Optimal. Leaf size=491 \[ -\frac {8 \sqrt {-a} c^{3/2} \sqrt {\frac {c x^2}{a}+1} \left (5 a e^2+4 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 e^4 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}+\frac {32 \sqrt {-a} c^{5/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (2 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 e^4 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {32 c^2 d \sqrt {a+c x^2} \left (2 a e^2+c d^2\right )}{35 e^3 \sqrt {d+e x} \left (a e^2+c d^2\right )^2}-\frac {4 c \sqrt {a+c x^2} \left (e x \left (5 a e^2+7 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{35 e^3 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac {2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}} \]
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Rubi [A] time = 0.48, antiderivative size = 491, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {733, 811, 835, 844, 719, 424, 419} \[ \frac {32 c^2 d \sqrt {a+c x^2} \left (2 a e^2+c d^2\right )}{35 e^3 \sqrt {d+e x} \left (a e^2+c d^2\right )^2}-\frac {8 \sqrt {-a} c^{3/2} \sqrt {\frac {c x^2}{a}+1} \left (5 a e^2+4 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 e^4 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}+\frac {32 \sqrt {-a} c^{5/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (2 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 e^4 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {4 c \sqrt {a+c x^2} \left (e x \left (5 a e^2+7 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{35 e^3 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac {2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 719
Rule 733
Rule 811
Rule 835
Rule 844
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx &=-\frac {2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}+\frac {(6 c) \int \frac {x \sqrt {a+c x^2}}{(d+e x)^{7/2}} \, dx}{7 e}\\ &=-\frac {4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (7 c d^2+5 a e^2\right ) x\right ) \sqrt {a+c x^2}}{35 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac {(4 c) \int \frac {3 a c d e-c \left (4 c d^2+5 a e^2\right ) x}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx}{35 e^3 \left (c d^2+a e^2\right )}\\ &=\frac {32 c^2 d \left (c d^2+2 a e^2\right ) \sqrt {a+c x^2}}{35 e^3 \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (7 c d^2+5 a e^2\right ) x\right ) \sqrt {a+c x^2}}{35 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}+\frac {(8 c) \int \frac {\frac {1}{2} a c e \left (c d^2+5 a e^2\right )-2 c^2 d \left (c d^2+2 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{35 e^3 \left (c d^2+a e^2\right )^2}\\ &=\frac {32 c^2 d \left (c d^2+2 a e^2\right ) \sqrt {a+c x^2}}{35 e^3 \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (7 c d^2+5 a e^2\right ) x\right ) \sqrt {a+c x^2}}{35 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac {\left (16 c^3 d \left (c d^2+2 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{35 e^4 \left (c d^2+a e^2\right )^2}+\frac {\left (4 c^2 \left (4 c d^2+5 a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{35 e^4 \left (c d^2+a e^2\right )}\\ &=\frac {32 c^2 d \left (c d^2+2 a e^2\right ) \sqrt {a+c x^2}}{35 e^3 \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (7 c d^2+5 a e^2\right ) x\right ) \sqrt {a+c x^2}}{35 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac {\left (32 a c^{5/2} d \left (c d^2+2 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{35 \sqrt {-a} e^4 \left (c d^2+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (8 a c^{3/2} \left (4 c d^2+5 a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{35 \sqrt {-a} e^4 \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=\frac {32 c^2 d \left (c d^2+2 a e^2\right ) \sqrt {a+c x^2}}{35 e^3 \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (7 c d^2+5 a e^2\right ) x\right ) \sqrt {a+c x^2}}{35 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}+\frac {32 \sqrt {-a} c^{5/2} d \left (c d^2+2 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 e^4 \left (c d^2+a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} c^{3/2} \left (4 c d^2+5 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{35 e^4 \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 3.22, size = 659, normalized size = 1.34 \[ \frac {2 \left (-e^2 \left (a+c x^2\right ) \left (-16 c^2 d (d+e x)^3 \left (2 a e^2+c d^2\right )-16 c d (d+e x) \left (a e^2+c d^2\right )^2+c (d+e x)^2 \left (15 a e^2+19 c d^2\right ) \left (a e^2+c d^2\right )+5 \left (a e^2+c d^2\right )^3\right )-\frac {4 c^2 (d+e x)^3 \left (-\sqrt {a} e (d+e x)^{3/2} \left (5 i a^{3/2} e^3+i \sqrt {a} c d^2 e+8 a \sqrt {c} d e^2+4 c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+4 \sqrt {c} d (d+e x)^{3/2} \left (2 a^{3/2} e^3+\sqrt {a} c d^2 e-2 i a \sqrt {c} d e^2-i c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+4 d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (2 a^2 e^2+a c \left (d^2+2 e^2 x^2\right )+c^2 d^2 x^2\right )\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{35 e^5 \sqrt {a+c x^2} (d+e x)^{7/2} \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} \sqrt {e x + d}}{e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 5277, normalized size = 10.75 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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