3.1.37 \(\int (c+d (a+b x))^{5/2} \, dx\) [37]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {33, 32} \begin {gather*} \frac {2 (d (a+b x)+c)^{7/2}}{7 b d} \end {gather*}

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)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 33

Int[((a_.) + (b_.)*(u_))^(m_), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m, x], x, u], x]
/; FreeQ[{a, b, m}, x] && LinearQ[u, x] && NeQ[u, x]

Rubi steps

\begin {align*} \int (c+d (a+b x))^{5/2} \, dx &=\frac {\text {Subst}\left (\int (c+d x)^{5/2} \, dx,x,a+b x\right )}{b}\\ &=\frac {2 (c+d (a+b x))^{7/2}}{7 b d}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 (c+a d+b d x)^{7/2}}{7 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*(a + b*x))^(5/2),x]

[Out]

(2*(c + a*d + b*d*x)^(7/2))/(7*b*d)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 3.22, size = 239, normalized size = 10.39 \begin {gather*} \text {Piecewise}\left [\left \{\left \{c^{\frac {5}{2}} x,b\text {==}0\text {\&\&}d\text {==}0\right \},\left \{x \left (a d+c\right )^{\frac {5}{2}},b\text {==}0\right \},\left \{c^{\frac {5}{2}} x,d\text {==}0\right \}\right \},\frac {2 a^3 d^2 \sqrt {a d+b d x+c}}{7 b}+\frac {6 a^2 c d \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 b d^2 x^2 \sqrt {a d+b d x+c}}{7}+\frac {6 a c^2 \sqrt {a d+b d x+c}}{7 b}+\frac {12 a c d x \sqrt {a d+b d x+c}}{7}+\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 {2 c^3 \sqrt {a d+b d x+c}}{7 b d}+\frac {6 c^2 x \sqrt {a d+b d x+c}}{7}\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Piecewise[{{c ^ (5 / 2) x, b == 0 && d == 0}, {x (a d + c) ^ (5 / 2), b == 0}, {c ^ (5 / 2) x, d == 0}}, 2 a ^
3 d ^ 2 Sqrt[a d + b d x + c] / (7 b) + 6 a ^ 2 c d Sqrt[a d + b d x + c] / (7 b) + 6 a ^ 2 d ^ 2 x Sqrt[a d
+ b d x + c] / 7 + 6 a b d ^ 2 x ^ 2 Sqrt[a d + b d x + c] / 7 + 6 a c ^ 2 Sqrt[a d + b d x + c] / (7 b) + 12
a c d x Sqrt[a d + b d x + c] / 7 + 2 b ^ 2 d ^ 2 x ^ 3 Sqrt[a d + b d x + c] / 7 + 6 b c d x ^ 2 Sqrt[a d + b
d x + c] / 7 + 2 c ^ 3 Sqrt[a d + b d x + c] / (7 b d) + 6 c ^ 2 x Sqrt[a d + b d x + c] / 7]

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Maple [A]
time = 0.09, size = 20, normalized size = 0.87

method	result	size
gosper	\(\frac {2 \left (b d x +a d +c \right )^{\frac {7}{2}}}{7 b d}\)	\(20\)
derivativedivides	\(\frac {2 \left (b d x +a d +c \right )^{\frac {7}{2}}}{7 b d}\)	\(20\)
default	\(\frac {2 \left (b d x +a d +c \right )^{\frac {7}{2}}}{7 b d}\)	\(20\)
trager	\(\frac {2 \left (b^{3} d^{3} x^{3}+3 a \,b^{2} d^{3} x^{2}+3 a^{2} b \,d^{3} x +3 b^{2} c \,d^{2} x^{2}+a^{3} d^{3}+6 a b c \,d^{2} x +3 a^{2} c \,d^{2}+3 b \,c^{2} d x +3 a \,c^{2} d +c^{3}\right ) \sqrt {b d x +a d +c}}{7 b d}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*(b*x+a))^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.25, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left ({\left (b x + a\right )} d + c\right )}^{\frac {7}{2}}}{7 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*((b*x + a)*d + c)^(7/2)/(b*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (19) = 38\).
time = 0.31, size = 104, normalized size = 4.52 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*(b*x+a))^(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 = 1.64, size = 270, normalized size = 11.74 \begin {gather*} \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} \end {gather*}

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*sqrt(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*sqrt(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 [B] Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (19) = 38\).
time = 0.01, size = 884, normalized size = 38.43 \begin {gather*} \frac {\frac {2 b^{3} d^{3} \left (\frac {1}{7} \sqrt {a d+b d x+c} \left (a d+b d x+c\right )^{3}-\frac {3}{5} \sqrt {a d+b d x+c} \left (a d+b d x+c\right )^{2} a d-\frac {3}{5} \sqrt {a d+b d x+c} \left (a d+b d x+c\right )^{2} c+\sqrt {a d+b d x+c} \left (a d+b d x+c\right ) a^{2} d^{2}+2 \sqrt {a d+b d x+c} \left (a d+b d x+c\right ) a d c+\sqrt {a d+b d x+c} \left (a d+b d x+c\right ) c^{2}-\sqrt {a d+b d x+c} a^{3} d^{3}-3 \sqrt {a d+b d x+c} a^{2} d^{2} c-3 \sqrt {a d+b d x+c} a d c^{2}-\sqrt {a d+b d x+c} c^{3}\right )}{d^{3} b^{3}}+\frac {6 a b^{2} d^{3} \left (\frac {1}{5} \sqrt {a d+b d x+c} \left (a d+b d x+c\right )^{2}-\frac {2}{3} \sqrt {a d+b d x+c} \left (a d+b d x+c\right ) a d-\frac {2}{3} \sqrt {a d+b d x+c} \left (a d+b d x+c\right ) c+\sqrt {a d+b d x+c} a^{2} d^{2}+2 \sqrt {a d+b d x+c} a d c+\sqrt {a d+b d x+c} c^{2}\right )}{d^{2} b^{2}}+\frac {6 b^{2} c d^{2} \left (\frac {1}{5} \sqrt {a d+b d x+c} \left (a d+b d x+c\right )^{2}-\frac {2}{3} \sqrt {a d+b d x+c} \left (a d+b d x+c\right ) a d-\frac {2}{3} \sqrt {a d+b d x+c} \left (a d+b d x+c\right ) c+\sqrt {a d+b d x+c} a^{2} d^{2}+2 \sqrt {a d+b d x+c} a d c+\sqrt {a d+b d x+c} c^{2}\right )}{d^{2} b^{2}}+2 a^{3} d^{3} \sqrt {a d+b d x+c}+\frac {6 a^{2} b d^{3} \left (\frac {1}{3} \sqrt {a d+b d x+c} \left (a d+b d x+c\right )-a d \sqrt {a d+b d x+c}-c \sqrt {a d+b d x+c}\right )}{b d}+6 a^{2} c d^{2} \sqrt {a d+b d x+c}+\frac {12 a b c d^{2} \left (\frac {1}{3} \sqrt {a d+b d x+c} \left (a d+b d x+c\right )-a d \sqrt {a d+b d x+c}-c \sqrt {a d+b d x+c}\right )}{b d}+6 a c^{2} d \sqrt {a d+b d x+c}+\frac {6 b c^{2} d \left (\frac {1}{3} \sqrt {a d+b d x+c} \left (a d+b d x+c\right )-a d \sqrt {a d+b d x+c}-c \sqrt {a d+b d x+c}\right )}{b d}+2 c^{3} \sqrt {a d+b d x+c}}{b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*(b*x+a))^(5/2),x)

[Out]

2/35*(35*(b*d*x + a*d + c)^(3/2)*a^2*d^2 - 35*(3*sqrt(b*d*x + a*d + c)*a*d - (b*d*x + a*d + c)^(3/2) + 3*sqrt(
b*d*x + a*d + c)*c)*a^2*d^2 - 21*(b*d*x + a*d + c)^(5/2)*a*d + 70*(b*d*x + a*d + c)^(3/2)*a*c*d - 70*(3*sqrt(b
*d*x + a*d + c)*a*d - (b*d*x + a*d + c)^(3/2) + 3*sqrt(b*d*x + a*d + c)*c)*a*c*d + 5*(b*d*x + a*d + c)^(7/2) -
21*(b*d*x + a*d + c)^(5/2)*c + 35*(b*d*x + a*d + c)^(3/2)*c^2 - 35*(3*sqrt(b*d*x + a*d + c)*a*d - (b*d*x + a*
d + c)^(3/2) + 3*sqrt(b*d*x + a*d + c)*c)*c^2 + 7*(15*sqrt(b*d*x + a*d + c)*a^2*d^2 - 10*(b*d*x + a*d + c)^(3/
2)*a*d + 30*sqrt(b*d*x + a*d + c)*a*c*d + 3*(b*d*x + a*d + c)^(5/2) - 10*(b*d*x + a*d + c)^(3/2)*c + 15*sqrt(b
*d*x + a*d + c)*c^2)*a*d + 7*(15*sqrt(b*d*x + a*d + c)*a^2*d^2 - 10*(b*d*x + a*d + c)^(3/2)*a*d + 30*sqrt(b*d*
x + a*d + c)*a*c*d + 3*(b*d*x + a*d + c)^(5/2) - 10*(b*d*x + a*d + c)^(3/2)*c + 15*sqrt(b*d*x + a*d + c)*c^2)*
c)/(b*d)

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Mupad [B]
time = 0.18, size = 93, normalized size = 4.04 \begin {gather*} \frac {6\,x\,\sqrt {c+d\,\left (a+b\,x\right )}\,{\left (c+a\,d\right )}^2}{7}+\frac {2\,\sqrt {c+d\,\left (a+b\,x\right )}\,{\left (c+a\,d\right )}^3}{7\,b\,d}+\frac {2\,b^2\,d^2\,x^3\,\sqrt {c+d\,\left (a+b\,x\right )}}{7}+\frac {6\,b\,d\,x^2\,\sqrt {c+d\,\left (a+b\,x\right )}\,\left (c+a\,d\right )}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*(a + b*x))^(5/2),x)

[Out]

(6*x*(c + d*(a + b*x))^(1/2)*(c + a*d)^2)/7 + (2*(c + d*(a + b*x))^(1/2)*(c + a*d)^3)/(7*b*d) + (2*b^2*d^2*x^3
*(c + d*(a + b*x))^(1/2))/7 + (6*b*d*x^2*(c + d*(a + b*x))^(1/2)*(c + a*d))/7

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