3.6.91 \(\int (d+e x)^{5/2} (a+c x^2) \, dx\) [591]

Optimal. Leaf size=63 \[ \frac {2 \left (c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^3}-\frac {4 c d (d+e x)^{9/2}}{9 e^3}+\frac {2 c (d+e x)^{11/2}}{11 e^3} \]

[Out]

2/7*(a*e^2+c*d^2)*(e*x+d)^(7/2)/e^3-4/9*c*d*(e*x+d)^(9/2)/e^3+2/11*c*(e*x+d)^(11/2)/e^3

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Rubi [A]
time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \begin {gather*} \frac {2 (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^3}+\frac {2 c (d+e x)^{11/2}}{11 e^3}-\frac {4 c d (d+e x)^{9/2}}{9 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)*(a + c*x^2),x]

[Out]

(2*(c*d^2 + a*e^2)*(d + e*x)^(7/2))/(7*e^3) - (4*c*d*(d + e*x)^(9/2))/(9*e^3) + (2*c*(d + e*x)^(11/2))/(11*e^3
)

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int (d+e x)^{5/2} \left (a+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2+a e^2\right ) (d+e x)^{5/2}}{e^2}-\frac {2 c d (d+e x)^{7/2}}{e^2}+\frac {c (d+e x)^{9/2}}{e^2}\right ) \, dx\\ &=\frac {2 \left (c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^3}-\frac {4 c d (d+e x)^{9/2}}{9 e^3}+\frac {2 c (d+e x)^{11/2}}{11 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 44, normalized size = 0.70 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (99 a e^2+c \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(99*a*e^2 + c*(8*d^2 - 28*d*e*x + 63*e^2*x^2)))/(693*e^3)

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Maple [A]
time = 0.39, size = 48, normalized size = 0.76

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (63 c \,e^{2} x^{2}-28 c d e x +99 e^{2} a +8 c \,d^{2}\right )}{693 e^{3}}\)	\(41\)
derivativedivides	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {4 c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\)	\(48\)
default	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {4 c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\)	\(48\)
trager	\(\frac {2 \left (63 e^{5} c \,x^{5}+161 c d \,e^{4} x^{4}+99 e^{5} a \,x^{3}+113 e^{3} d^{2} c \,x^{3}+297 a d \,e^{4} x^{2}+3 d^{3} e^{2} c \,x^{2}+297 a \,d^{2} e^{3} x -4 c \,d^{4} e x +99 a \,d^{3} e^{2}+8 c \,d^{5}\right ) \sqrt {e x +d}}{693 e^{3}}\)	\(109\)
risch	\(\frac {2 \left (63 e^{5} c \,x^{5}+161 c d \,e^{4} x^{4}+99 e^{5} a \,x^{3}+113 e^{3} d^{2} c \,x^{3}+297 a d \,e^{4} x^{2}+3 d^{3} e^{2} c \,x^{2}+297 a \,d^{2} e^{3} x -4 c \,d^{4} e x +99 a \,d^{3} e^{2}+8 c \,d^{5}\right ) \sqrt {e x +d}}{693 e^{3}}\)	\(109\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^3*(1/11*c*(e*x+d)^(11/2)-2/9*c*d*(e*x+d)^(9/2)+1/7*(a*e^2+c*d^2)*(e*x+d)^(7/2))

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Maxima [A]
time = 0.31, size = 48, normalized size = 0.76 \begin {gather*} \frac {2}{693} \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} c - 154 \, {\left (x e + d\right )}^{\frac {9}{2}} c d + 99 \, {\left (c d^{2} + a e^{2}\right )} {\left (x e + d\right )}^{\frac {7}{2}}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*(x*e + d)^(11/2)*c - 154*(x*e + d)^(9/2)*c*d + 99*(c*d^2 + a*e^2)*(x*e + d)^(7/2))*e^(-3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (50) = 100\).
time = 1.87, size = 104, normalized size = 1.65 \begin {gather*} -\frac {2}{693} \, {\left (4 \, c d^{4} x e - 8 \, c d^{5} - 9 \, {\left (7 \, c x^{5} + 11 \, a x^{3}\right )} e^{5} - {\left (161 \, c d x^{4} + 297 \, a d x^{2}\right )} e^{4} - {\left (113 \, c d^{2} x^{3} + 297 \, a d^{2} x\right )} e^{3} - 3 \, {\left (c d^{3} x^{2} + 33 \, a d^{3}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/693*(4*c*d^4*x*e - 8*c*d^5 - 9*(7*c*x^5 + 11*a*x^3)*e^5 - (161*c*d*x^4 + 297*a*d*x^2)*e^4 - (113*c*d^2*x^3
+ 297*a*d^2*x)*e^3 - 3*(c*d^3*x^2 + 33*a*d^3)*e^2)*sqrt(x*e + d)*e^(-3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (63) = 126\).
time = 0.31, size = 218, normalized size = 3.46 \begin {gather*} \begin {cases} \frac {2 a d^{3} \sqrt {d + e x}}{7 e} + \frac {6 a d^{2} x \sqrt {d + e x}}{7} + \frac {6 a d e x^{2} \sqrt {d + e x}}{7} + \frac {2 a e^{2} x^{3} \sqrt {d + e x}}{7} + \frac {16 c d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 c d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 c d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 c d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 c d e x^{4} \sqrt {d + e x}}{99} + \frac {2 c e^{2} x^{5} \sqrt {d + e x}}{11} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a x + \frac {c x^{3}}{3}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a*d**3*sqrt(d + e*x)/(7*e) + 6*a*d**2*x*sqrt(d + e*x)/7 + 6*a*d*e*x**2*sqrt(d + e*x)/7 + 2*a*e**2
*x**3*sqrt(d + e*x)/7 + 16*c*d**5*sqrt(d + e*x)/(693*e**3) - 8*c*d**4*x*sqrt(d + e*x)/(693*e**2) + 2*c*d**3*x*
*2*sqrt(d + e*x)/(231*e) + 226*c*d**2*x**3*sqrt(d + e*x)/693 + 46*c*d*e*x**4*sqrt(d + e*x)/99 + 2*c*e**2*x**5*
sqrt(d + e*x)/11, Ne(e, 0)), (d**(5/2)*(a*x + c*x**3/3), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (50) = 100\).
time = 1.98, size = 380, normalized size = 6.03 \begin {gather*} \frac {2}{3465} \, {\left (231 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} c d^{3} e^{\left (-2\right )} + 297 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} c d^{2} e^{\left (-2\right )} + 3465 \, \sqrt {x e + d} a d^{3} + 3465 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a d^{2} + 33 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} c d e^{\left (-2\right )} + 693 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a d + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} c e^{\left (-2\right )} + 99 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*c*d^3*e^(-2) + 297*(5*(x*e + d)^
(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*c*d^2*e^(-2) + 3465*sqrt(x*e + d
)*a*d^3 + 3465*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*d^2 + 33*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d +
378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*c*d*e^(-2) + 693*(3*(x*e + d)^(5/2)
- 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*d + 5*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*
e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*c*e^(-2) + 99*
(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a)*e^(-1)

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Mupad [B]
time = 0.06, size = 44, normalized size = 0.70 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{7/2}\,\left (63\,c\,{\left (d+e\,x\right )}^2+99\,a\,e^2+99\,c\,d^2-154\,c\,d\,\left (d+e\,x\right )\right )}{693\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)*(d + e*x)^(5/2),x)

[Out]

(2*(d + e*x)^(7/2)*(63*c*(d + e*x)^2 + 99*a*e^2 + 99*c*d^2 - 154*c*d*(d + e*x)))/(693*e^3)

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