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

Optimal. Leaf size=23 \[ \frac{2 (d (a+b x)+c)^{7/2}}{7 b d} \]

[Out]

(2*(c + d*(a + b*x))^(7/2))/(7*b*d)

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Rubi [A]  time = 0.0271118, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 (d (a+b x)+c)^{7/2}}{7 b d} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*(a + b*x))^(5/2),x]

[Out]

(2*(c + d*(a + b*x))^(7/2))/(7*b*d)

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Rubi in Sympy [A]  time = 2.14283, size = 17, normalized size = 0.74 \[ \frac{2 \left (c + d \left (a + b x\right )\right )^{\frac{7}{2}}}{7 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(c + d*(a + b*x))**(7/2)/(7*b*d)

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Mathematica [A]  time = 0.0429123, size = 23, normalized size = 1. \[ \frac{2 (d (a+b x)+c)^{7/2}}{7 b d} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c + d*(a + b*x))^(7/2))/(7*b*d)

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Maple [A]  time = 0.003, size = 20, normalized size = 0.9 \[{\frac{2}{7\,db} \left ( bdx+ad+c \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*(b*x+a))^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.30901, size = 26, normalized size = 1.13 \[ \frac{2 \,{\left ({\left (b x + a\right )} d + c\right )}^{\frac{7}{2}}}{7 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*((b*x + a)*d + c)^(7/2)/(b*d)

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Fricas [A]  time = 0.232988, size = 140, normalized size = 6.09 \[ \frac{2 \,{\left (b^{3} d^{3} x^{3} + a^{3} d^{3} + 3 \, a^{2} c d^{2} + 3 \, a c^{2} d + c^{3} + 3 \,{\left (a b^{2} d^{3} + b^{2} c d^{2}\right )} x^{2} + 3 \,{\left (a^{2} b d^{3} + 2 \, a b c d^{2} + b c^{2} d\right )} x\right )} \sqrt{b d x + a d + c}}{7 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(b^3*d^3*x^3 + a^3*d^3 + 3*a^2*c*d^2 + 3*a*c^2*d + c^3 + 3*(a*b^2*d^3 + b^2*
c*d^2)*x^2 + 3*(a^2*b*d^3 + 2*a*b*c*d^2 + b*c^2*d)*x)*sqrt(b*d*x + a*d + c)/(b*d
)

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Sympy [A]  time = 173.73, size = 270, normalized size = 11.74 \[ \begin{cases} c^{\frac{5}{2}} x & \text{for}\: b = 0 \wedge d = 0 \\x \left (a d + c\right )^{\frac{5}{2}} & \text{for}\: b = 0 \\c^{\frac{5}{2}} x & \text{for}\: d = 0 \\\frac{2 a^{3} d^{2} \sqrt{a d + b d x + c}}{7 b} + \frac{6 a^{2} d^{2} x \sqrt{a d + b d x + c}}{7} + \frac{6 a^{2} c d \sqrt{a d + b d x + c}}{7 b} + \frac{6 a b d^{2} x^{2} \sqrt{a d + b d x + c}}{7} + \frac{12 a c d x \sqrt{a d + b d x + c}}{7} + \frac{6 a c^{2} \sqrt{a d + b d x + c}}{7 b} + \frac{2 b^{2} d^{2} x^{3} \sqrt{a d + b d x + c}}{7} + \frac{6 b c d x^{2} \sqrt{a d + b d x + c}}{7} + \frac{6 c^{2} x \sqrt{a d + b d x + c}}{7} + \frac{2 c^{3} \sqrt{a d + b d x + c}}{7 b d} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((c**(5/2)*x, Eq(b, 0) & Eq(d, 0)), (x*(a*d + c)**(5/2), Eq(b, 0)), (c*
*(5/2)*x, Eq(d, 0)), (2*a**3*d**2*sqrt(a*d + b*d*x + c)/(7*b) + 6*a**2*d**2*x*sq
rt(a*d + b*d*x + c)/7 + 6*a**2*c*d*sqrt(a*d + b*d*x + c)/(7*b) + 6*a*b*d**2*x**2
*sqrt(a*d + b*d*x + c)/7 + 12*a*c*d*x*sqrt(a*d + b*d*x + c)/7 + 6*a*c**2*sqrt(a*
d + b*d*x + c)/(7*b) + 2*b**2*d**2*x**3*sqrt(a*d + b*d*x + c)/7 + 6*b*c*d*x**2*s
qrt(a*d + b*d*x + c)/7 + 6*c**2*x*sqrt(a*d + b*d*x + c)/7 + 2*c**3*sqrt(a*d + b*
d*x + c)/(7*b*d), True))

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GIAC/XCAS [A]  time = 0.217603, size = 385, normalized size = 16.74 \[ \frac{2 \,{\left (35 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} a^{2} d^{2} + 70 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} a c d + 35 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} c^{2} - 14 \,{\left (5 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} a d - 3 \,{\left (b d x + a d + c\right )}^{\frac{5}{2}} + 5 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} c\right )} a d - 14 \,{\left (5 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} a d - 3 \,{\left (b d x + a d + c\right )}^{\frac{5}{2}} + 5 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} c\right )} c + \frac{35 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} a^{2} b^{12} d^{14} - 42 \,{\left (b d x + a d + c\right )}^{\frac{5}{2}} a b^{12} d^{13} + 70 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} a b^{12} c d^{13} + 15 \,{\left (b d x + a d + c\right )}^{\frac{7}{2}} b^{12} d^{12} - 42 \,{\left (b d x + a d + c\right )}^{\frac{5}{2}} b^{12} c d^{12} + 35 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}} b^{12} c^{2} d^{12}}{b^{12} d^{12}}\right )}}{105 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(35*(b*d*x + a*d + c)^(3/2)*a^2*d^2 + 70*(b*d*x + a*d + c)^(3/2)*a*c*d + 3
5*(b*d*x + a*d + c)^(3/2)*c^2 - 14*(5*(b*d*x + a*d + c)^(3/2)*a*d - 3*(b*d*x + a
*d + c)^(5/2) + 5*(b*d*x + a*d + c)^(3/2)*c)*a*d - 14*(5*(b*d*x + a*d + c)^(3/2)
*a*d - 3*(b*d*x + a*d + c)^(5/2) + 5*(b*d*x + a*d + c)^(3/2)*c)*c + (35*(b*d*x +
a*d + c)^(3/2)*a^2*b^12*d^14 - 42*(b*d*x + a*d + c)^(5/2)*a*b^12*d^13 + 70*(b*d
*x + a*d + c)^(3/2)*a*b^12*c*d^13 + 15*(b*d*x + a*d + c)^(7/2)*b^12*d^12 - 42*(b
*d*x + a*d + c)^(5/2)*b^12*c*d^12 + 35*(b*d*x + a*d + c)^(3/2)*b^12*c^2*d^12)/(b
^12*d^12))/(b*d)