3.7.3 \(\int \frac {(a+c x^2)^2}{(d+e x)^{5/2}} \, dx\) [603]

Optimal. Leaf size=123 \[ -\frac {2 \left (c d^2+a e^2\right )^2}{3 e^5 (d+e x)^{3/2}}+\frac {8 c d \left (c d^2+a e^2\right )}{e^5 \sqrt {d+e x}}+\frac {4 c \left (3 c d^2+a e^2\right ) \sqrt {d+e x}}{e^5}-\frac {8 c^2 d (d+e x)^{3/2}}{3 e^5}+\frac {2 c^2 (d+e x)^{5/2}}{5 e^5} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 123, 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 \sqrt {d+e x} \left (a e^2+3 c d^2\right )}{e^5}+\frac {8 c d \left (a e^2+c d^2\right )}{e^5 \sqrt {d+e x}}-\frac {2 \left (a e^2+c d^2\right )^2}{3 e^5 (d+e x)^{3/2}}+\frac {2 c^2 (d+e x)^{5/2}}{5 e^5}-\frac {8 c^2 d (d+e x)^{3/2}}{3 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^2/(d + e*x)^(5/2),x]

[Out]

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

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Mathematica [A]
time = 0.06, size = 96, normalized size = 0.78 \begin {gather*} \frac {2 \left (-5 a^2 e^4+10 a c e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+c^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-5*a^2*e^4 + 10*a*c*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + c^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d
*e^3*x^3 + 3*e^4*x^4)))/(15*e^5*(d + e*x)^(3/2))

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Maple [A]
time = 0.46, size = 117, normalized size = 0.95

method	result	size
risch	\(\frac {2 c \left (3 c \,e^{2} x^{2}-14 c d e x +30 e^{2} a +73 c \,d^{2}\right ) \sqrt {e x +d}}{15 e^{5}}-\frac {2 \left (-12 c d e x +e^{2} a -11 c \,d^{2}\right ) \left (e^{2} a +c \,d^{2}\right )}{3 e^{5} \left (e x +d \right )^{\frac {3}{2}}}\)	\(84\)
gosper	\(-\frac {2 \left (-3 c^{2} e^{4} x^{4}+8 c^{2} d \,x^{3} e^{3}-30 a c \,e^{4} x^{2}-48 d^{2} e^{2} x^{2} c^{2}-120 a c d \,e^{3} x -192 c^{2} d^{3} e x +5 a^{2} e^{4}-80 a c \,d^{2} e^{2}-128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\)	\(106\)
trager	\(-\frac {2 \left (-3 c^{2} e^{4} x^{4}+8 c^{2} d \,x^{3} e^{3}-30 a c \,e^{4} x^{2}-48 d^{2} e^{2} x^{2} c^{2}-120 a c d \,e^{3} x -192 c^{2} d^{3} e x +5 a^{2} e^{4}-80 a c \,d^{2} e^{2}-128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\)	\(106\)
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+4 a c \,e^{2} \sqrt {e x +d}+12 c^{2} d^{2} \sqrt {e x +d}+\frac {8 c d \left (e^{2} a +c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\)	\(117\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+4 a c \,e^{2} \sqrt {e x +d}+12 c^{2} d^{2} \sqrt {e x +d}+\frac {8 c d \left (e^{2} a +c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^5*(1/5*c^2*(e*x+d)^(5/2)-4/3*c^2*d*(e*x+d)^(3/2)+2*a*c*e^2*(e*x+d)^(1/2)+6*c^2*d^2*(e*x+d)^(1/2)+4*c*d*(a*
e^2+c*d^2)/(e*x+d)^(1/2)-1/3*(a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)/(e*x+d)^(3/2))

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Maxima [A]
time = 0.34, size = 117, normalized size = 0.95 \begin {gather*} \frac {2}{15} \, {\left ({\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} - 20 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d + 30 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} \sqrt {x e + d}\right )} e^{\left (-4\right )} - \frac {5 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4} - 12 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} {\left (x e + d\right )}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{\frac {3}{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)^(5/2),x, algorithm="maxima")

[Out]

2/15*((3*(x*e + d)^(5/2)*c^2 - 20*(x*e + d)^(3/2)*c^2*d + 30*(3*c^2*d^2 + a*c*e^2)*sqrt(x*e + d))*e^(-4) - 5*(
c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4 - 12*(c^2*d^3 + a*c*d*e^2)*(x*e + d))*e^(-4)/(x*e + d)^(3/2))*e^(-1)

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Fricas [A]
time = 1.21, size = 117, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (192 \, c^{2} d^{3} x e + 128 \, c^{2} d^{4} + {\left (3 \, c^{2} x^{4} + 30 \, a c x^{2} - 5 \, a^{2}\right )} e^{4} - 8 \, {\left (c^{2} d x^{3} - 15 \, a c d x\right )} e^{3} + 16 \, {\left (3 \, c^{2} d^{2} x^{2} + 5 \, a c d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{15 \, {\left (x^{2} e^{7} + 2 \, d x e^{6} + d^{2} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(192*c^2*d^3*x*e + 128*c^2*d^4 + (3*c^2*x^4 + 30*a*c*x^2 - 5*a^2)*e^4 - 8*(c^2*d*x^3 - 15*a*c*d*x)*e^3 +
16*(3*c^2*d^2*x^2 + 5*a*c*d^2)*e^2)*sqrt(x*e + d)/(x^2*e^7 + 2*d*x*e^6 + d^2*e^5)

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Sympy [A]
time = 11.41, size = 121, normalized size = 0.98 \begin {gather*} - \frac {8 c^{2} d \left (d + e x\right )^{\frac {3}{2}}}{3 e^{5}} + \frac {2 c^{2} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{5}} + \frac {8 c d \left (a e^{2} + c d^{2}\right )}{e^{5} \sqrt {d + e x}} + \frac {\sqrt {d + e x} \left (4 a c e^{2} + 12 c^{2} d^{2}\right )}{e^{5}} - \frac {2 \left (a e^{2} + c d^{2}\right )^{2}}{3 e^{5} \left (d + e x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*c**2*d*(d + e*x)**(3/2)/(3*e**5) + 2*c**2*(d + e*x)**(5/2)/(5*e**5) + 8*c*d*(a*e**2 + c*d**2)/(e**5*sqrt(d
+ e*x)) + sqrt(d + e*x)*(4*a*c*e**2 + 12*c**2*d**2)/e**5 - 2*(a*e**2 + c*d**2)**2/(3*e**5*(d + e*x)**(3/2))

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Giac [A]
time = 2.36, size = 133, normalized size = 1.08 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} e^{20} - 20 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d e^{20} + 90 \, \sqrt {x e + d} c^{2} d^{2} e^{20} + 30 \, \sqrt {x e + d} a c e^{22}\right )} e^{\left (-25\right )} + \frac {2 \, {\left (12 \, {\left (x e + d\right )} c^{2} d^{3} - c^{2} d^{4} + 12 \, {\left (x e + d\right )} a c d e^{2} - 2 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} e^{\left (-5\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*(x*e + d)^(5/2)*c^2*e^20 - 20*(x*e + d)^(3/2)*c^2*d*e^20 + 90*sqrt(x*e + d)*c^2*d^2*e^20 + 30*sqrt(x*e
+ d)*a*c*e^22)*e^(-25) + 2/3*(12*(x*e + d)*c^2*d^3 - c^2*d^4 + 12*(x*e + d)*a*c*d*e^2 - 2*a*c*d^2*e^2 - a^2*e
^4)*e^(-5)/(x*e + d)^(3/2)

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Mupad [B]
time = 0.40, size = 122, normalized size = 0.99 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}-\frac {\frac {2\,a^2\,e^4}{3}+\frac {2\,c^2\,d^4}{3}-\left (8\,c^2\,d^3+8\,a\,c\,d\,e^2\right )\,\left (d+e\,x\right )+\frac {4\,a\,c\,d^2\,e^2}{3}}{e^5\,{\left (d+e\,x\right )}^{3/2}}+\frac {\left (12\,c^2\,d^2+4\,a\,c\,e^2\right )\,\sqrt {d+e\,x}}{e^5}-\frac {8\,c^2\,d\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2/(d + e*x)^(5/2),x)

[Out]

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

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