3.7.4 \(\int \frac {(a+c x^2)^2}{(d+e x)^{7/2}} \, dx\) [604]

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

[Out]

-2/5*(a*e^2+c*d^2)^2/e^5/(e*x+d)^(5/2)+8/3*c*d*(a*e^2+c*d^2)/e^5/(e*x+d)^(3/2)+2/3*c^2*(e*x+d)^(3/2)/e^5-4*c*(
a*e^2+3*c*d^2)/e^5/(e*x+d)^(1/2)-8*c^2*d*(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 \left (a e^2+3 c d^2\right )}{e^5 \sqrt {d+e x}}+\frac {8 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac {2 \left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac {2 c^2 (d+e x)^{3/2}}{3 e^5}-\frac {8 c^2 d \sqrt {d+e x}}{e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^2/(d + e*x)^(7/2),x]

[Out]

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

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Mathematica [A]
time = 0.06, size = 96, normalized size = 0.78 \begin {gather*} -\frac {2 \left (3 a^2 e^4+2 a c e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )+c^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^2/(d + e*x)^(7/2),x]

[Out]

(-2*(3*a^2*e^4 + 2*a*c*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + c^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40
*d*e^3*x^3 - 5*e^4*x^4)))/(15*e^5*(d + e*x)^(5/2))

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Maple [A]
time = 0.51, size = 110, normalized size = 0.89

method	result	size
gosper	\(-\frac {2 \left (-5 c^{2} e^{4} x^{4}+40 c^{2} d \,x^{3} e^{3}+30 a c \,e^{4} x^{2}+240 d^{2} e^{2} x^{2} c^{2}+40 a c d \,e^{3} x +320 c^{2} d^{3} e x +3 a^{2} e^{4}+16 a c \,d^{2} e^{2}+128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\)	\(106\)
trager	\(-\frac {2 \left (-5 c^{2} e^{4} x^{4}+40 c^{2} d \,x^{3} e^{3}+30 a c \,e^{4} x^{2}+240 d^{2} e^{2} x^{2} c^{2}+40 a c d \,e^{3} x +320 c^{2} d^{3} e x +3 a^{2} e^{4}+16 a c \,d^{2} e^{2}+128 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\)	\(106\)
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-8 c^{2} d \sqrt {e x +d}-\frac {4 c \left (e^{2} a +3 c \,d^{2}\right )}{\sqrt {e x +d}}+\frac {8 c d \left (e^{2} a +c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}}{e^{5}}\)	\(110\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-8 c^{2} d \sqrt {e x +d}-\frac {4 c \left (e^{2} a +3 c \,d^{2}\right )}{\sqrt {e x +d}}+\frac {8 c d \left (e^{2} a +c \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}}{e^{5}}\)	\(110\)
risch	\(-\frac {2 c^{2} \left (-e x +11 d \right ) \sqrt {e x +d}}{3 e^{5}}-\frac {2 \left (30 a c \,e^{4} x^{2}+90 d^{2} e^{2} x^{2} c^{2}+40 a c d \,e^{3} x +160 c^{2} d^{3} e x +3 a^{2} e^{4}+16 a c \,d^{2} e^{2}+73 c^{2} d^{4}\right )}{15 e^{5} \sqrt {e x +d}\, \left (x^{2} e^{2}+2 d x e +d^{2}\right )}\)	\(125\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^2/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 119, normalized size = 0.97 \begin {gather*} \frac {2}{15} \, {\left (5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{2} - 12 \, \sqrt {x e + d} c^{2} d\right )} e^{\left (-4\right )} - \frac {{\left (3 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 30 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} {\left (x e + d\right )}^{2} + 3 \, a^{2} e^{4} - 20 \, {\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 {5}{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)^(7/2),x, algorithm="maxima")

[Out]

2/15*(5*((x*e + d)^(3/2)*c^2 - 12*sqrt(x*e + d)*c^2*d)*e^(-4) - (3*c^2*d^4 + 6*a*c*d^2*e^2 + 30*(3*c^2*d^2 + a
*c*e^2)*(x*e + d)^2 + 3*a^2*e^4 - 20*(c^2*d^3 + a*c*d*e^2)*(x*e + d))*e^(-4)/(x*e + d)^(5/2))*e^(-1)

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Fricas [A]
time = 1.54, size = 126, normalized size = 1.02 \begin {gather*} -\frac {2 \, {\left (320 \, c^{2} d^{3} x e + 128 \, c^{2} d^{4} - {\left (5 \, c^{2} x^{4} - 30 \, a c x^{2} - 3 \, a^{2}\right )} e^{4} + 40 \, {\left (c^{2} d x^{3} + a c d x\right )} e^{3} + 16 \, {\left (15 \, c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{15 \, {\left (x^{3} e^{8} + 3 \, d x^{2} e^{7} + 3 \, d^{2} x e^{6} + d^{3} 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)^(7/2),x, algorithm="fricas")

[Out]

-2/15*(320*c^2*d^3*x*e + 128*c^2*d^4 - (5*c^2*x^4 - 30*a*c*x^2 - 3*a^2)*e^4 + 40*(c^2*d*x^3 + a*c*d*x)*e^3 + 1
6*(15*c^2*d^2*x^2 + a*c*d^2)*e^2)*sqrt(x*e + d)/(x^3*e^8 + 3*d*x^2*e^7 + 3*d^2*x*e^6 + d^3*e^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (119) = 238\).
time = 0.64, size = 592, normalized size = 4.81 \begin {gather*} \begin {cases} - \frac {6 a^{2} e^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {32 a c d^{2} e^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {80 a c d e^{3} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {60 a c e^{4} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {256 c^{2} d^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {640 c^{2} d^{3} e x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {480 c^{2} d^{2} e^{2} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {80 c^{2} d e^{3} x^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {10 c^{2} e^{4} x^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a^{2} x + \frac {2 a c x^{3}}{3} + \frac {c^{2} x^{5}}{5}}{d^{\frac {7}{2}}} & \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)**(7/2),x)

[Out]

Piecewise((-6*a**2*e**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x))
- 32*a*c*d**2*e**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 80*
a*c*d*e**3*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 60*a*c*e*
*4*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 256*c**2*d**4/
(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 640*c**2*d**3*e*x/(15*
d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 480*c**2*d**2*e**2*x**2/(1
5*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 80*c**2*d*e**3*x**3/(15*
d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 10*c**2*e**4*x**4/(15*d**2
*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)), Ne(e, 0)), ((a**2*x + 2*a*c*x**
3/3 + c**2*x**5/5)/d**(7/2), True))

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Giac [A]
time = 3.17, size = 130, normalized size = 1.06 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{2} e^{10} - 12 \, \sqrt {x e + d} c^{2} d e^{10}\right )} e^{\left (-15\right )} - \frac {2 \, {\left (90 \, {\left (x e + d\right )}^{2} c^{2} d^{2} - 20 \, {\left (x e + d\right )} c^{2} d^{3} + 3 \, c^{2} d^{4} + 30 \, {\left (x e + d\right )}^{2} a c e^{2} - 20 \, {\left (x e + d\right )} a c d e^{2} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*c^2*e^10 - 12*sqrt(x*e + d)*c^2*d*e^10)*e^(-15) - 2/15*(90*(x*e + d)^2*c^2*d^2 - 20*(x*e
+ d)*c^2*d^3 + 3*c^2*d^4 + 30*(x*e + d)^2*a*c*e^2 - 20*(x*e + d)*a*c*d*e^2 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*e^(-5)
/(x*e + d)^(5/2)

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Mupad [B]
time = 0.42, size = 105, normalized size = 0.85 \begin {gather*} -\frac {2\,\left (3\,a^2\,e^4+16\,a\,c\,d^2\,e^2+40\,a\,c\,d\,e^3\,x+30\,a\,c\,e^4\,x^2+128\,c^2\,d^4+320\,c^2\,d^3\,e\,x+240\,c^2\,d^2\,e^2\,x^2+40\,c^2\,d\,e^3\,x^3-5\,c^2\,e^4\,x^4\right )}{15\,e^5\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2/(d + e*x)^(7/2),x)

[Out]

-(2*(3*a^2*e^4 + 128*c^2*d^4 - 5*c^2*e^4*x^4 + 40*c^2*d*e^3*x^3 + 240*c^2*d^2*e^2*x^2 + 16*a*c*d^2*e^2 + 30*a*
c*e^4*x^2 + 320*c^2*d^3*e*x + 40*a*c*d*e^3*x))/(15*e^5*(d + e*x)^(5/2))

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