*c*d^3 + 9*a*d^4)*x^3 + 3*(5*b*c^
2*d^2 + 9*a*c*d^3)*x^2 + (b*c^3*d + 27*a*c^2*d^2)*x)*sqrt(d*x + c)/d^2

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Sympy [A]  time = 3.06738, size = 194, normalized size = 4.62 \[ \begin{cases} \frac{2 a c^{3} \sqrt{c + d x}}{7 d} + \frac{6 a c^{2} x \sqrt{c + d x}}{7} + \frac{6 a c d x^{2} \sqrt{c + d x}}{7} + \frac{2 a d^{2} x^{3} \sqrt{c + d x}}{7} - \frac{4 b c^{4} \sqrt{c + d x}}{63 d^{2}} + \frac{2 b c^{3} x \sqrt{c + d x}}{63 d} + \frac{10 b c^{2} x^{2} \sqrt{c + d x}}{21} + \frac{38 b c d x^{3} \sqrt{c + d x}}{63} + \frac{2 b d^{2} x^{4} \sqrt{c + d x}}{9} & \text{for}\: d \neq 0 \\c^{\frac{5}{2}} \left (a x + \frac{b x^{2}}{2}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(d*x+c)**(5/2),x)

[Out]

Piecewise((2*a*c**3*sqrt(c + d*x)/(7*d) + 6*a*c**2*x*sqrt(c + d*x)/7 + 6*a*c*d*x
**2*sqrt(c + d*x)/7 + 2*a*d**2*x**3*sqrt(c + d*x)/7 - 4*b*c**4*sqrt(c + d*x)/(63
*d**2) + 2*b*c**3*x*sqrt(c + d*x)/(63*d) + 10*b*c**2*x**2*sqrt(c + d*x)/21 + 38*
b*c*d*x**3*sqrt(c + d*x)/63 + 2*b*d**2*x**4*sqrt(c + d*x)/9, Ne(d, 0)), (c**(5/2
)*(a*x + b*x**2/2), True))

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GIAC/XCAS [A]  time = 0.22607, size = 308, normalized size = 7.33 \[ \frac{2 \,{\left (105 \,{\left (d x + c\right )}^{\frac{3}{2}} a c^{2} + 42 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a c + \frac{21 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} b c^{2}}{d} + \frac{6 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} d^{12} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c d^{12} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} d^{12}\right )} b c}{d^{13}} + \frac{3 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} d^{12} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c d^{12} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} d^{12}\right )} a}{d^{12}} + \frac{{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} d^{24} - 135 \,{\left (d x + c\right )}^{\frac{7}{2}} c d^{24} + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} d^{24} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3} d^{24}\right )} b}{d^{25}}\right )}}{315 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x + c)^(5/2),x, algorithm="giac")

[Out]

2/315*(105*(d*x + c)^(3/2)*a*c^2 + 42*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*
a*c + 21*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*b*c^2/d + 6*(15*(d*x + c)^(7/
2)*d^12 - 42*(d*x + c)^(5/2)*c*d^12 + 35*(d*x + c)^(3/2)*c^2*d^12)*b*c/d^13 + 3*
(15*(d*x + c)^(7/2)*d^12 - 42*(d*x + c)^(5/2)*c*d^12 + 35*(d*x + c)^(3/2)*c^2*d^
12)*a/d^12 + (35*(d*x + c)^(9/2)*d^24 - 135*(d*x + c)^(7/2)*c*d^24 + 189*(d*x +
c)^(5/2)*c^2*d^24 - 105*(d*x + c)^(3/2)*c^3*d^24)*b/d^25)/d