### 3.607 $$\int \sqrt{d+e x} (a+c x^2)^3 \, dx$$

Optimal. Leaf size=204 $\frac{6 c^2 (d+e x)^{11/2} \left (a e^2+5 c d^2\right )}{11 e^7}-\frac{8 c^2 d (d+e x)^{9/2} \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac{6 c (d+e x)^{7/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac{12 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^7}+\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^3}{3 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7}-\frac{12 c^3 d (d+e x)^{13/2}}{13 e^7}$

[Out]

(2*(c*d^2 + a*e^2)^3*(d + e*x)^(3/2))/(3*e^7) - (12*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(5/2))/(5*e^7) + (6*c*(c*d
^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(7/2))/(7*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(9/2))/(9*e^7)
+ (6*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(11/2))/(11*e^7) - (12*c^3*d*(d + e*x)^(13/2))/(13*e^7) + (2*c^3*(d + e*
x)^(15/2))/(15*e^7)

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Rubi [A]  time = 0.0826531, antiderivative size = 204, normalized size of antiderivative = 1., 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 = {697} $\frac{6 c^2 (d+e x)^{11/2} \left (a e^2+5 c d^2\right )}{11 e^7}-\frac{8 c^2 d (d+e x)^{9/2} \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac{6 c (d+e x)^{7/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac{12 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^7}+\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^3}{3 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7}-\frac{12 c^3 d (d+e x)^{13/2}}{13 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[d + e*x]*(a + c*x^2)^3,x]

[Out]

(2*(c*d^2 + a*e^2)^3*(d + e*x)^(3/2))/(3*e^7) - (12*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(5/2))/(5*e^7) + (6*c*(c*d
^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(7/2))/(7*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(9/2))/(9*e^7)
+ (6*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(11/2))/(11*e^7) - (12*c^3*d*(d + e*x)^(13/2))/(13*e^7) + (2*c^3*(d + e*
x)^(15/2))/(15*e^7)

Rule 697

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 \sqrt{d+e x} \left (a+c x^2\right )^3 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3 \sqrt{d+e x}}{e^6}-\frac{6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{e^6}+\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{5/2}}{e^6}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{7/2}}{e^6}+\frac{3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{e^6}-\frac{6 c^3 d (d+e x)^{11/2}}{e^6}+\frac{c^3 (d+e x)^{13/2}}{e^6}\right ) \, dx\\ &=\frac{2 \left (c d^2+a e^2\right )^3 (d+e x)^{3/2}}{3 e^7}-\frac{12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{5 e^7}+\frac{6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac{8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{9/2}}{9 e^7}+\frac{6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac{12 c^3 d (d+e x)^{13/2}}{13 e^7}+\frac{2 c^3 (d+e x)^{15/2}}{15 e^7}\\ \end{align*}

Mathematica [A]  time = 0.144417, size = 170, normalized size = 0.83 $\frac{2 (d+e x)^{3/2} \left (1287 a^2 c e^4 \left (8 d^2-12 d e x+15 e^2 x^2\right )+15015 a^3 e^6+39 a c^2 e^2 \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )+c^3 \left (1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-1536 d^5 e x+1024 d^6-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[d + e*x]*(a + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(3/2)*(15015*a^3*e^6 + 1287*a^2*c*e^4*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 39*a*c^2*e^2*(128*d^4 - 1
92*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + c^3*(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^2*x^2
- 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*x^6)))/(45045*e^7)

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Maple [A]  time = 0.046, size = 205, normalized size = 1. \begin{align*}{\frac{6006\,{c}^{3}{x}^{6}{e}^{6}-5544\,{c}^{3}d{x}^{5}{e}^{5}+24570\,a{c}^{2}{e}^{6}{x}^{4}+5040\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-21840\,a{c}^{2}d{e}^{5}{x}^{3}-4480\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+38610\,{a}^{2}c{e}^{6}{x}^{2}+18720\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-30888\,{a}^{2}cd{e}^{5}x-14976\,a{c}^{2}{d}^{3}{e}^{3}x-3072\,{c}^{3}{d}^{5}ex+30030\,{a}^{3}{e}^{6}+20592\,{a}^{2}c{d}^{2}{e}^{4}+9984\,{d}^{4}{e}^{2}a{c}^{2}+2048\,{c}^{3}{d}^{6}}{45045\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^3*(e*x+d)^(1/2),x)

[Out]

2/45045*(e*x+d)^(3/2)*(3003*c^3*e^6*x^6-2772*c^3*d*e^5*x^5+12285*a*c^2*e^6*x^4+2520*c^3*d^2*e^4*x^4-10920*a*c^
2*d*e^5*x^3-2240*c^3*d^3*e^3*x^3+19305*a^2*c*e^6*x^2+9360*a*c^2*d^2*e^4*x^2+1920*c^3*d^4*e^2*x^2-15444*a^2*c*d
*e^5*x-7488*a*c^2*d^3*e^3*x-1536*c^3*d^5*e*x+15015*a^3*e^6+10296*a^2*c*d^2*e^4+4992*a*c^2*d^4*e^2+1024*c^3*d^6
)/e^7

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Maxima [A]  time = 1.14468, size = 282, normalized size = 1.38 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{3} - 20790 \,{\left (e x + d\right )}^{\frac{13}{2}} c^{3} d + 12285 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 20020 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 19305 \,{\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 54054 \,{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 15015 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{45045 \, e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*c^3 - 20790*(e*x + d)^(13/2)*c^3*d + 12285*(5*c^3*d^2 + a*c^2*e^2)*(e*x + d)^(1
1/2) - 20020*(5*c^3*d^3 + 3*a*c^2*d*e^2)*(e*x + d)^(9/2) + 19305*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*(e*
x + d)^(7/2) - 54054*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d)^(5/2) + 15015*(c^3*d^6 + 3*a*c^2*d^4*
e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(e*x + d)^(3/2))/e^7

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Fricas [A]  time = 1.8093, size = 581, normalized size = 2.85 \begin{align*} \frac{2 \,{\left (3003 \, c^{3} e^{7} x^{7} + 231 \, c^{3} d e^{6} x^{6} + 1024 \, c^{3} d^{7} + 4992 \, a c^{2} d^{5} e^{2} + 10296 \, a^{2} c d^{3} e^{4} + 15015 \, a^{3} d e^{6} - 63 \,{\left (4 \, c^{3} d^{2} e^{5} - 195 \, a c^{2} e^{7}\right )} x^{5} + 35 \,{\left (8 \, c^{3} d^{3} e^{4} + 39 \, a c^{2} d e^{6}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{4} e^{3} + 312 \, a c^{2} d^{2} e^{5} - 3861 \, a^{2} c e^{7}\right )} x^{3} + 3 \,{\left (128 \, c^{3} d^{5} e^{2} + 624 \, a c^{2} d^{3} e^{4} + 1287 \, a^{2} c d e^{6}\right )} x^{2} -{\left (512 \, c^{3} d^{6} e + 2496 \, a c^{2} d^{4} e^{3} + 5148 \, a^{2} c d^{2} e^{5} - 15015 \, a^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3003*c^3*e^7*x^7 + 231*c^3*d*e^6*x^6 + 1024*c^3*d^7 + 4992*a*c^2*d^5*e^2 + 10296*a^2*c*d^3*e^4 + 1501
5*a^3*d*e^6 - 63*(4*c^3*d^2*e^5 - 195*a*c^2*e^7)*x^5 + 35*(8*c^3*d^3*e^4 + 39*a*c^2*d*e^6)*x^4 - 5*(64*c^3*d^4
*e^3 + 312*a*c^2*d^2*e^5 - 3861*a^2*c*e^7)*x^3 + 3*(128*c^3*d^5*e^2 + 624*a*c^2*d^3*e^4 + 1287*a^2*c*d*e^6)*x^
2 - (512*c^3*d^6*e + 2496*a*c^2*d^4*e^3 + 5148*a^2*c*d^2*e^5 - 15015*a^3*e^7)*x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 4.06417, size = 265, normalized size = 1.3 \begin{align*} \frac{2 \left (- \frac{6 c^{3} d \left (d + e x\right )^{\frac{13}{2}}}{13 e^{6}} + \frac{c^{3} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{6}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right )}{11 e^{6}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (- 12 a c^{2} d e^{2} - 20 c^{3} d^{3}\right )}{9 e^{6}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} + 15 c^{3} d^{4}\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 6 a^{2} c d e^{4} - 12 a c^{2} d^{3} e^{2} - 6 c^{3} d^{5}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}\right )}{3 e^{6}}\right )}{e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3*(e*x+d)**(1/2),x)

[Out]

2*(-6*c**3*d*(d + e*x)**(13/2)/(13*e**6) + c**3*(d + e*x)**(15/2)/(15*e**6) + (d + e*x)**(11/2)*(3*a*c**2*e**2
+ 15*c**3*d**2)/(11*e**6) + (d + e*x)**(9/2)*(-12*a*c**2*d*e**2 - 20*c**3*d**3)/(9*e**6) + (d + e*x)**(7/2)*(
3*a**2*c*e**4 + 18*a*c**2*d**2*e**2 + 15*c**3*d**4)/(7*e**6) + (d + e*x)**(5/2)*(-6*a**2*c*d*e**4 - 12*a*c**2*
d**3*e**2 - 6*c**3*d**5)/(5*e**6) + (d + e*x)**(3/2)*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c*
*3*d**6)/(3*e**6))/e

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Giac [A]  time = 1.36843, size = 301, normalized size = 1.48 \begin{align*} \frac{2}{45045} \,{\left (1287 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a^{2} c e^{\left (-2\right )} + 39 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} a c^{2} e^{\left (-4\right )} +{\left (3003 \,{\left (x e + d\right )}^{\frac{15}{2}} - 20790 \,{\left (x e + d\right )}^{\frac{13}{2}} d + 61425 \,{\left (x e + d\right )}^{\frac{11}{2}} d^{2} - 100100 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{3} + 96525 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{4} - 54054 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{5} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{6}\right )} c^{3} e^{\left (-6\right )} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3}\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*c*e^(-2) + 39*(315*(x*e
+ d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(
3/2)*d^4)*a*c^2*e^(-4) + (3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 1001
00*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*c^
3*e^(-6) + 15015*(x*e + d)^(3/2)*a^3)*e^(-1)