3.7.1 \(\int \frac {(a+c x^2)^2}{\sqrt {d+e x}} \, dx\) [601]

Optimal. Leaf size=125 \[ \frac {2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}{e^5}-\frac {8 c d \left (c d^2+a e^2\right ) (d+e x)^{3/2}}{3 e^5}+\frac {4 c \left (3 c d^2+a e^2\right ) (d+e x)^{5/2}}{5 e^5}-\frac {8 c^2 d (d+e x)^{7/2}}{7 e^5}+\frac {2 c^2 (d+e x)^{9/2}}{9 e^5} \]

[Out]

-8/3*c*d*(a*e^2+c*d^2)*(e*x+d)^(3/2)/e^5+4/5*c*(a*e^2+3*c*d^2)*(e*x+d)^(5/2)/e^5-8/7*c^2*d*(e*x+d)^(7/2)/e^5+2
/9*c^2*(e*x+d)^(9/2)/e^5+2*(a*e^2+c*d^2)^2*(e*x+d)^(1/2)/e^5

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Rubi [A]
time = 0.03, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {711} \begin {gather*} \frac {4 c (d+e x)^{5/2} \left (a e^2+3 c d^2\right )}{5 e^5}-\frac {8 c d (d+e x)^{3/2} \left (a e^2+c d^2\right )}{3 e^5}+\frac {2 \sqrt {d+e x} \left (a e^2+c d^2\right )^2}{e^5}+\frac {2 c^2 (d+e x)^{9/2}}{9 e^5}-\frac {8 c^2 d (d+e x)^{7/2}}{7 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^2/Sqrt[d + e*x],x]

[Out]

(2*(c*d^2 + a*e^2)^2*Sqrt[d + e*x])/e^5 - (8*c*d*(c*d^2 + a*e^2)*(d + e*x)^(3/2))/(3*e^5) + (4*c*(3*c*d^2 + a*
e^2)*(d + e*x)^(5/2))/(5*e^5) - (8*c^2*d*(d + e*x)^(7/2))/(7*e^5) + (2*c^2*(d + e*x)^(9/2))/(9*e^5)

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 \frac {\left (a+c x^2\right )^2}{\sqrt {d+e x}} \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^2}{e^4 \sqrt {d+e x}}-\frac {4 c d \left (c d^2+a e^2\right ) \sqrt {d+e x}}{e^4}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^{3/2}}{e^4}-\frac {4 c^2 d (d+e x)^{5/2}}{e^4}+\frac {c^2 (d+e x)^{7/2}}{e^4}\right ) \, dx\\ &=\frac {2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}{e^5}-\frac {8 c d \left (c d^2+a e^2\right ) (d+e x)^{3/2}}{3 e^5}+\frac {4 c \left (3 c d^2+a e^2\right ) (d+e x)^{5/2}}{5 e^5}-\frac {8 c^2 d (d+e x)^{7/2}}{7 e^5}+\frac {2 c^2 (d+e x)^{9/2}}{9 e^5}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 96, normalized size = 0.77 \begin {gather*} \frac {2 \sqrt {d+e x} \left (315 a^2 e^4+42 a c e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(315*a^2*e^4 + 42*a*c*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + c^2*(128*d^4 - 64*d^3*e*x + 48*d^2*
e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4)))/(315*e^5)

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Maple [A]
time = 0.42, size = 107, normalized size = 0.86

method	result	size
gosper	\(\frac {2 \sqrt {e x +d}\, \left (35 c^{2} e^{4} x^{4}-40 c^{2} d \,x^{3} e^{3}+126 a c \,e^{4} x^{2}+48 d^{2} e^{2} x^{2} c^{2}-168 a c d \,e^{3} x -64 c^{2} d^{3} e x +315 a^{2} e^{4}+336 a c \,d^{2} e^{2}+128 c^{2} d^{4}\right )}{315 e^{5}}\)	\(106\)
trager	\(\frac {2 \sqrt {e x +d}\, \left (35 c^{2} e^{4} x^{4}-40 c^{2} d \,x^{3} e^{3}+126 a c \,e^{4} x^{2}+48 d^{2} e^{2} x^{2} c^{2}-168 a c d \,e^{3} x -64 c^{2} d^{3} e x +315 a^{2} e^{4}+336 a c \,d^{2} e^{2}+128 c^{2} d^{4}\right )}{315 e^{5}}\)	\(106\)
risch	\(\frac {2 \sqrt {e x +d}\, \left (35 c^{2} e^{4} x^{4}-40 c^{2} d \,x^{3} e^{3}+126 a c \,e^{4} x^{2}+48 d^{2} e^{2} x^{2} c^{2}-168 a c d \,e^{3} x -64 c^{2} d^{3} e x +315 a^{2} e^{4}+336 a c \,d^{2} e^{2}+128 c^{2} d^{4}\right )}{315 e^{5}}\)	\(106\)
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 \left (e^{2} a +c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (e^{2} a +c \,d^{2}\right )^{2} \sqrt {e x +d}}{e^{5}}\)	\(107\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 \left (e^{2} a +c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (e^{2} a +c \,d^{2}\right )^{2} \sqrt {e x +d}}{e^{5}}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^2/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/e^5*(1/9*c^2*(e*x+d)^(9/2)-4/7*c^2*d*(e*x+d)^(7/2)+1/5*(2*(a*e^2+c*d^2)*c+4*d^2*c^2)*(e*x+d)^(5/2)-4/3*(a*e^
2+c*d^2)*c*d*(e*x+d)^(3/2)+(a*e^2+c*d^2)^2*(e*x+d)^(1/2))

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Maxima [A]
time = 0.28, size = 126, normalized size = 1.01 \begin {gather*} \frac {2}{315} \, {\left (42 \, {\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 c e^{\left (-2\right )} + {\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^{2} e^{\left (-4\right )} + 315 \, \sqrt {x e + d} a^{2}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/315*(42*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*c*e^(-2) + (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^2*e^(-4)
+ 315*sqrt(x*e + d)*a^2)*e^(-1)

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Fricas [A]
time = 3.13, size = 99, normalized size = 0.79 \begin {gather*} -\frac {2}{315} \, {\left (64 \, c^{2} d^{3} x e - 128 \, c^{2} d^{4} - 7 \, {\left (5 \, c^{2} x^{4} + 18 \, a c x^{2} + 45 \, a^{2}\right )} e^{4} + 8 \, {\left (5 \, c^{2} d x^{3} + 21 \, a c d x\right )} e^{3} - 48 \, {\left (c^{2} d^{2} x^{2} + 7 \, a c d^{2}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-2/315*(64*c^2*d^3*x*e - 128*c^2*d^4 - 7*(5*c^2*x^4 + 18*a*c*x^2 + 45*a^2)*e^4 + 8*(5*c^2*d*x^3 + 21*a*c*d*x)*
e^3 - 48*(c^2*d^2*x^2 + 7*a*c*d^2)*e^2)*sqrt(x*e + d)*e^(-5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (121) = 242\).
time = 15.28, size = 330, normalized size = 2.64 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{2} d}{\sqrt {d + e x}} - 2 a^{2} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {4 a c d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {4 a c \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {2 c^{2} d \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{4}} - \frac {2 c^{2} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}}}{e} & \text {for}\: e \neq 0 \\\frac {a^{2} x + \frac {2 a c x^{3}}{3} + \frac {c^{2} x^{5}}{5}}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**2/(e*x+d)**(1/2),x)

[Out]

Piecewise(((-2*a**2*d/sqrt(d + e*x) - 2*a**2*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 4*a*c*d*(d**2/sqrt(d + e*x)
+ 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 4*a*c*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*
x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 2*c**2*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)*
*(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 - 2*c**2*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x
) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4)/
e, Ne(e, 0)), ((a**2*x + 2*a*c*x**3/3 + c**2*x**5/5)/sqrt(d), True))

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Giac [A]
time = 2.22, size = 126, normalized size = 1.01 \begin {gather*} \frac {2}{315} \, {\left (42 \, {\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 c e^{\left (-2\right )} + {\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^{2} e^{\left (-4\right )} + 315 \, \sqrt {x e + d} a^{2}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/315*(42*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*c*e^(-2) + (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^2*e^(-4)
+ 315*sqrt(x*e + d)*a^2)*e^(-1)

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Mupad [B]
time = 0.04, size = 114, normalized size = 0.91 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}-\frac {\left (8\,c^2\,d^3+8\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^2\,\sqrt {d+e\,x}}{e^5}+\frac {\left (12\,c^2\,d^2+4\,a\,c\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}-\frac {8\,c^2\,d\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2/(d + e*x)^(1/2),x)

[Out]

(2*c^2*(d + e*x)^(9/2))/(9*e^5) - ((8*c^2*d^3 + 8*a*c*d*e^2)*(d + e*x)^(3/2))/(3*e^5) + (2*(a*e^2 + c*d^2)^2*(
d + e*x)^(1/2))/e^5 + ((12*c^2*d^2 + 4*a*c*e^2)*(d + e*x)^(5/2))/(5*e^5) - (8*c^2*d*(d + e*x)^(7/2))/(7*e^5)

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